The probability of flipping a coin and getting tails is 50%.
This is because in a fair coin toss, there is an equal chance of getting heads or tails. This means that the probability of each outcome is 50%.
To prove this experimentally, you flipped the coin 16 times and got heads 7 times. This means that out of the 16 times, you got heads 7 times and tails 9 times. Therefore, the probability of getting tails is 9/16, which is the same as 50%.
In general, the probability of flipping a coin and getting tails is always 50%, no matter how many times the coin is flipped or what the result of previous flips were. This is because each flip of the coin is an independent event and has the same probability of being heads or tails.
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Write an explicit function to model the value of the nth term in the sequence f(1)=4
The explicit functions are:
Arithmetic sequence with common difference d=3: f(n) = 3n +1
Geometric sequence with common ratio r=2: f(n) = 2^n + 2
Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2
There are infinitely many possible sequences that satisfy the condition f(1)=4, so the function to model the value of the nth term in the sequence will depend on the specific pattern or rule that the sequence follows.
Here are some examples of possible sequences and the corresponding explicit functions:
Arithmetic sequence with common difference d=3: f(n) = 4 + 3(n-1) = 3n + 1
Geometric sequence with common ratio r=2: f(n) = 4 x 2^(n-1) = 2^n+2
Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2
Fibonacci sequence with initial values f(1)=f(2)=1: f(n) = ((1+sqrt(5))/2)^n/sqrt(5) - ((1-sqrt(5))/2)^n/sqrt(5)
Note that these are just examples, and there are many other possible functions that could model a sequence with f(1)=4
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There are 5 students in Mrs. Templeton's art class. Each student has 2 paintbrushes on his desk. Which of the following shows how to find the total number of paintbrushes in the classroom?
1. 5 divided by 2
2. 5 times 5
3. 2 times 2
4. 5 times 2
Answer:
D
Step-by-step explanation:
To find the total number of paintbrushes in the classroom, we need to multiply the number of paintbrushes on each student's desk by the number of students.
Since there are 5 students and each student has 2 paintbrushes, we can use the multiplication operation to find the total number of paintbrushes:
5 students × 2 paintbrushes per student = 10 paintbrushes
Therefore, the correct option is 4. 5 times 2.
Maximize p = 6x + 9y + 3. 3z + 12w subject to
(a) 1. 2x + y + z + w ≤ 121. 5
(b) 2. 2x + y − z − w ≥ 30
(c) 1. 2x + y + z + 1. 2w ≥ 31. 5
(d) x ≥ 0, y ≥ 0, z ≥ 0, w ≥ 0.
Round all answers to two decimal places
The maximum value of p is 107.09 and it is obtained when x = 1.44, y = 31.02, z = 0, and w = 0.
To maximize the objective function p = 6x + 9y + 3.3z + 12w
We can use the simplex method.
First, we need to convert the inequalities into equalities by introducing slack variables:
(a) 1.2x + y + z + w + s1 = 121.5
(b) 2x + y − z − w + s2 = 30
(c) 2x + y + z + 1.2w + s3 = 31.5
We can then write the augmented matrix for the problem:
x y z w s1 s2 s3 b
1.2 1 1 1 1 0 0 121.5
2 1 -1 -1 0 1 0 30
2 1 1 1 0 0 1 31.5
-6 -9 -3.3 -12 0 0 0 0
We choose the most negative coefficient in the bottom row, which is -12. We then select the pivot element in the column corresponding to this coefficient, which is 121.5 in the first row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:
x y z w s1 s2 s3 b
1 0.83 0.17 0.33 0.67 -0.50 -0.17 100.67
0 0.17 -1.33 -1.33 -0.83 0.50 -0.17 12.33
0 0.17 0.33 0.33 -0.67 -0.50 0.83 10.17
0 -6.50 -12.90 -12.00 4.00 4.50 1.00 408.00
Next, we choose the most negative coefficient in the bottom row, which is -12.9. We select the pivot element in the column corresponding to this coefficient, which is 0.33 in the third row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:
x y z w s1 s2 s3 b
1 0.00 0.85 0.40 1.44 -0.54 -0.08 107.09
0 0.00 -1.22 -1.67 -1.00 0.38 0.08 -12.93
0 1.00 2.45 2.33 -2.00 1.50 0.33 31.02
0 0.00 -2.19 -5.60 10.00 10.50 1.83 630.00
The objective function value at this point is p = 107.09.
