Graphs of Polynomial Functions

Answer: To find the length of the other side of the rectangle, we can use the formula for the area of a rectangle, which is:

area = length x width

We know that the area is 2314 square miles and one side is 26 miles. Let's plug these values into the formula and solve for the other side:

2314 = 26 x width

Divide both sides by 26:

width = 2314/26

Simplify:

width = 89

Therefore, the other side of the rectangle is 89 miles.

Step-by-step explanation:

The town with the greatest average rainfall was Milton; the average rainfall in Southport was 0.75 inches more than in Fairview, and the actual rainfall in Fairview was 4.15 inches.

To identify this only observe the table and identify the highest value, in this case, the highest value is 4. 20 in average for Milton.

4.09 - 3.35 = 0.75 inches

3.35 (information registered) + 0.8 = 4.15 inches

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Answer:

The statement is correct

Step-by-step explanation:

Use the equation of a circle:

x² + y² = r²

x² + y² = 3²

then enter the coordinates of the point (2; √5):

2² + (√5)² = 9

4 + 5 = 9 (that is correct)

The point (2, √5) lies on the circle centered at the origin with radius 3 is True statement.

The standard equation of a circle is given by:

(x-h)² + (y-k)² = r²

Where (h,k) is the coordinates of center of the circle and r is the radius.

We have,

The point  (2, √5) lies on the circle centered at the origin with radius 3.

We know the equation of circle at origin is

x² + y² = r², where r is the radius of circle

So, x² + y² = 3²

Now. to check the point put x= 2 and y= √5 as

x² + y²

= 2² + √5²

= 4 +5

= 9

Thus, the given statement is Correct.

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Function is continuous across its domain is B.

The domain of a function is the set of numbers that can go into a given function. In other words, it is the set of x-values that you can put into any given equation. The set of possible y-values is called the range. If you want to know how to find the domain of a function in a variety of situations,

If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of [tex]f[/tex].

The function f(x) = 0.52 is a constant function, and constant functions are always continuous across their domain.

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Answer:

  63.0°

Step-by-step explanation:

You want the measure of an angle in a right triangle with hypotenuse 11 and adjacent side 5.

The cosine relation is ...

  Cos = Adjacent/Hypotenuse

  cos(x) = 5/11

Then the angle is found using the inverse cosine function:

  x = arccos(5/11) ≈ 63.0°

The value of x is about 63.0°.

__

Additional comment

The calculator's angle mode needs to be set to "degrees."

[tex]\begin{array}{llll} 19. & \text{x}= \frac{9\sqrt{6}}{2}, & \text{y}= \frac{9\sqrt{6}}{2}, & \text{z}= 18\\\\21. & \text{x}= 6, & \text{y}= 6\sqrt{3}, & \text{z}= 6\sqrt{6}\\\\23. & \text{x}= 10\sqrt{3}, & \text{y}= 5\sqrt{3}, & \text{z}= 15\\\\\end{array}[/tex]

===========================================================

Work Shown:

Problem 19

The triangle up top is a 30-60-90 triangle. The short leg is opposite the smallest angle 30 degrees. Double the short leg to get the hypotenuse.

z = 2*9 = 18.

The long leg is found like so

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{long leg} = 9\sqrt{3}\\\\[/tex]

These two formulas apply to 30-60-90 triangles only.

The long leg of the 30-60-90 triangle up top is the hypotenuse of the 45-45-90 triangle at the bottom.

For 45-45-90 triangles, we can say:

[tex]\text{hypotenuse} = (\text{leg})*\sqrt{2}\\\\\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}\\\\\text{leg} = \frac{\text{hypotenuse}*\sqrt{2}}{2}\\\\\text{leg} = \frac{9\sqrt{3}*\sqrt{2}}{2}\\\\\text{leg} = \frac{9\sqrt{3*2}}{2}\\\\\text{leg} = \frac{9\sqrt{6}}{2}\\\\[/tex]

This represents the lengths of x and y. For any 45-45-90 triangle, the two legs are the same length (the right triangle is isosceles).

