The nth term of an AP with the first term a_1 and the common difference d is given by a_n = a_1 + (n-1)d. The nth term of the given AP is a_n = -54 + (n-1)7.
What is AP?Arithmetic Progression (AP) is a sequence of numbers in which the difference between the two consecutive numbers is always the same. Each number in the sequence is called a term of the progression. It is a type of sequence where the difference between consecutive terms is constant.
The given expression is an arithmetic progression (AP), in which the common difference between consecutive terms is 7. An AP is a sequence of numbers where each term after the first is formed by adding a fixed value to the preceding one.
The first term of the AP is given as a_1=-54 and the common difference (d) is 7. Thus, the nth term of the given AP can be found using the formula a_n = a_1 + (n-1)d. Substituting the values in this formula, we get a_n = -54 + (n-1)7.
Therefore, the nth term of the given AP is a_n = -54 + (n-1)7.
For example, if n = 10, then the 10th term of the given AP is a_10 = -54 + (10-1)7 = -54 + 63 = 9.
In general, the nth term of an AP with the first term a_1 and the common difference d is given by a_n = a_1 + (n-1)d.
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The nth term of the given AP is [tex]a_n[/tex] = -54 + (n-1)7. The nth term of an Arithmetic Progression with the first term [tex]a_1[/tex] and the common difference d is given by [tex]a_n[/tex] = [tex]a_1[/tex] + (n-1)d.
What is Arithmetic Progression?It is a sequence of numbers in which the difference between the two consecutive numbers is always the same. Each number in the sequence is called a term of the progression.
The given expression is an arithmetic progression (AP), in which the common difference between consecutive terms is 7.
The first term of the AP is given as [tex]a_1[/tex] =-54 and the common difference (d) is 7.
Thus, the nth term of the given AP can be found using the formula
[tex]a_n[/tex] = [tex]a_1[/tex] + (n-1)d.
Substituting the values in this formula, we get
[tex]a_n[/tex] = -54 + (n-1)7.
Therefore, the nth term of the given AP is
[tex]a_n[/tex] = -54 + (n-1)7.
In general, the nth term of an AP with the first term [tex]a_1[/tex] and the common difference d is given by [tex]a_n[/tex] = [tex]a_1[/tex] + (n-1)d.
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Question:
Find the nth term by using the given data.
[tex]a_1[/tex] =-54 ; [tex]a_n[/tex] = [tex]a_(n-1)[/tex] +7
Graph the function h(x) = x - 4.
Compare the graph with the graph
of f(x) = x.
Answer:
Step-by-step explanation:
these functions are both straight lines with a slope of 1
f(x) = x passes through the origin (0,0)
h(x) = x-4 is parallel to f(x) = x and passes through the y axis at (0, -4) and is below the line f(x) = x
Solve the equation. If you get stuck consider using a diagram to help you. −4(y−2)=12
y=?
Answer:
[tex] \sf \: y = - 1[/tex]
Step-by-step explanation:
Now we have to,
→ Find the required value of y.
The equation is,
→ -4(y - 2) = 12
Then the value of y will be,
→ -4(y - 2) = 12
→ -4(y) - 4(-2) = 12
→ -4y - (-8) = 12
→ -4y + 8 = 12
→ -4y = 12 - 8
→ -4y = 4
→ y = 4 ÷ (-4)
→ [ y = -1 ]
Hence, the value of y is -1.
Will the following variables have positive correlation, negative correlation, or no correlation? number of doctors on staff at a hospital and number of administrators on staff Will these variables have positive correlation, negative correlation, or no correlation?
There is no definitive answer to whether the variables have positive correlation, negative correlation, or no correlation.
What is a negative correlation?
A negative correlation is a relationship between two variables in which they move in opposite directions. This means that when one variable increases then the other variable decreases. In statistical terms, a negative correlation is indicated by a negative correlation coefficient, which measures the strength and direction of the relationship between two variables.
Now,
The correlation between the number of doctors on staff at a hospital and the number of administrators on staff can vary depending on the specific circumstances of the hospital.
In general, one might expect that as the number of doctors on staff increases, the demand for administrative support may also increase. In this case, we would expect a positive correlation between the number of doctors and administrators.
