The number of people in the auditorium is 3 hours after the doors open is the same as the number of people in the auditorium 5 hours after the doors open.
Answers
A function notation for two hours after the open, there are 108 people in the auditorium is N(2) = 108.
A function notation for the number of people in the auditorium 3 hours after the doors open is the same as the number of people in the auditorium 5 hours after the doors open is N(3) = N(5).
What is a function?In Mathematics, a function refers to a mathematical expression which can be used for defining and showing the relationship that exist between two or more variables in a data set.
This ultimately implies that, a function typically shows the relationship between input values (x-values or domain) and output values (y-values or range) of a data set, as well as showing how the elements in a table are uniquely paired (mapped).
Based on the information provided, the number of people in the auditorium can be represented by this function notation;
N(t)
Where:
t represents number of hours.
After 2 hours, we have:
N(2) = 108.
For the last statement, we have:
N(3) = N(5).
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Related Questions
Find the value of x
A. 135
B. 40
C. 50
D. 45
Answers
Hope this helps :)
You just need to complete the right bottom angle, you get 45. Every triangle has 180º in total,you already have 95 and 45, the get 180 you need 40.
Could use some help on this proof. I can't figure it out.Use induction on the size of S to show that if S is a finite set, then |2s| = 2|S|*Note: Here, |S| means the cardinality of S.
Answers
We have shown that |2S| = 2|S| for all finite sets S, by induction on the size of S.
To prove that if S is a finite set, then |2S| = 2|S| using mathematical induction, we need to show that the statement is true for a base case and then show that it holds for all possible cases.Base Case:When S has only one element, say {x}, then the power set of S, 2S, contains two elements: {} and {x}. Hence, |2S| = 2 = 2|S|.Inductive Hypothesis:Assume that for any finite set S of size k, the statement |2S| = 2|S| holds true.Inductive Step:Consider a set S' of size k+1. Let x be any element in S', and let S = S' \ {x} be the set obtained by removing x from S'. By the inductive hypothesis, we know that |2S| = 2|S|.Now consider 2S' = {A : A ⊆ S'} to be the power set of S'. For any set A in 2S', there are two possibilities:A does not contain x, in which case A is a subset of S and there are 2|S| possible choices for A.A contains x, in which case we can write A as A = {x} ∪ B for some subset B of S. There are 2|S| possible choices for B, so there are 2|S| possible choices for A.Therefore, |2S'| = 2|S| + 2|S| = 2(2|S|) = 2|S'|, which completes the induction step.Thus, we have shown that |2S| = 2|S| for all finite sets S, by induction on the size of S.For more such question on induction
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Use polar coordinates to calculate the area of the region. R = {(x, y) | x^2 + y^2 ≤ 100, x ≥ 6}
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The area of the region with polar coordinates R = {(x, y) | x² + y² ≤ 100, x ≥ 6} is approximately 197.39 square units.
To calculate the area, first, rewrite the given inequalities in polar coordinates: r² ≤ 100 and rcos(θ) ≥ 6. Next, find the bounds for r and θ. Since r² ≤ 100, r must be between 0 and 10.
For the second inequality, rcos(θ) ≥ 6, divide by r (assuming r ≠ 0) to get cos(θ) ≥ 6/r. To satisfy this inequality, θ must be between arccos(6/r) and π for r in [6, 10].
Now, integrate the area using polar coordinates with the following formula: A = 0.5 * ∫(from 6 to 10) ∫(from arccos(6/r) to π) (r^2) dθ dr. After evaluating the integral, you get the area A ≈ 197.39 square units.
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find a 95 confidence interval for the mean surgery time for this procedure. round the answers to two decimal places. The 95% confidence interval is _____ , ______
Answers
To find a 95% confidence interval for the mean surgery time for this procedure, you need to use the following formula:
CI = X-bar ± (t * (s / √n))
where:
- CI is the confidence interval
- X-bar is the sample mean
- t is the t-score, which corresponds to the desired confidence level (95%) and the degrees of freedom (n-1)
- s is the sample standard deviation
- n is the sample size
1. Calculate the sample mean (X-bar), sample standard deviation (s), and sample size (n).
2. Find the appropriate t-score using a t-table or calculator for 95% confidence level and (n-1) degrees of freedom.
3. Plug the values into the formula and calculate the interval.
4. Round the answers to two decimal places.
The 95% confidence interval for the mean surgery time is (lower limit, upper limit). Make sure to provide the specific values for your data in the formula.