--------------------------------

Problem 21

Focus on the 30-60-90 triangle up top.

The hypotenuse of 12 divides in half to get x = 12/2 = 6 as the short leg.

The long leg is [tex]6\sqrt{3}[/tex]

This is also the leg of the 45-45-90 triangle down below. Therefore, we have [tex]y = 6\sqrt{3}[/tex]

Multiply the leg by sqrt(2) to find the hypotenuse.

[tex]\text{hypotenuse} = \text{leg}*\sqrt{2}\\\\\text{z} = 6\sqrt{3}*\sqrt{2}\\\\\text{z} = 6\sqrt{3*2}\\\\\text{z} = 6\sqrt{6}\\\\[/tex]

--------------------------------

Problem 23

Focus on the triangle on the right. This is a 45-45-90 triangle.

The hypotenuse is 15sqrt(2), which must mean each leg is 15 units long. Therefore, z = 15. The unmarked vertical leg is also 15 units long.

This vertical side is the leg of the 30-60-90 triangle on the left. It's the long leg.

[tex]\text{long leg} = (\text{short leg})*\sqrt{3}\\\\\text{short leg} = \frac{\text{long leg}}{\sqrt{3}}\\\\\text{short leg} = \frac{(\text{long leg})*\sqrt{3}}{3}\\\\\text{y} = \frac{15\sqrt{3}}{3}\\\\\text{y} = 5\sqrt{3}\\\\[/tex]

Double this short leg to get the hypotenuse of the 30-60-90 triangle.

[tex]x = 2y\\\\x = 2*5\sqrt{3}\\\\x = 10\sqrt{3}\\\\[/tex]

Answer:

Question 19:

Question 21:

Question 23:

Step-by-step explanation:

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : 1 : √2.  Therefore, the formula for the ratio of the sides is b : b : b√2 where:

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.  Therefore, the formula for the ratio of the sides is c: c√3 : 2c where:

Side z is the hypotenuse of a 30-60-90 triangle where the leg opposite the 30° angle measures 9 units. Therefore, c = 9.

[tex]\implies z=2c =2 \cdot 9=18\; \sf units[/tex]

Therefore, the other leg of the same triangle (opposite the 60° angle) measures c√3 = 9√3 units.

Sides x and y are the congruent legs of a 45-45-90 triangle with hypotenuse measuring 9√3 units. Therefore b√2 = 9√3, so b = (9√6)/2.

[tex]\implies x=\dfrac{9 \sqrt{6}}{2}\; \sf units[/tex]

[tex]\implies y=\dfrac{9 \sqrt{6}}{2}\; \sf units[/tex]

Side x is the side opposite the 30° angle in a 30-60-90 triangle with a hypotenuse of 12 units. Therefore, 2c = 12 so c = 6.

[tex]\implies x=6\; \sf units[/tex]

Therefore, the other leg of the same triangle (opposite the 60° angle) measures c√3 = 6√3 units.

Side y is one of the congruent legs of a 45-45-90 triangle where the other congruent leg measures 6√3 units.

[tex]\implies y=6\sqrt{3}\; \sf units[/tex]

Side z is the hypotenuse of a 45-45-90 triangle where the congruent legs measure 6√3 units. Therefore b = 6√3.

[tex]\implies z=b\sqrt{2} = 6 \sqrt{3} \sqrt{2}=6\sqrt{6}\; \sf units[/tex]

Side z is one of the congruent legs of a 45-45-90 triangle where the hypotenuse measures 15√2 units. Therefore, b√2 = 15√2, so b = 15.

[tex]\implies z=b=15\; \sf units[/tex]

Side x is the hypotenuse of a 30-60-90 triangle where the leg opposite the 60° measures 15 units. Therefore, c√3 = 15, so c = 5√3.