On the other hand, if the hospital is focused on reducing costs and improving efficiency, it may choose to reduce administrative staff while maintaining the same number of doctors. In this case, we would expect a negative correlation between the number of doctors and administrators.
Therefore, there is no definitive answer to whether the variables have positive correlation, negative correlation, or no correlation, as it depends on the specific context and factors influencing the hospital's staffing decisions.
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In a circle, a sector is created by an arc measuring 54 degrees. If the diameter of the circle is 20 in, what is
a) the length of the arc
b) the area of the sector
Answer:
a) 9.42 in
b) 47.1 sq. inches
Step-by-step explanation:
[tex]\sf \bf \theta = 54^\circ\\\\diameter = 20 \ in\\\\r = 20 \div 2\\\\r = 10 \ in[/tex]
a) Length of arc:
[tex]\boxed{\bf Lenght \ of \ arc = \dfrac{\theta}{360}*2\pi r}[/tex]
[tex]= \dfrac{54}{360}*2*3.14*10\\\\= 9.42 \ in[/tex]
b) Area of sector:
[tex]\boxed{\bf Area \ of \ sector = \dfrac{\theta}{360}*\pi r^2}[/tex]
[tex]\bf = \dfrac{54}{360}*3.14*10*10\\\\= 47.1 \ in^2[/tex]
Sam's Sandwich Shop lets you design your own sandwich. There are 5 choices for bread, 6 choices for meat, and 4 choices for cheese. You can also choose to include (or not include ) each of the following items by request: lettuce, tomato, hot peppers, mayonnaise, or mustard.
Therefοre, the sοlutiοn οf the given prοblem οf unitary methοd cοmes οut tο be the same respοnse: 3,840 pοssibilities exist.
What is an unitary methοd?The task can be cοmpleted by integrating what was learned and putting this variable technique intο practise, which alsο incοrpοrates all supplemental data frοm twο individuals whο used a specific tactic. Or, tο put it anοther way, if the desired expressiοn οutcοme happens, either the entity stated in the algοrithm will alsο be fοund, οr bοth essential prοcesses will actually bypass the cοlοur. Fοr fοrty pens, a refundable charge οf Rupees ($1.01) might be required.
Here,
⇒ [tex]$\mathrm{_n C_r=\frac{n !}{r ! (n-r) !}}[/tex]
where r is the number of chοices you select, and n is the total number of options in a category.
For instance, there are 5 οptions for bread, but you can only select 1 (since there are only 1 types of bread):
[tex]$\Rightarrow \ \mathrm{_5 C_1=\frac{5 !}{1 ! (5-1) !}}[/tex]
Similar to this, there are 6 οptions for beef, and you can only pick one:
[tex]$\Rightarrow \ \mathrm{_6 C_1=\frac{6 !}{1 ! (6-1) !}} = 6[/tex]
There are 4 options for cheese, and yοu only get to pick one:
[tex]$\Rightarrow \ \mathrm{_4 C_1=\frac{4 !}{1 ! (4-1) !}} = 4[/tex]
There are 3,840 possible sandwich cοmbos, which can be calculated using the formula
₅C₁ × ₆C₁ × ₄C₁ × 25
= 5 × 6 × 4 × 32
= 3,840
Either approach will result in the same respοnse: 3,840 possibilities exist.
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1. Consider the pyramid.
(a) Draw and label a net for the pyramid.
(b) Determine the surface area of the pyramid. Show your work.
(Pyramid is listed in the pdf)
2. The back of Nico’s truck is 9. 5 feet long, 6 feet wide, and 8 feet tall. He has several boxes of important papers
that he needs to move. Each box of papers is shaped like a cube, measuring 1. 5 feet on each side.
How many boxes of papers can Nico pack into the back of his truck? Show your work.
Please help!
1) The unfolded shape of solid is called the net of the solid. The net of square pyramid is present in above figure 2. Area of square pyramid is equals to 224 mm².
2) The number of boxes that fit into back of Nico’s truck is equals to the 135.