The solution is x = 1.44, y = 31.02, z = 0, w = 0, and the maximum value of p is 107.09.
Therefore, the maximum value of p is 107.09, when x = 1.44, y = 31.02, z = 0, and w = 0.
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Write a formula for the number of seconds, in any number
of minute
Number of seconds = Number of minutes x 60
What is a minute?A minute is a unit of time measurement that is equal to 60 seconds. It is commonly used to measure short periods of time, such as the duration of a conversation or the time it takes to complete a task.
What is time measurement?Time measurement is the process of quantifying the duration of an event or the interval between two events. It involves measuring the elapsed time between a start point and an end point, or between two points in time.
To convert any number of minutes into seconds, we can use the formula:
Number of seconds = Number of minutes x 60
This formula works because there are 60 seconds in one minute. So, to convert minutes to seconds, we simply multiply the number of minutes by 60 to get the equivalent number of seconds.
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Billy ran 52 yards. How many feet did he run?
Answer:
The answer to your problem is, 156
First lets find out how many feet or in one yard ?
Well 1 yard = 3 feet.
Now that we know that we can use our knowledge to answer the next question:
If Bill ran 52 yards, how many feet did he ran?
We can do math to solve our answer. :
1 x 52 = 52.
Now we need to multiply 52 x 3.
That is a little bit to much, so lets make the problem easier!
By breaking up the problem \/ to make it more easier.
50 x 3. ( 5 x 3 = 15, just add a zero! ) = 150
3 x 2 = 6.
Next add:
150 + 6 = 156.
Thus the answer to your problem is, 156
For sample proportion 0.450 and sample size 31, what is the
standard error of the distribution of sample proportions? Enter
your answer with 3 decimal places.
The standard error of the distribution of sample proportions is approximately 0.090.
The formula to find the standard error of the distribution of sample proportions is given by:Standard error = √(p(1-p)/n)Where,p = Sample proportionn = Sample sizeGiven that the sample proportion is 0.450 and the sample size is 31, we can substitute the values into the formula to find the standard error.Standard error = √(0.450(1-0.450)/31) ≈ 0.090So, the standard error of the distribution of sample proportions is approximately 0.090.
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DW
Problem 2: Given are a segment AB and a ray CD. Use compass, straightedge, and pencil,
to construct a point X on CD such that
CX = 2 1/2 AB
m is parallel to AB, and passing through points C.
Therefore, m II AB.
In geometry, a line segment is a part of a line bounded by two distinct endpoints and contains every point of the line between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed segment includes two endpoints, an open segment does not include both endpoints; a half-open segment has exactly one end. In geometry, a line segment is usually represented using a line above the two end symbols (like AB).
We use the basic rules of construction to draw lines according to the problem.
Draw a straight line AB and take a point C outside of it. Using a ruler and compass to draw the AB line, follow the steps below:
Draw a line AB, and take a point C outside this line. Take any point P on AB.Connect C to P.Take P as the center of the circle, take a suitable radius, draw an arc and cut AB in D, and PC in E.With C as the center and the same radius as in the previous step, draw an arc FG, and PC to H.Adjust the compass to the length of DE. Without changing the compass aperture, draw an arc HG at point I with H as the center.Draw a line l connecting points C and I as shown in the figure.Therefore, the line l is parallel to the line AB.
Complete Question:
Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only.
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What can baby lizards do that baby snakes can’t
Baby lizards can run, climb and in some cases even change color to match their surroundings, while baby snakes are generally limited to crawling and slithering.
Baby lizards have developed legs, claws and a tail to help them navigate their environment, while baby snakes have lost their legs during evolution and have developed a long, slender body to help them move around.
Additionally, some baby lizards are born with a protective membrane around their eggs that allows them to move around freely, while baby snakes are usually born inside an egg that they must break out of before hatching. Overall, baby lizards have more mobility and flexibility than baby snakes, which gives them an advantage in many environments.
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25 POINTS
Where do the graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°)?
300°
150° and 210°
60°
30° and 330°
The graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 3 minus cosine of the quantity x over 2 intersect on the interval [0, 360°) at: D. 30° and 330°.
What is a point of intersection?In Mathematics and Geometry, a point of intersection simply refers to the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.