[tex]\implies x=2c=2 \cdot 5\sqrt{3}=10\sqrt{3}\; \sf units[/tex]

Side y is the side opposite the 30° angle in a 30-60-90 triangle where the hypotenuse measures 10√3 units. Therefore, 2c = 10√3, so c = 5√3.

[tex]\implies y=c=5\sqrt{3}\; \sf units[/tex]

Answer:

The slope of the line is 60.

Step-by-step explanation:

We are given a table of values, and h and c are substitutes from x and y. Judging by this, if we know the formula [tex]\frac{y2-y1}{x2-x1}[/tex] then we can evaluate the slope, we just have to take two of the coordinate points from the table.

Lets take (3,205) and (6,385) and plug it into the formula

[tex]\frac{385-205}{6-3}[/tex]

Evaluate

[tex]\frac{180}{3}[/tex]

Simplify the fraction

[tex]\frac{60}{1}=60[/tex]

The slope of the line is 60.

Hence, x=-2 and y=4 are the answers to the system of equations.

For the purpose of connecting algebra and geometry with the aid of line and curve graphs, coordinate geometry is necessary. Finding points on a plane is a crucial component of mathematics. It also has a number of uses in other scientific fields such dimensional geometry, calculus, and trigonometry. Use y=-2x in place of x in the first equation:

5x - 9(-2x) = -46

Simplify and find x's value:

5x + 18x = -46

23x = -46

x = -2

We can use the second equation to get y now that we know the value of x:

y = -2x = -2(-2) = 4

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1/2cm in the scale drawing represents one foot on the actual building.

What is the scale factor?

The ratio of the scale of an original thing to a new object that is a representation of it but of a different size is known as a scale factor (bigger or smaller). When both the original dimensions and the new dimensions are known, the scale factor may be determined. The following is a basic formula to determine a figure's scale factor: Scale factor is equal to the difference between the dimensions of the old and new shapes.

Here, we have

Given: The scale drawing of a building has a height of 10 centimeters. The actual building is 20 feet high.

we have to find how many centimeters in the scale drawing represent one foot on the actual building.

The answer will be 1/2 because 1 centimeter is equal to 2 feet in reality. But since we want to know the answer of how many centimeters is in 1 foot, we will divide that in half, to get an answer of 1/2 centimeter.

Hence, 1/2cm in the scale drawing represents one foot on the actual building.

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[tex]\cfrac{31\pi }{4}\implies \cfrac{8\pi +8\pi +8\pi +4\pi +3\pi }{4}\implies 2\pi +2\pi +2\pi +\pi +\cfrac{3\pi }{4}[/tex]

so the angle θ  is really 3 revolutions plus π and then a little bit more, Check the picture below for the red reference angle.

Answer:

≈ $5194,7

Step-by-step explanation:

First, find out how much did the price decrease:

[tex] \frac{7421 \times 30\%}{100\%} = 2226.3[/tex]

Now, subtract this number from the previous one (the price before):

$7421,00 - $2226,30 = $5194,70 ≈ $5194,7

The two points on the line to graph the function is (3, 4/3) and (-6, 10/3).

A line is a one-dimensional, straight shape that can go on forever in both directions. It is made up of an infinite number of points that are organized in a straight line.

We are unable to offer two points on the line for graphing in the absence of a line or function.

We consider the function is, [tex]f(x) = 2 - \frac{2}{9} x[/tex]

To graph the function [tex]f(x) = 2 - \frac{2}{9} x[/tex] , we can choose any two values of x, plug them into the equation to find the corresponding values of y, and then plot the points (x, y) on the coordinate plane. Here are two possible sets of points we can use:

Set 1:

Let x = 0:

f(0) = 2 - (2/9)(0) = 2

Therefore, one point on the line is (0, 2).

Let x = 9:

f(9) = 2 - (2/9)(9) = 0

and another point on the line is (9, 0).