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complete question:
find a 95 confidence interval for the mean surgery time for this procedure. what quantities do you need to calculate the 95 confidence interval ?
Arcs and Angle Relationships in circles , help fast pls
Answers
The value of x in the quadrilateral is 4.
We are given that;
Four angles of quadrilateral 5x, 102, 4y-12, 3x+8
Now,
The sum of all interior angles of quadrilateral is 360degree
So, 5x + 102 + 4y-12 + 3x+8 = 360
Also opposite angles are equal
5x= 3x+8
2x=8
x=4
Therefore, by the properties of quadrilateral the answer will be 4.
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suppose f(x) is continuous on [2,7] and −4≤f′(x)≤2 for all x in (2,7). use the mean value theorem to estimate f(7)−f(2).
Answers
We find that -20 ≤ f(7) - f(2) ≤ 10. So, using the Mean Value Theorem, we can estimate the difference between f(7) and f(2) to be in the range of [-20, 10].
We are given that f(x) is continuous on [2,7] and its derivative, f'(x), is between -4 and 2 for all x in (2,7). We are asked to use the Mean Value Theorem (MVT) to estimate f(7) - f(2).
First, let's recall the Mean Value Theorem. If a function is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
Now, let's apply the MVT to our problem. We have:
f'(c) = (f(7) - f(2)) / (7 - 2)
We know that -4 ≤ f'(x) ≤ 2 for all x in (2,7). Therefore, -4 ≤ f'(c) ≤ 2 for some c in (2,7). Using this inequality, we get two separate inequalities:
-4 ≤ (f(7) - f(2)) / 5
2 ≥ (f(7) - f(2)) / 5
Now, we can multiply both sides of each inequality by 5:
-20 ≤ f(7) - f(2)
10 ≥ f(7) - f(2)
Thus, we find that -20 ≤ f(7) - f(2) ≤ 10. So, using the Mean Value Theorem, we can estimate the difference between f(7) and f(2) to be in the range of [-20, 10].
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s there a vector field G on 3 such that curlG =xyz, −y^6z^5, y^5z^6?YesNoExplain.There ---Select--- is is no such G because div(curl G) ? = ≠ 0.
Answers
Expression is not equal to zero, we cannot find a vector field G such that curl G = (xyz, -y⁶z⁵, y⁵z⁶). Therefore, the answer is no.
Describe more about why this answer is no?There is no such G because the divergence of curl G is not equal to zero. The divergence of curl G is given by the scalar triple product identity:
div(curl G) = dot(grad, curl G)
Using this identity and the given components of curl G, we have:
div(curl G) = x(∂/∂x)(-y⁶z⁵) + y(∂/∂y)(y⁵z⁶) + z(∂/∂z)(xyz)
div(curl G) = -6xy⁵z⁵ + 5y⁶z⁵ + xz
Since this expression is not equal to zero, we cannot find a vector field G such that curl G = (xyz, -y⁶z⁵, y⁵z⁶). Therefore, the answer is no.
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Let (X,Y) be uniformly distributed on the triangleD with vertices (1,0), (2,0) and (0,1), as in Example 10.19. (a) Find the conditional probability P(X ≤ 1 2|Y =y). You might first deduce the answer from Figure 10.2 and then check your intuition with calculation. (b) Verify the averaging identity for P(X ≤ 1 2). That is, check that P(X ≤ 1 2)=:[infinity] −[infinity] P(X ≤ 1 2|Y =y)fY(y)dy.
Answers
Louise is buying wallpaper. It costs $7.98 per meter. She needs 150 feet. How much
will the wallpaper cost? Round to the nearest half dollar.
$365.00
$364.00
$364.94
$364.40
Answers
Answer:
$365.00.
Step-by-step explanation:
Suppose n is a vector normal to the tangent plane of the surface F(x,y,z) = 0 at a point. How is n related to thegradient of F at that point?Choose the correct answer below..A. The gradient of F is a multiple of nB. The gradient of F is equal to n.C. The gradient of F is orthogonal to nD. The gradient of F is not related to n
Answers
If n is a vector normal to the tangent plane of the surface F(x,y,z) = 0 at a point, then option (C) the gradient of F is orthogonal to n.