In this context, we can logically deduce that the point of intersection of the graphs would be at function f(x) = function g(x):
[tex]cos(\frac{x}{2} )=\sqrt{3} -cos(\frac{x}{2} )\\\\\sqrt{3}=cos(\frac{x}{2} ) + cos(\frac{x}{2} )\\\\\sqrt{3}=2cos(\frac{x}{2} )\\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2} \\\\cos(\frac{x}{2} )=\frac{\sqrt{3}}{2}\times \frac{\sqrt{3}}{\sqrt{3}} \\\\cos(\frac{x}{2} )= \frac{3}{2\sqrt{3}}[/tex]
x/2 = 15
x = 15(2)
x = 30
x/2 = 165
x = 2(165)
x = 330.
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Trey is driving to Philadelphia. Suppose that the distance to his destination (in miles) is a linear function of his total driving time (in minutes). Trey has 69 miles
to his destination after 15 minutes of driving, and he has 47.4 miles to his destination after 39 minutes of driving. How many miles will he have to his
destination after 51 minutes of driving?
Chapter 5 Lesson 1 Adding and Subtracting Polynomials
1. Quadratic monomials.
2. Biquadratic five-term polynomials.
3. Quadratic trinomials.
4. x³ + 3x² - 5x - 4
5. -[tex]x^{5}[/tex] + 4[tex]x^{4}[/tex] +2x³ + 2x - 7
6. - x² + 5x + 9
7. y² - 3y - 9
8. 5(x³ + x)
9. 2x² + 2x -5
What are polynomials?Algebraic expressions called polynomials only have non-negative integer powers for their variables. A polynomial is, for instance, 5x² - x + 1. The polynomial 3x³ + 4x + 5/x + 6[tex]x^{3/2}[/tex] is not a polynomial since one of the powers of "x" is a fraction and the other is negative.
Expressions with one or more terms that have a non-zero coefficient are called polynomials. Variables, exponents, and constants make up polynomial terms. The "leading term" refers to the first term of the polynomial in standard form.
Here in the given question,
We can see the highest degree of the variable and we can determine the name of each polynomial.
Likewise, we can just arrange the expressions as per the highest value of the power of the variable.
And simplify the expression by adding or subtracting the like terms.
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The effectiveness of a new antibacterial cream called Formulation NS is being tested. From previous research, it is known that without medication, the mean healing time (defined as the time for the scab to fall off) is 7. 6 days. A random sample of 250 college students apply Formulation NS to their wounds. The mean healing time for these students is 7. 5 days with a standard deviation of 0. 9 days. Test at the 5% significance level if applying Formulation NS speeds (takes less time) healing than foregoing treatment. Use the critical value approach
The t-statistic is less than the critical value, we fail to reject the null hypothesis
The hypothesis testing to be conducted here is a two-tailed test for the population mean, using the formula:
H0: μ = 7.6 days
Ha: μ ≠ 7.6 days
The critical value is ± 1.960, with a confidence level of 95%.
The sample mean is 7.5 days and the sample standard deviation is 0.9 days. Using the formula, we can calculate the t-statistic:
t = (7.5 - 7.6) / (0.9 / √250) = -1.105
Since the t-statistic is less than the critical value, we fail to reject the null hypothesis, suggesting that Formulation NS does not significantly speed up the healing process compared to the foregoing treatment.
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how to find slope of x + y = 9
Answer:
m = -1
Step-by-step explanation:
Let's change the equation to y = mx + b
m = the slope
x + y = 9
Subtract x both sides
y = -x + 9
m = -1
So, the slope is -1
During the summer, every student became 5% taller. Eric was x before summer, after summer he was 151.2 cm.
Let's assume Eric's height before summer is x cm. After summer, he became 5% taller, which means his new height is 1.05x cm. We also know that his new height is 151.2 cm. So we can set up an equation:
1.05x = 151.2
To solve for x, we can divide both sides by 1.05:
x = 151.2 / 1.05
x = 144 cm
Therefore, Eric's height before summer was 144 cm.
Please help mee
For the following question, find the value of the variable(s). If your answer is not an integer, leave it in simples radical form
hope it helps you
option d
HELP ME PLEASE!!!! THIS IS DUE TODAY!!!
Answer:
Yes, the triangles are similar.
Step-by-step explanation:
The triangles are similar because of the three angle measurements.
1) One of the measurements are labeled, and the other two are implied.
2) The measurement of angles HMG and JMK are equal because they are vertical angles.