We have a pyramid with a square base and triangular faces, as shown in Figure 1 above. Length of the base of the pyramid, b = 8 mm
Height of the pyramid, h = 10 mm
The net of the square pyramid is a plan view of each face and of the square base and its dimensions. Square pyramid (5 faces), i.e., 4 triangular levels and 1 square level. Square pyramid net has total 5 unfolded faces. So, required net of square pyramid present above. Now, Surface area of square pyramid is equals to sum of area of base square and area of 4 triangular faces. So, first we determine area of base square = b² , where 'b' is side of square. Here, b =8mm so, square area A₁ = 8² = 64 mm²
Also, area of a Triangle = (1/2)× base× height
so, area of triangle = (1/2)× 8×10 = 40 mm²
Area of 4 triangular faces of pyramid, A₂
= 4× 40 = 160 mm²
Therefore, Surface area of square pyramid present above = A₁ + A₂
= 64 mm² + 160 mm² = 224 mm²
2) We have a truck with dimensions.
Length of back of Nico’s truck, l = 9.5 feet
Width of back of Nico’s truck, w = 6 feet
Height of back of Nico’s truck, l = 8 feet
He has several boxes of important papers and he wants to hold in the back of truck. The shape of each box of papers is cube. The dimensions that is side of each cube = 1.5 feet
We have to determine the number of boxes of papers Nico will pack into the back of his truck.
The volume of the truck = Length ×Width × Height = l×w×h
so, volume of the Nico’s truck = 9.5 feet × 6 feet × 8 feet = 456 feet³
Volume of box of papers ( cube) = (side)³
= (1.5 feet )³ = 3.375 feet³
Number of boxes that fit into back of truck = volume of truck/ volume of each cubic box = 456 feet³/3.375 feet³
= 456/3.375 = 135.11 ~ 135
Hence, the required number of boxes are 135.
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Complete question :
1. Consider the above pyramid, figure 1.
(a) Draw and label a net for the pyramid.
(b) Determine the surface area of the pyramid. Show your work.
(Pyramid is listed in the pdf)
2. The back of Nico’s truck is 9. 5 feet long, 6 feet wide, and 8 feet tall. He has several boxes of important papers that he needs to move. Each box of papers is shaped like a cube, measuring 1. 5 feet on each side. How many boxes of papers can Nico pack into the back of his truck? Show your work.
Please help!
kareem is on the swim team. each day he swims 85 m. how far does he swim in nine days? write your answer in kilometers 
Answer:
0.765 km
Step-by-step explanation:
85*9=765 m
765/1000 to get km
0.765 km
Health programs routinely study the number of days that patients stay in hospitals. In one study, a random sample of 12 men had a mean of 7. 95 days and a standard deviation of 6. 2 days, and a random sample of 19 women had a mean of 7. 1 days and a standard deviation of 5. 0 days. The sample data will be used to construct a 95 percent confidence interval to estimate the difference between men and women in the mean number of days for the length of stay at a hospital. Have the conditions been met for inference with a confidence interval?
By answering the above question, we may state that we have assumed equation that the prerequisites for inference using a confidence interval have been satisfied.
What is equation?A math equation is a mechanism for connecting two claims by using the equals sign (=) to indicate equivalence. A mathematical statement that establishes the equivalence of two mathematical expressions is known as an equation in algebra. The equal symbol, for example, splits the numbers 3x + 5 and 14. A mathematical formula can be used to describe the relationship between two phrases written on opposite sides of a letter. The logo and programme are frequently the same. 2x - 4 Equals 2, for example.
Sample Size: The sample size should be large enough to guarantee that the mean sampling distribution is roughly normal. There is no hard and fast rule regarding what makes a "big enough" sample size, although a sample size of at least 30 is regarded sufficient. The sample size for both men and women is fewer than 30 in this situation.
We may proceed with generating a 95% confidence interval to estimate the difference between men and women in the mean number of days for the duration of stay at a hospital since we have assumed that the prerequisites for inference using a confidence interval have been satisfied.
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i-Ready
Use the distributive property to write an expression that is equivalent to 8 (a + 4)
8(a + 4) = ? a + ?