The gradient of F at a point (x₀, y₀, z₀) is defined as the vector (∂F/∂x, ∂F/∂y, ∂F/∂z) evaluated at that point. This gradient vector is perpendicular (or orthogonal) to the level surface of F passing through that point.
The tangent plane to the surface F(x, y, z) = 0 at a point (x₀, y₀, z₀) is defined as the plane that touches the surface at that point and is perpendicular to the normal vector at that point.
Thus, if n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point (x₀, y₀, z₀), then n is perpendicular to the tangent plane. Since the gradient vector of F at the point (x₀, y₀, z₀) is perpendicular to the tangent plane, it is also perpendicular to n.
Therefore, the correct answer is (C) The gradient of F is orthogonal to n.
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Given the four rational numbers below, come up with the greatest sum, difference, product, and quotient, using two of the numbers for each operation. Numbers may be used more than once. Show your work.
Answers
Answer:
[tex]\textsf{Greatest sum}=25.1=25\frac{1}{10}[/tex]
[tex]\textsf{Greatest difference}=30.9=30\frac{9}{10}[/tex]
[tex]\textsf{Greatest product}=122.1=122\frac{1}{10}[/tex]
[tex]\textsf{Greatest quotient}=2.8\overline{03}=2\frac{53}{66}[/tex]
Step-by-step explanation:
Method 1First, rewrite each number as an improper fraction with the common denominator of 10.
[tex]6.6=\dfrac{66}{10}[/tex]
[tex]-4\frac{3}{5}=-\dfrac{4 \cdot 5+3}{5}=-\dfrac{23}{5}=-\dfrac{23 \cdot 2}{5 \cdot 2}=-\dfrac{46}{10}[/tex]
[tex]18\frac{1}{2}=\dfrac{18 \cdot 2+1}{2}=\dfrac{37}{2}=\dfrac{37\cdot 5}{2\cdot 5}=\dfrac{185}{10}[/tex]
[tex]-12.4=-\dfrac{124}{10}[/tex]
Now order the improper fractions from smallest to largest:
[tex]-\dfrac{124}{10},\;\;-\dfrac{46}{10},\;\;\dfrac{66}{10},\;\;\dfrac{185}{10}[/tex]
The greatest sum can be found by adding the largest two numbers:
[tex]\implies \dfrac{66}{10}+\dfrac{185}{10}=\dfrac{66+185}{10}=\dfrac{251}{10}=25.1=25\frac{1}{10}[/tex]
The greatest difference can be found by subtracting the smaller number from the largest number:
[tex]\implies \dfrac{185}{10}-\left(-\dfrac{124}{10}\right)=\dfrac{185+124}{10}=\dfrac{309}{10}=30.9=30\frac{9}{10}[/tex]
The greatest product can be found by multiplying the largest two numbers:
[tex]\implies \dfrac{66}{10}\cdot \dfrac{185}{10}=\dfrac{66\cdot 185}{10 \cdot 10}=\dfrac{12210}{100}=122.1=122\frac{1}{10}[/tex]
The greatest quotient can be found by dividing the largest number by the smallest number, given the two numbers have the same sign.
[tex]\implies \dfrac{185}{10} \div \dfrac{66}{10}= \dfrac{185}{10} \cdot \dfrac{10}{66}=\dfrac{185}{66}=2\frac{53}{66}[/tex]
[tex]\hrulefill[/tex]
Method 2Rewrite all the numbers as decimals:
[tex]6.6[/tex]
[tex]-4\frac{3}{5}=-4.6[/tex]
[tex]18\frac{1}{2}=18.5[/tex]
[tex]-12.4[/tex]
Now order the decimals from smallest to largest:
[tex]-12.4, \;\; -4.6, \;\;6.6,\;\;18.5[/tex]
The greatest sum can be found by adding the largest two numbers:
[tex]\begin{array}{rr}&6.6\\+&18.5\\\cline{2-2} &25.1\\ \cline{2-2}&^1^1\;\;\;\end{array}[/tex]
The greatest difference can be found by subtracting the smallest number from the largest number:
[tex]18.5-(-12.4)=18.5+12.4[/tex]
[tex]\begin{array}{rr}&18.5\\+&12.4\\\cline{2-2} &30.9\\ \cline{2-2}&^1\;\;\;\;\end{array}[/tex]
The greatest product can be found by multiplying the largest two numbers:
[tex]\begin{array}{rr}&18.5\\\times&6.6\\\cline{2-2} &11.10\\+&111.00\\ \cline{2-2}&122.10\end{array}[/tex]
The greatest quotient can be found by dividing the largest number by the smallest number, given the two numbers have the same sign.