3) Lastly, because two of the angles are the same, we know that the last angles, GJK and HGJ are the same because the three angles must have a sum of 180°.
The ratio of the cost of one metre of polyester fabric to the cost of one metre
of cotton fabric is 2: 7
Complete the table
The cost of one metre of cotton fabric is £ 7. The cost of one metre polyester fabric is £ 2.
The ratio of the cost of one metre of polyester fabric to the cost of one metre of cotton fabric is 2: 7.
1 m polyester / 1 m cotton = 2 / 7
1 m cotton = 7/2 × 1 m polyester
for 2 m of polyester fabric, the cost is £ 4
for 1 m of polyester fabric, the cost is £ 4/2 = £ 2
for 1 m of cotton fabric = 7/2 × 1 m polyester fabric = 7/2 × £2 = £ 7
for 1 m of cotton fabric, the cost is £ 7
For 2m, polyester fabric 2 × £2 = £ 4
cotton fabric 2 × £7 = £ 14
For 6m, polyester fabric 6 × £2 = £ 12
cotton fabric 6 × £7 = £ 42
For 8m, polyester fabric 8 × £2 = £ 16
cotton fabric 8 × £7 = £ 56
For 9m, polyester fabric 9 × £2 = £ 18
cotton fabric 9 × £7 = £ 63
The complete table
2m 6m 8m 9m
polyester fabric £ 4 £ 12 £ 16 £ 18
cotton fabric £ 14 £ 42 £ 56 £ 63
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Peace's average mark on her 5 maths tests was 88. If her lowest score was dropped, her new average would be 90. What is her lowest mark?
Her average score on the four remaining tests is indeed 90, which confirms that her lowest score was 80.
Let's assume Peace's lowest score on the five math tests is x.
According to the problem statement, her average mark on all five tests is 88. This means that the sum of her scores on all five tests is:
5 * 88 = 440
If her lowest score was dropped, then the sum of her scores on the remaining four tests would be:
4 * 90 = 360
We know that the sum of her scores on all five tests is 440, so we can write an equation:
440 - x = 360
Solving for x, we get:
x = 80
Therefore, her lowest mark was 80. We can check this by finding her average score after dropping her lowest score:
(88 + 88 + 88 + 88 + 90) / 5 = 90
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1. What is the finance charge on June 11 (monthly periodic rate: 1.3)
2. What is the new card balance on June 12th
1. The finance charge on June 11 is $1.98.
2. The new card balance on June 12th is $341.30
What is the finance charge on June 11?To calculate the finance charge and new card balance, we first need to calculate the average daily balance for the billing cycle.
May 13 Charge toys: $129.79 (balance: $129.79)
May 15 Payment $50 (balance: $79.79)
June 1 Charge clothing $135.95 (balance: $215.74)
June 8 Charge Housewares $37.63 (balance: $253.37)
Billing cycle: May 13 to June 11 (30 days)
Average daily balance:
= (129.79 x 18) + (79.79 x 16) + (215.74 x 10) + (253.37 x 6) / 30
= $152.49
The Finance charge is computed as:
= Average daily balance * Monthly periodic rate.
= 152.49 x (1.3/100)
= $1.98.
What is the new card balance on June 12th?To calculate the new card balance on June 12th, we need to add the finance charge and any new charges to the previous balance and subtract any payments made.
Previous balance (as of June 1st) = $215.74
New charges (since June 1st) = $135.95 + $37.63 = $173.58
Payments made (since May 15th) = $50
The New card balance on June 12th will be:
= $215.74 + $173.58 + $1.98 - $50
= $341.30
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Vector v is defined by the components 3, 5;. Vector w is defined by the components negative 1, 4;. Determine the angle θ, in degrees, formed between vector v and vector w, where 0° < θ ≤ 180°.
Answer:250
Step-by-step explanation:
please help me in this
Answer: At a greengrocer, two bananas and one apple cost $1.16 .
Than the equation becomes
2x + y = 1.16
one banana and one apple cost 0.71.
Than the equation becomes
x + y = 0.71
Subtracting x + y = 0.71 from 2x + y = 1.16
2x - x + y - y = 1.16 - 0.71
x = 0.45
Put in the equation x + y = 0.71
0.45 + y = 0.71
y = 0.71 - 0.45
y = 0.26
The cost of the one apple is 0.26
Step-by-step explanation:
MRS. JAMISON'S CLASS
MR. ZIMMERMAN'S CLASS
.