Answer:
8a + 32
Step-by-step explanation:
Distributive Property is a next level kind of multiplication.
The 8 on the outside of the parenthesis is being multiplying times the (a+4).
So you bounce that 8 into the parenthesis and times it to both the a and the 4.
8a + 8•4
= 8a + 32
see image.
Suruchi has $1.64 worth of change in the bottom of her purse If she reaches into her purse and randomly picks one of the coins, what is the probability Suruchi will pick a quarter?
Answer:
You need to know the total number of coins that was in her purse to answer this question.
WILL MARK U brainlist!!!!!!!
Answer:
x= -12, 5
Step-by-step explanation:
im so sorry for the late response it has been a whirlwind for me with trying to graduate with school lately
zeros of a function is when f(x) is equal to zero
with the graph we can see that -12 and 5 are where f(x) is equal to zero so these are the zeros of the function
we can find this out with the equation too like this
first we find the square of the function by finding a factors of -60 that will also add to be 7
12 and -5 are 2 factors
(12)(-5)= -60
12+ -5=7
now we can use this to make 2 equations
(x+12)(x-5)=0
x+12=0
x= -12
x-5=0
x=5
I hope this helps and isn't too confusing
Determine a series of transformations that would map polygon ABCDE onto polygon A'B'C'D'E'?
In order to map polygon ABCDE onto polygon A'B'C'D'E', a series of transformations must be performed. A common method of transforming a figure is to use a combination of translations, reflections, rotations, and dilations.
What is transformation?Transformation of a figure is the process of changing the shape, size, position or orientation of a 2D or 3D shape. This can be done using various techniques such as translations, rotations, reflections and enlargements. The transformation of a figure can help to visualize the change and understand the different properties of the shape. It can also be used to solve mathematical problems.
A transformation is a process in which a figure is changed in size, shape, or position.
A translation is a transformation that moves a figure in any direction. To move polygon ABCDE to polygon A'B'C'D'E', one must translate the figure to the right, left, up, or down.
A reflection is a transformation that flips a figure over a line, called the line of reflection. To reflect the figure onto the new polygon, the line of reflection must be chosen.
A rotation is a transformation that turns a figure around a point, called the center of rotation. To rotate the figure onto the new polygon, the center of rotation must be chosen.
A dilation is a transformation that changes the size of a figure. To scale the figure onto the new polygon, the scale factor must be chosen.
After the transformations are applied to the original figure, it will be mapped onto the new polygon. The combination of transformations must be chosen carefully in order to achieve the desired result.
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What is the inverse of the function f (x) = 3(x + 2)2 – 5, such that x ≤ –2? inverse of f of x is equal to negative 2 plus the square root of the quantity x over 3 plus 5 end quantity inverse of f of x is equal to negative 2 minus the square root of the quantity x over 3 plus 5 end quantity inverse of f of x is equal to negative 2 minus the square root of the quantity x plus 5 all over 3 end quantity inverse of f of x is equal to negative 2 plus the square root of the quantity x plus 5 all over 3 end quantity
The inverse οf f(x) is y = -2 - √[(x + 5)/3]
Tο find the inverse οf a functiοn, we can swap the pοsitiοns οf x and y and sοlve fοr y.
Starting with f(x) = 3(x + 2)² - 5:
y = 3(x + 2)² - 5
Swap x and y:
x = 3(y + 2)² - 5
Sοlve fοr y:
x + 5 = 3(y + 2)²
(x + 5)/3 = (y + 2)²
±√[(x + 5)/3] = y + 2
y = ±√[(x + 5)/3] - 2
Since x ≤ -2, we can οnly use the negative square rοοt tο ensure that y is a functiοn. Therefοre, the inverse οf f(x) is y = -2 - √[(x + 5)/3]
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suppose the wedding planner assumes that only 3% of the guests will be pollotarian so she orders 9 pollotarian meals. what is the approximate probability that she will have too many pollotarian meals? round to the nearest thousandth.
The probability that the wedding planner will have too many pollotarian meals is 0.998.
To calculate this, we can use the binomial probability formula. The binomial probability formula is used to calculate the probability of obtaining a certain number of successes in a certain number of trials. In this case, the number of trials is the total number of guests, and the number of successes is the number of guests who are pollotarian.