Now, we can plot these two points on the coordinate plane and draw a straight line passing through them to graph the function [tex]f(x) = 2 - \frac{2}{9} x[/tex]

Set 2:

Let x = 3:

f(3) = 2 - (2/9)(3) = 2 - (2/3) = 4/3

So one point on the line is (3, 4/3).

Let x = -6:

f(-6) = 2 - (2/9)(-6) = 2 + (4/3) = 10/3

So another point on the line is (-6, 10/3).

Now, we can plot these two points on the coordinate plane and draw a straight line passing through them to graph the function [tex]f(x) = 2 - \frac{2}{9} x[/tex]

Therefore, the two points on the line to graph the function is (3, 4/3) and (-6, 10/3).

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The sides of the triangle in order from shortest to longest are b < a < c

In triangle ABC, the side opposite angle A is denoted by a,

To determine the order of the sides from shortest to longest, we need to use the triangle inequality theorem which implies that

The side opposite the largest angle is the largest side and vice versa

So, we have

b from B = 38 degrees

a from A = 54 degrees

c from C = 88 degrees

Therefore, the order of the sides from shortest to longest is: b < a < c

In other words, the side opposite angle B is the shortest, followed by the side opposite angle A, and the side opposite angle C is the longest.

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the length of blue rectangle 2 to the width of green rectangle 6 is: 2/6 = 0.33 (rounded to two decimal places)

How to find the length of blue rectangle?

To find the length of blue rectangle 2 in relation to the length of green rectangle 4, we need to divide the length of blue rectangle 2 by the length of green rectangle 4.

So, the length of blue rectangle 2 to the length of green rectangle 4 is:

2/4 = 0.5

To find the length of blue rectangle 2 in relation to the width of green rectangle 6, we need to divide the length of blue rectangle 2 by the width of green rectangle 6.

So, the length of blue rectangle 2 to the width of green rectangle 6 is:

2/6 = 0.33 (rounded to two decimal places)

Note that the ratio of the length of blue rectangle 2 to the length of green rectangle 4 is different from the ratio of the length of blue rectangle 2 to the width of green rectangle 6.

A rectangle is a two-dimensional shape that has four sides and four right angles. It is a type of quadrilateral, which means a four-sided polygon.

In a rectangle, the opposite sides are parallel and have equal length. This makes it different from a square, which is also a rectangle but has four equal sides.

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The cost per photo at which revenue is maximized is about $127.95.

We must first calculate the relevant y-value for the quadratic function at

x = 6.5 in order to get the pricing per photo that optimizes revenue (i.e., the revenue). When we enter x = 6.5, we obtain:

[tex]R(6.5) = 0.3(6.5)^2 + 3.9(6.5) + 97.5 =[/tex]$127.95

Before any costs or expenses are subtracted, revenue is the income derived from the sale of goods or services, or from any other use of capital or assets, related to an organization's primary operations. Also, it is known as sales or turnover.

from the question:

We can apply quadratic regression to identify the quadratic function that represents the revenue as a function of how many images are sold. This will provide us with the form's function:

[tex]R(x) = ax^2 + bx + c[/tex]

When x is the number of sold images, R(x), a, b, and c are coefficients we need to discover, and R(x) is the revenue.

We may construct an equation system to find the coefficients using the provided data:

(1) [tex]a(3)^2 + b(3) + c = 135[/tex]

(2)[tex]a(5)^2 + b(5) + c = 195[/tex]

(3) [tex]a(10)^2 + b(10) + c = 240[/tex]

When we simplify each equation, we obtain:

(1) 9a + 3b + c = 135

(2) 25a + 5b + c = 195

(3) 100a + 10b + c = 240

Any approach will work to solve this system of equations. We'll apply the substitution technique in this case.