.
.
o +
:
2 3
5 6
NUMBER OF TVS PER HOUSEHOLD
0 1 2 3
5 6
NUMBER OF TVs PER HOUSEHOLD
7
The mode number of TVs per household for both Mrs. Jamison's class and and Mr. Zimmerman's class is 3.
O True
False
The given statement "The mode number of TVs per household for both Mrs. Jamison's class and and Mr. Zimmerman's class is 3." is true. The mode of TVs per household is 3 in both Mrs. Jamison's class and and Mr. Zimmerman's class.
Mode is a statistical measure that represents the value that appears most frequently in a dataset. It is one of the measures of central tendency, along with mean and median.
In Mrs. Jamison's class, the mode number of TVs per household is 3 because it appears twice, while all other numbers appear only once. Similarly, in Mr. Zimmerman's class, the mode number of TVs per household is 3 because it appears twice, while all other numbers appear only once.
Therefore, the statement "The mode number of TVs per household for both Mrs. Jamison's class and Mr. Zimmerman's class is 3" is true.
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Khaled calculates the mean of five different prime numbers. His answer is an integer, what is the smallest possible integer he could have obtained.
Answer:
6
Step-by-step explanation:
[tex] \frac{2 + 3 + 5 + 7 + 13}{5} = \frac{30}{5} = 6[/tex]
About 74% of all female heart transplant patients will survive for at least 3 years. Seventy female heart transplant patients are randomly selected. What is the probability that the sample proportion surviving for at least 3 years will be less than 70% ? Assume the sampling distribution of sample proportions is a normal distribution. The mean of the sample proportion is equal to the population proportion and the standard deviation is equal to {n/pq}.
The probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.
What is a probability?Probability is a branch of statistics that deals with the study of random events and their likelihood of occurrence.
First, calculate the standard deviation of the sampling distribution of sample proportions using the formula.
σ = [tex]\sqrt{[(p*q)/n]}[/tex], where p is the population proportion, q = (1 - p), and n is the sample size.
In this case, p = 0.74, q = 0.26, and n = 70
Therefore, σ = [tex]\sqrt{[(0.74*0.26)/70]}[/tex] = 0.066
Next, we need to standardize the sample proportion using the formula,
z = (X - p) / σ, where X is the sample mean, p is the population proportion, and σ is the standard deviation of the sampling distribution.
In this case, X = 0.70, p = 0.74, and σ = 0.066
Thus, z = (0.70 - 0.74) / 0.066 = -0.606
Using a standard normal distribution table, we find that the cumulative probability for a z-score of -0.606 is 0.2734.
Therefore, the probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.
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Which equation would help find the measure of ∠m? x 52 78 = 180 52 x = 180 78 180 52 = 78 x 180 x = 78 52
Answer:
A
Step-by-step explanation:
solve the equation negative 2y plus 6 equals negative 12
Answer:
y = -9
Step-by-step explanation:
Answer:
3
Step-by-step explanation:
since 2y is adding to positive 6 we take 6 to the right side where there its like term 12. we subtract 6 from 12 which is giving us 6. we are remaining with 2y=6 we divide both sides by 2 giving us y=3
Explain the importance of the unit circle in trigonometry.
Answer: The unit circle is an essential tool in trigonometry because it helps in understanding and visualizing the relationships between angles and the values of the sine, cosine, and tangent functions.
The unit circle is a circle with a radius of one unit and centered at the origin of a coordinate plane. It is divided into 360 degrees or 2π radians. By placing this circle on the coordinate plane, we can easily determine the sine and cosine values of angles in standard position.
For any given angle θ, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and the cosine value is the x-coordinate of that same point. The tangent function, which is the ratio of sine to cosine, can also be determined using the unit circle.
The unit circle also helps in understanding the periodicity of the sine and cosine functions. Since the circumference of the unit circle is 2π, the sine and cosine functions repeat themselves after every 2π radians or 360 degrees. This periodicity allows for the use of trigonometric identities and formulas to simplify and solve complex trigonometric equations.
In summary, the unit circle is an essential tool in trigonometry as it provides a visual representation of angles and their corresponding sine, cosine, and tangent values, and allows for the use of trigonometric identities and formulas to solve complex problems.
Find the missing side length and angles of △ABC given that
m∠B=42∘, a=10, and c=23.