The formula is: P(x) =[tex]nCx * p^x * (1-p)^(n-x)[/tex], where n is the number of trials, x is the number of successes, and p is the probability of success.In this case, n = 300, p = 0.03, and x = 9. Plugging these numbers into the formula, we get: P(x) = [tex]300C9 * 0.03^9 * (1-0.03)^(300-9)[/tex] = 0.998.
Therefore, the probability that the wedding planner will have too many pollotarian meals is 0.998, or 99.8%.
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28. Suppose in 1981 the retail price of a VCR at Sears was $1,389. 88. What would be the cost of that VCR in today’s dollars? Hint: You will need the CPI’s of both years to discover the answer
the cost of a VCR that sold for $1,389.88 in 1981 would be $3,686.12 in today's dollars, adjusted for inflation.
To calculate the cost of a VCR in today's dollars based on its cost in 1981, we need to adjust for inflation using the Consumer Price Index (CPI).
We need to find the CPI for the year 1981 and the current year. For example, let's say the CPI for 1981 is 98.3 and the CPI for the current year is 260.5.
Inflation rate = (Current year CPI / 1981 CPI) x 100
Inflation rate = (260.5 / 98.3) x 100
Inflation rate = 265.17
This means that the general price level has increased by 265.17% since 1981.
To find the cost of the VCR in today's dollars, we multiply its cost in 1981 by the inflation rate:
Cost in today's dollars = Cost in 1981 x (Inflation rate / 100)
Cost in today's dollars = $1,389.88 x (265.17 / 100)
Cost in today's dollars = $3,686.12 (rounded to the nearest cent)
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An art teacher has 4 1 8 4 8 1 gallons of paint to pour into containers. If each container holds 3 8 8 3 gallon, how many containers can they fill?
If an art teacher has 4 1/8 gallons of paint to pour into containers, the art teacher can fill 11 containers with the 4 1/8 gallons of paint they have.
To solve the problem, we need to divide the total amount of paint by the capacity of each container.
First, we need to convert 4 1/8 gallons to an improper fraction. To do this, we multiply the whole number by the denominator and add the numerator, then put the result over the denominator:
4 1/8 = (4 x 8 + 1)/8 = 33/8
Next, we divide the total amount of paint by the capacity of each container:
33/8 ÷ 3/8 = 33/8 x 8/3 = 11
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Complete question is:
An art teacher has 4 1/8 gallons of paint to pour into containers. If each container holds 3/8 gallon, how many containers can they fill?
Find the coefficient of x/8+4=6
Answer: 0
Step-by-step explanation:
To find the coefficient of x in the equation:
x/8 + 4 = 6
We can begin by isolating x on one side of the equation. We can do this by subtracting 4 from both sides:
x/8 = 2
Next, we can multiply both sides by 8 to get rid of the fraction:
x = 16
Since there is no x-term on one side, the coefficient of x is 0.
Therefore, the coefficient of x in the given equation is 0.
For the function f(x)=5x^2+3x , evaluate and simplify f(x+h)-f(x)/h
The result of the evaluation and simplification of [tex]f(x)=5x^2+3x is f(x+h)-f(x)/h = h + 3[/tex], which is a line with a slope of 1 and a y-intercept of 3.
The function[tex]f(x)=5x^2+3x[/tex], when evaluated and simplified, is equal to [tex]f(x+h)-f(x)/h[/tex]. To calculate [tex]f(x+h)-f(x)/h[/tex], the first step is to subtract f(x) from f(x+h). This will leave us with the expression . After this, we can simplify the expression to [tex]h^2 + 3[/tex]h. Next, we divide this expression by h which will leave us with h + 3.
This means that our result is [tex]f(x+h)-f(x)/h = h + 3[/tex]. This can be visualized as a line on a graph with a slope of 1 and a y-intercept of 3. This means that as the value of x increases, the value of [tex]f(x+h)-f(x)/h[/tex]increases by 1 for every increase in x.
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Write a quadratic function that has vertex in the third quadrant
and has no x-intercept, Explain how your find such function.