Equation (1) gives us:

c = 135 - 9a - 3b

When we enter this into equation (2), we obtain:

25a + 5b + (135 - 9a - 3b) = 195

Simplifying, we get:

16a + 2b = 12

From equation (3), we get:

c = 240 - 100a - 10b

Substituting this into equation (1), we get:

9a + 3b + (240 - 100a - 10b) = 135

Simplifying, we get:

-91a - 7b = -105

Solving the system of equations 16a + 2b = 12 and -91a - 7b = -105, we get:

a = 0.3

b = 3.9

c = 97.5

Therefore, the quadratic function that models the revenue as a function of the number of photos sold is:

[tex]R(x) = 0.3x^2 + 3.9x + 97.5[/tex]

We must determine the x-value that corresponds to the parabola's vertex in order to determine the number of images that must be sold in order to maximize revenue. The vertex's x-value is determined by:

x = -b / 2a

Plugging in the values for a and b, we get:

x = -3.9 / 2(0.3) = -6.5

This indicates that the vertex of the parabola lies at x = 6.5 because the number of images sold cannot be negative. Darlena should therefore sell 6.5 photographs in order to increase her money.

We must first calculate the relevant y-value for the quadratic function at x = 6.5 in order to get the pricing per photo that optimizes revenue (i.e., the revenue). When we enter x = 6.5, we obtain:

[tex]R(6.5) = 0.3(6.5)^2 + 3.9(6.5) + 97.5 ≈ $127.95[/tex]

As a result, $127.95 per photo is the price that maximizes sales.

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Mean is the average. Add the minutes together and divide by 10.

[tex]80 \div10 = 8[/tex]

The answer is C

The y-coordinate value of Point C if the area of ABC is 9 units squared is: 5 or -1

Point A is at (x, 2) and B is at (x + 6, 2). Since AB must lie on the line y=2  and be 6 units long. Point C is on the line x = -3 . So let C be at (-3, y).

Since ΔABC is a right angle, then point C must have the same x-coordinate as point A. Therefore, A(-3, 2) and B(2, 2).

The area of ΔABC is 9. So,

9 = 1/2 (b)(h)

where b is the base and h is the height.

so b = 6 and h = AC

Solving this for AC gives

9 = 1/2 (6)(AC)

18/6 = AC

3 = AC

Point C must lie 3 units above point A or 3 units below the point A. If it lies 3 units above, then it has a y-coordinate of 2 + 3 = 5.

If it lies 3 units below, it has a y-coordinate 2 - 3 = -1.

Therefore, y-coordinate is 5 or -1.

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Answer:

The volume is 10 × 20 × 12 = 2,400 square inches.

Answer:

1. C

2. B

Step-by-step explanation:

A. A(-2,5) => A' (-2 + 2, 5) or A' (0,5)

B. A(-2,5) => A' (- -2, 5) or A' (2,5)

Answer:

Step-by-step explanation:

the problem that would have to be plugged into the calculator would be as following

Answer:

The value of t that satisfies the equation and makes triangle ABC an isosceles triangle is 9 cm.

Step-by-step explanation:

Since triangle ABC is an isosceles triangle, we know that the lengths of AB and AC are equal. Using the lengths given in the diagram, we can set up an equation:

AB = AC

4t - 7 = 2t + 11

Now we can solve for t:

4t - 7 = 2t + 11

2t = 18

t = 9

Therefore, the value of t that satisfies the equation and makes triangle ABC an isosceles triangle is 9 cm.

Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

By answering the above question, we may state that Hence, neither expressions French nor German are spoken by nearly 97.7% of the world's population.

In mathematics, you can multiply, divide, add, or take away. The following is how an expression is put together: Numeric value, expression, and math operator The elements of a mathematical expression include numbers, parameters, and functions. It is feasible to use contrasting words and expressions. Any mathematical statement containing variables, numbers, and a mathematical action between them is known as an expression, often known as an algebraic expression. As an example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

In the image you included, a pie chart illustrating the distribution of the global languages spoken by the entire population can be seen.