Round each answer to the nearest tenth if necessary.
23.m∠C = 48°m∠A = 90°
Given that the triangle ABC, m∠B = 42∘, a = 10, and c = 23. Now we have to find the missing side length and angles of △ABC. Step-by-step explanation: The formula to find the missing side length of a triangle is given by the Pythagoras theorem. Hence, using the Pythagoras theorem, we get;b2 = c2 - a2 = 23² - 10²b² = 529b = √529b = 23Length of b is 23.We know that the sum of all three angles of a triangle is equal to 180°. Therefore, using this concept, we can find the remaining angles of the triangle as follows;m∠A + m∠B + m∠C = 180°m∠A + 42° + m∠C = 180°m∠A + m∠C = 138°Also, we know that the sum of the two opposite angles of a triangle is equal to the third angle. Therefore, we can use this concept to find one of the remaining angles.m∠A = 180° - m∠C - m∠Bm∠A = 180° - m∠C - 42°m∠A = 138° - m∠C/ Thus,m∠A + m∠C = 138°And,m∠A = 138° - m∠COn solving these two equations, we get,m∠C = 48°m∠A = 90°Using the Pythagoras theorem we find the missing side length of a triangle as given below;b² = c² - a² = 23² - 10²b² = 529b = √529b = 23The length of the missing side is 23.Therefore, the missing side length and angles of △ABC are:Length of b is 23.m∠C = 48°m∠A = 90°
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PLEASE SOMEBODY HELP: Generate a symbolic rule for locating the point that divides a line segment into two parts so that the ratio of the lengths is m: n, with the point closer to the left endpoint.
Step-by-step explanation:
Let the coordinates of the left endpoint of the line segment be (x1, y1) and the coordinates of the right endpoint be (x2, y2). Let the point dividing the line segment be (x, y). Then the distance between the left endpoint and (x, y) is mx/(m+n) and the distance between (x, y) and the right endpoint is nx/(m+n).
Using the distance formula, we have:
[tex]distance \: between \: (x1, y1) \: and \: (x, y) = \sqrt{((x - x1)^2 + (y - y1)^2)} = \frac{mx}{(m+n)} [/tex]
[tex]distance \: between \: (x, y) \: and \: (x2, y2) = \sqrt{((x2 - x)^2 + (y2 - y)^2)} = \frac{nx}{(m+n)} [/tex]
Squaring both equations, we get:
[tex](x - x1)^2 + (y - y1)^2 = \frac{mx}{(m+n)^2}[/tex]
[tex](x2 - x)^2 + (y2 - y)^2 = \frac{nx}{(m+n)^2}[/tex]
Expanding the squares, we get:
[tex]x^2 - 2x1x + x1^2 + y^2 - 2y1y + y1^2 = \frac{mx}{(m+n)^2} [/tex]
[tex]x^2 - 2x2x + x2^2 + y^2 - 2y2y + y2^2 = \frac{nx}{(m+n)^2}[/tex]
Rearranging and simplifying, we get:
[tex]x = \frac{(mx2 + nx1)}{(m+n)} \: and \: y = \frac{(my2 + ny1)}{(m+n)}[/tex]
Therefore, the point that divides the line segment into two parts so that the ratio of the lengths is m: n and the point is closer to the left endpoint is:
[tex](x, y) = \frac{(mx2 + nx1)}{(m+n)}, \frac{(my2 + ny1)}{(m+n)}[/tex]
The area of cross-section of a solid cylinder is 803. 84 ft2, and the height of the solid is 12. 25 ft. Find the volume of the solid
cylinder using the Cavalieri's Principle.
9800. 25 ft
9587. 15 f3
9847. 04 ft3
9807. 50 ft
V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3. Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
According to Cavalieri's principle, if two solids have the same height and every plane section passing through one solid along the height has the same area as the corresponding plane section passing through the other solid, the two solids must also have the same volume.
A circle of radius r has the same cross-section as a solid cylinder. A = r2 calculates the circle's area.
Therefore, πr^2 = 803.84 r^2 = 803.84/π\sr = √(803.84/π)
The solid cylinder is 12.25 feet tall.
We may get the volume of the solid cylinder using the volume of a cylinder formula, V = r2h: V = π(√(803.84/π))^2(12.25) = 9587.15 ft^3
Hence, following Cavalieri's theory, the solid cylinder has a volume of around 9587.15 ft3.
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