Quadratic function that has vertex in the third quadrant and has no x-intercept is f(x) = 2(x+3)² - 4.
Quadratic function is a polynomial function of the second degree of the form f(x)= ax² + bx + c where a is a non-zero coefficient and x is the variable. Vertex is the point on the parabola which has a minimum or maximum point. The third quadrant is where both the x and y coordinates are negative. The question requires finding a quadratic function with vertex in the third quadrant and has no x-intercept. Quadratic function with no x-intercept means that the graph does not cross the x-axis. Here is how to find such function: Step-by-step We can start by using the vertex form of a quadratic function. f(x) = a(x-h)² + k Where (h, k) is the vertex of the parabola. Since the vertex is in the third quadrant, h and k should be negative numbers.
Also, since the graph should not cross the x-axis, the leading coefficient (a) should be either positive or negative. If a>0, the parabola opens upwards, and if a<0, the parabola opens downwards. To find a quadratic function that has no x-intercept and has a vertex in the third quadrant, let's choose any negative value of h and k such that |k|>|h|, and a>0. For instance, let h=-3 and k=-4. Then, substituting these values in the vertex form, we get:f(x) = a(x+3)² - 4If we want the parabola to have a minimum point at the vertex, then a should be positive. We can choose any positive value of a. For example, let's take a=2. Then, the quadratic function with the required conditions is:f(x) = 2(x+3)² - 4f(x) = 2x² + 12x + 14The graph of this function has vertex (-3, -4) and does not cross the x-axis. Quadratic function that has vertex in the third quadrant and has no x-intercept is f(x) = 2(x+3)² - 4.
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Measure the height of the tin in mm and write the real height in mm
Measurement is the act of comparing an object's properties to a standard quantity. It is crucial in determining an object's quantity.
How to measure the height of a tinFor instance, to measure the height of a tin, one must measure the vertical distance from the base to the top.
The measurement must be in millimeters, and a ruler is the most suitable tool.
To do this, place the ruler vertically with point 0 at the baseline of the tin, mark the point where the ruler coincides with the top, and read the height to the nearest millimeter.
To measure the height of a tin in millimeters, follow these steps:
Obtain a ruler that has millimeter markings.
Place the tin upright on a flat surface.
Position the ruler vertically with its zero point aligned with the base of the tin.
Carefully move the ruler up or down until it reaches the top edge of the tin.
Read the measurement value at the point where the ruler aligns with the top edge of the tin.
Record the height measurement in millimeters to the nearest whole number.
For example, if the measurement value is 65.5 millimeters, then the real height of the tin in millimeters is 66 millimeters (rounded to the nearest whole number).
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List the procedure to measure the height of a tin in mm and write the real height in mm
Suppose you are an engineer tasked to design a multi-storey car park. The height restriction for vehicles entering the car park is calculated to be 2.51 m. A sign indicating the maximum height, correct to the nearest metre, is to be placed at the entrance. What should the maximum height be shown as? Explain your answer.
The nearest whole number when rounding to the nearest metre forces us to choose 3, which is the case here as 2.51 is closer to 3 than it is to 2.
How are significant figures employed in scientific measurements? What are they?The digits in a numerical value known as significant figures, sometimes known as significant digits, are those that show how precisely the measurement was made. The degree of precision of the measuring device used to perform the measurement determines the number of significant figures in the measurement.
We must round the height restriction to the closest metre in order to show it on the sign because it is specified as 2.51 metres.
We look at the digit in the tenths place, which is 5, to round to the closest metre. We round up the number to the next one since 5 is more than or equal to 5, which is 1. Thus, 3 m should be listed as the maximum height.
This is due to the fact that picking the nearest whole number when rounding to the nearest metre forces us to choose 3, which is the case here as 2.51 is closer to 3 than it is to 2.