We need to figure out what portion of the world's population does not speak either French or German in order to respond to your query.

Around 1.2% of the world's population speaks French, according to the pie chart, and 1.1% speaks German, according to the same data.

Hence, it is possible to determine the proportion of people on Earth who do not speak either French or German as follows:

German: 1.1%; French: 1.2%; 100% = 97.7%

Hence, neither French nor German are spoken by nearly 97.7% of the world's population.

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Answer:There’s no equation or numbers in your word problem

Step-by-step explanation:

The square yards the company needs to mow for the yard shape below is 58.88 square yards.

A square is a two-dimensional figure that has four sides and all four sides are equal.

The area of a square is given as side²

We have,

From the figure,

The yard shape has two shapes.

Triangle and a square.

Now,

Area of the triangle shape.

= 1/2 x 6.4 x 5.6

= 17.92 square yards

Area of the square shape.

= 6.4²

= 40.96 square yards

Now,

The square yards the company needs to mow for the yard shape below.

= 17.92 + 40.96

= 58.88 square yards.

Thus,

The square yards the company needs to mow for the yard shape below is 58.88 square yards.

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The value of y is 5 when x = -31/3.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.

There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form.

Thus, from the given question, we can see that:

The value of y is -3 when x = -17/5.

The value of y is 0 when x = 1/2.

The equation that represents this relationship is y = (2x - 1) / (x + 6).

However, the value of y cannot be 2 for any value of x that satisfies this equation.

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Answer:

The total surface area of the triangular prism is 556 square units.

Step-by-step explanation:

The given 3D figure is a triangular prism.

The formula to find the total surface area of a triangular prism is:

[tex]\boxed{\textsf{Total Surface Area}= \sf 2 A_B+(a+b+c)h}[/tex]

where:

Therefore, the surface area of a triangular prism is the sum of the area of two congruent triangular bases and three rectangles.

The area of a triangle can be calculated by halving the product of the length of its base and height.  

The base of the triangles is 17 units and the height is 8 units.

Therefore, the area of each triangular base is:

[tex]\begin{aligned}\implies \sf Area\;of\;triangular\;base\;(A_B)&=\sf\dfrac{1}{2} \cdot17 \cdot8\\&=\sf 68\;square\;units\end{aligned}[/tex]

Therefore, the values to substitute into the total surface area formula are:

[tex]\begin{aligned}\implies \textsf{Total Surface Area}&= \sf 2 A_B+(a+b+c)h\\&= \sf 2 (68)+(11+14+17)10\\&=\sf 2 \left(68\right)+(42)10\\&=\sf 136+420\\&=\sf 556 \; square\;units\end{aligned}[/tex]

Therefore, the total surface area of the given triangular prism is 556 square units.

Answer:

5/36 is the simplest answer

Step-by-step explanation:

i did the math and searched it up i hope this helps

The distance travelled in the first 14 seconds would then be 3.57 x 14 = 49.98 metres.the acceleration would be 7.14 - 3.57 = 3.57 metres per second squared.

Distance is the numerical measurement of how far apart two objects are. It is measured in units such as kilometers, meters, miles, feet, inches, and more. Distance can be calculated in one, two, or three dimensions. Distance is a key factor in many real-world applications, such as navigation, transport, and communication. Distance can also be used to describe relationships, such as the connection between two people.

A)To work out the distance travelled on a cycle in the first 14 seconds, we must first determine the velocity of the cycle. This can be done by dividing the distance travelled by the time taken. For example, if the cycle travelled 50 metres in 14 seconds, the velocity would be 50 / 14 = 3.57 metres per second. The distance travelled in the first 14 seconds would then be 3.57 x 14 = 49.98 metres.

B) To work out the acceleration speed in the first 4 seconds, we must first determine the change in velocity over the period of time. We can do this by subtracting the initial velocity from the final velocity. For example, if the cycle started at a velocity of 3.57 metres per second, and increased to 7.14 metres per second after 4 seconds, the acceleration would be 7.14 - 3.57 = 3.57 metres per second squared.