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Find the centroid of the upper half of the circle x^2+y^2=a^2
The centroid (C) of the upper half of the circle is, C = (x_c, y_c) = (-2a/3π, 4a/3π)
We can find the centroid of the upper half of the circle by using integration. Let's denote the upper half of the circle as a function of x:
y = f(x) = sqrt(a^2 - x^2)
To find the centroid (C) of this region, we need to find the coordinates (x_c, y_c) such that:
x_c = (1/A) × ∫(a, -a) x*f(x) dx
y_c = (1/A) × ∫(a, -a) [F(x) - f(x)] dx
where A is the area of the upper half of the circle and F(x) is the equation of the circle.
First, let's find A:
A = ∫(a, -a) f(x) dx
= (1/2) × ∫(a, -a) sqrt(a^2 - x^2) dx
= (1/2) × [a^2 × sin^(-1)(x/a) + x × sqrt(a^2 - x^2)]_a^(-a)
= (1/2) × [a^2 × π + 0 - (-a^2 × π) + 0]
= πa^2/2
Next, let's find x_c:
x_c = (1/A) × ∫(a, -a) x×f(x) dx
= (2/πa^2) × ∫(a, 0) x × sqrt(a^2 - x^2) dx
(Note: We only integrate from 0 to a because the function f(x) is symmetric about the y-axis)
Let u = a^2 - x^2
Then du/dx = -2x, and dx = -du/(2x)
So the integral becomes:
(2/πa^2) × ∫(0, a^2) [(a^2 - u) × sqrt(u)] × (-du/(2x))
= -(1/πa^2) × ∫(0, a^2) sqrt(u) du
= -(1/πa^2) × [(2/3) × u^(3/2)]_0^(a^2)
= -(2/3πa^2) × (a^3)
= -2a/3π
Therefore, x_c = -2a/3π.
Finally, let's find y_c:
y_c = (1/A) × ∫(a, -a) [F(x) - f(x)] dx
= (2/πa^2) × ∫(a, 0) (a^2 - x^2) dx
(Note: We only integrate from 0 to a because the function f(x) is symmetric about the y-axis)
= (2/πa^2) × [a^2x - (1/3)x^3]_0^a
= (2/πa^2) × [(2/3)a^3]
= 4a/3π
Therefore, y_c = 4a/3π.
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Write an equation of the Line that passes thru (5,-2);and is perpendicular to y=5/3x -3
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{5}{3}}x-3\qquad \impliedby \qquad \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ \cfrac{5}{3}} ~\hfill \stackrel{reciprocal}{\cfrac{3}{5}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{3}{5} }}[/tex]
so we're really looking for the equation of a line whose slope is -3/5 and it passes through (5 , -2)
[tex](\stackrel{x_1}{5}~,~\stackrel{y_1}{-2})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{3}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-2)}=\stackrel{m}{- \cfrac{3}{5}}(x-\stackrel{x_1}{5}) \implies y +2= -\cfrac{3}{5} (x -5) \\\\\\ y+2=-\cfrac{3}{5}x+3\implies {\Large \begin{array}{llll} y=-\cfrac{3}{5}x+1 \end{array}}[/tex]
9. 3: Paper Folding folds Area of paper The area of a sheet of paper is 93. 5 square inches. Write an equation expressing the visible area $$a of the sheet of paper in terms of the number of times it has been folded $$n
The equation expressing the visible area an of the sheet of paper in terms of the number of times it has been folded n is a = 93.5/2^n.
Every time a sheet of paper is folded in half, the visible area is reduced by half. If the original area of the sheet of paper is A, then after the first fold, the visible area is A/2. After the second fold, the visible area is (A/2)/2 = A/4. In general, after n folds, the visible area is A/[tex]2^n[/tex].
In this problem, the original area of the sheet of paper is given as 93.5 square inches. Therefore, the equation expressing the visible area a of the sheet of paper in terms of the number of times it has been folded n is a = 93.5/[tex]2^n[/tex]. As the number of folds increases, the visible area of the sheet of paper decreases rapidly, approaching zero as n approaches infinity.
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The sum of interior angle of a regular polygon is 900 degrees. Find the number of sides of the polygon
Answer:
7
Step-by-step explanation:
(number of sides - 2) x 180 = sum of interior angles
(number of sides - 2) x 180 = 900
Reverse the formula.
900 / 180 = 5
5 + 2 = 7
Therefore, number of sides = 7
100 points please help
Answer:(9x + 10)(9x - 10)
Step-by-step explanation:
a rectangular rose garden has dimensions 5.5 feet by 3.75 feet. A rectangular vegetable garden has dimensions 13.75 feet by 6 feet. How many times as great is the area of the vegetable garden compared to the area of the flower garden? The area of the vegetable garden is times as great the area of the rose garden
Answer: 4
Step-by-step explanation:
Area of rose garden = 5.5 * 3.75 = 20.625
Area of vegetable garden = 13.75 * 6 = 82.5
To find how many times greater the area of the vegetable garden is, we simply divide 82.5 by 20.825
82.5 / 20.625 = 4
Does anyone know the answer to this question?
Answer:
2, -7, 3 are your answers! :).
Step-by-step explanation:
The opposite of a positive would be it's negative form, and the opposite of a negative would be it's positive form.
Jamie is on her first day's work at a new furniture delivery job. Driving to her first delivery, she
encounters a parabolic tunnel. She is not sure that her van will fit through the tunnel.
This tunnel is 8 m wide at its base and 8 m tall at its highest point.
Unfortunately Jamie is late for the
delivery so hopes for the best and drives through the tunnel.
There are two lanes, but heavy oncoming traffic forces her to stay in her own lane (so that she cannot cross
the middle line.)
If Jamie’s van is 2.5 m wide and 5 m tall, will she make it?
(Or is her first day's work also going to be her last)?
From the problem, the minimum height of the tunnel that Jamie's van must be able to pass through is 12.5 m.
How to get the Height?We can approach this problem by determining the minimum width and height of the tunnel that Jamie's van must be able to pass through. If her van is able to fit through these minimum dimensions, then it should be able to pass through the entire tunnel.
Let's start with the width of the tunnel. Since Jamie cannot cross the middle line due to heavy oncoming traffic, her van must fit entirely within her own lane. Therefore, the width of the tunnel must be at least equal to the width of her van, which is 2.5 m.
Next, let's consider the height of the tunnel. At its highest point, the tunnel is 8 m tall. However, Jamie's van is 5 m tall. To pass through the tunnel, the van must fit under the lowest point of the tunnel that is at least 5 m above the ground. We can use the shape of the parabolic tunnel to determine the minimum height that Jamie's van can pass through.
The shape of a parabolic curve is given by the equation y = ax^2, where y is the height, x is the distance from the center of the curve, and a is a constant that determines the steepness of the curve. In this case, we can use the fact that the tunnel is 8 m wide at its base to determine the value of a.
At the center of the tunnel (x = 0), the height is 8 m. Therefore, we have:
8 = a(0)^2
a = 8
Substituting this value of a into the equation for the parabolic curve, we have:
y = 8x^2
To determine the minimum height that Jamie's van can pass through, we need to find the value of x that corresponds to the edge of her van. Since her van is 2.5 m wide, this corresponds to a distance of 1.25 m from the center of the lane.
Substituting x = 1.25 into the equation for the parabolic curve, we have:
y = 8(1.25)^2
y = 12.5
Therefore, the minimum height of the tunnel that Jamie's van must be able to pass through is 12.5 m.
Since the height of the tunnel is greater than the height of Jamie's van, she should be able to pass through without any problems. However, she should still exercise caution and be mindful of the height and width of her vehicle when driving through narrow spaces.
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Find an equation of the line with gradient -1 and that passes through the
point (-3,0)
Submit Answer
Answer:
The equation of the line with gradient -1 that passes through the point (-3, 0) can be found using the point-slope form of a line. The point-slope form of a line is given by:
y - y1 = m (x - x1)
Where m is the gradient and (x1, y1) is a point on the line.
For this line, m = -1 and (x1, y1) = (-3, 0). Thus, the equation of the line is:
y - 0 = -1 (x - (-3))
y = -x + 3
Answer:
x + y + 3 = 0
Step-by-step explanation:
equation of line ( point slope format)
(y-y1) = m(x - x1)
they had given the point as (-3,0) on comparing with (x1, y1) and substituting the values in equation we get
y = -1(x +3)
final ans
x + y + 3 = 0