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complete question with diagram is

Question A) Work out the distance travelled on a cycle in the first 14 seconds. Q B) Work out thr acceleration speed in the first 4 seconds.

diagram shows trapezium in graph with velocity at y axiz and time at x axis

the average nicotine content of one kind of cigarette exceeds that of another kind by 0.20 milligram at the 0.05 level of significance.

To test the hypothesis that the average nicotine content of one kind of cigarette exceeds that of another kind by 0.20 milligram, we need to compare the means of two independent samples. We can use a two-sample t-test for this purpose. The null hypothesis is that the difference between the means is equal to 0.20 milligram, and the alternative hypothesis is that it is not equal to 0.20 milligram.

The test statistic is given by:

t = ([tex]x_{1}[/tex] - [tex]x_{2}[/tex] - d) / [tex]\sqrt{s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2} }[/tex]

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and d is the hypothesized difference in means (0.20 milligram in this case).

Substituting the given values, we have:

x1 = 2.61 milligrams

[tex]x_{1}[/tex] = 2.38 milligrams

[tex]s_{1}[/tex] = 0.12 milligram

[tex]s_{2}[/tex] = 0.14 milligram

n1 = 50 cigarettes

n2 = 40 cigarettes

d = 0.20 milligram

t = (2.61 - 2.38 - 0.20) / [tex]\sqrt{0.12^{2}/50+0.14^{2}/40 }[/tex]= 4.57

The degrees of freedom for the test are given by:

[tex]df= (s_{1}^{2} /n_{1} +s_{2}^{2}/n_{2})^{2}/(s_{1}^{2} /s_{1})^{2}/_{1}-1 +(s_{2}^{2}/n_{2}^{2} /n_{2}-1[/tex])

Substituting the given values, we have:

df = [tex]( 0.12^{2}/50+0.14^{2}/40)^{2}/((0.12^{2}/50)^{2}/49+(0.14^{2}/40)^{2}/39= 86.36[/tex](rounded to two decimal places)

Using a t-distribution table or a calculator, we can find the critical values for a two-tailed test at the 0.05 level of significance with 86 degrees of freedom. For a two-tailed test, the critical values are ±1.987.

Since the calculated t-value of 4.57 is greater than the critical value of 1.987, we reject the null hypothesis.

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The given expression is divisible by 6 without calculating

A number can be divided if it divides evenly (leaving no remainder) into another integer. For instance, 2 divides 34 equally, making 34 divisible by 2. 34 is not divided by 3, though, because doing so would leave us with a remainder.

We can prove that [tex]$5^7 + 5^6$[/tex] is divisible by 6 using modular arithmetic.

First, we can rewrite the expression as [tex]$5^6(5+1)$[/tex], which simplifies to [tex]$5^6 \cdot 6$[/tex].

Since 5 is not divisible by 2, we can use the fact that [tex]$a \equiv b \pmod{m}$[/tex] implies [tex]$ac \equiv bc \pmod{m}$[/tex] for any integer c to simplify our expression to [tex]$(-1)^6 \cdot 6$[/tex], or simply [tex]6 (since $5 \equiv -1 \pmod{6}$)[/tex].

Therefore, [tex]$5^7 + 5^6$[/tex] is divisible by 6.

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The cost of having a car towed 14 miles is $103.

What is function ?

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range) with the property that each input is related to exactly one output. Functions are often represented by a formula or rule that describes how to compute the output value from the input value.

The towing cost, C, can be expressed as a function of miles towed, x, using the given information as follows:

C(x) = 75 + 2x

where x is the number of miles towed.

To find the cost of towing a car 14 miles, we can substitute x = 14 into the function:

C(14) = 75 + 2(14)

= 75 + 28

= $103

Therefore, the cost of having a car towed 14 miles is $103.

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