[tex]\implies \dfrac{18.5}{6.6}=\dfrac{185}{66}=2.8\overline{03}[/tex]
What is the difference of the geometric mean and the arithmetic mean of 18 and 128
Answers
Answer:
Step-by-step explanation:
The arithmetic mean of 18 and 128 is (18+128)/2 = 73.
The geometric mean of 18 and 128 is the square root of their product: √(18*128) = √(2304) = 48.
So, the difference between the geometric mean and the arithmetic mean is:
48 - 73 = -25.
Therefore, the difference of the geometric mean and the arithmetic mean of 18 and 128 is -25.
If -ve root of G is taken then
G = -48
and diff of G and A will be -48 - 73 = -121
TELL WHETHER THE TRIANGLE IS A RIGHT TRIANGLE
Answers
Answer:
use Pythagorean theorem
Step-by-step explanation:
:)
a^2+b^2=c^2 if equal triangle is right!
Answer:
7. No. 8. Yes.
Step-by-step explanation:
Use the pythagorean theorem.
[tex]a^{2} + b^2 = c^2[/tex]
Where a and b are legs and c is the hypotenuse.
7.
[tex]3^2 + 7^2 \neq \sqrt{57}^2 \\9 + 49 = 58 \neq 57[/tex]
So it isn't a right triangle.
8.
[tex](5\sqrt{5})^2 = 11^2 + 2^2\\(\sqrt{125})^2 = 121 + 4\\125 = 125[/tex]
So this is a right triangle.
The probability density function f(x) for a uniform random variable X defined over the interval [2, 10] is
a. 4 b. 8 c. 0.20 d. None of these choices.
Answers
The probability density function f(x) for a uniform random variable X defined over the interval [2, 10] is:
d. None of these choices.
Step 1: Identify the interval limits, a and b.
a = 2, b = 10
Step 2: Calculate the width of the interval.
Width = b - a = 10 - 2 = 8
Step 3: Determine the probability density function for a uniform distribution.
f(x) = 1 / (b - a)
Step 4: Substitute the values of a and b in the formula.
f(x) = 1 / (10 - 2)
Step 5: Simplify the expression.
f(x) = 1 / 8 = 0.125
So, the correct answer is none of these choices (d).
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Find a linear differential operator that annihilates the given function. (Use D for the differential operator.)For,1+6x - 2x^3and,e^-x + 2xe^x - x^2e^x
Answers
The linear differential operator that annihilates [tex]e^{-x} + 2xe^x - x^2e^x is (D - 2)(D - 1)(D + 1).[/tex]
How to find linear differential operator?For [tex]1+6x - 2x^3:[/tex]
The first derivative is [tex]6 - 6x^2[/tex], and the second derivative is -12x. Since the second derivative is a constant multiple of the original function, we can use the differential operator D - 2 to annihilate the function:
[tex](D - 2)(1 + 6x - 2x^3) = D(1 + 6x - 2x^3) - 2(1 + 6x - 2x^3)[/tex]
[tex]= (6 - 6x^2) - 2 - 12x + 4x^3 - 2[/tex]
[tex]= 4x^3 - 6x^2 - 12x + 4[/tex]
[tex]For e^{-x} + 2xe^x - x^2e^x:[/tex]
The first derivative is [tex]2e^x - 2xe^x - x^2e^x,[/tex] and the second derivative is [tex]-2e^x + 2xe^x + 2e^x - 2xe^x - 2xe^x - x^2e^x[/tex]. Simplifying, we get:
[tex](D - 2)(D - 1)(D + 1)(e^{-x} + 2xe^x - x^2e^x) = (D - 1)(D + 1)(2e^x - 2xe^x - x^2e^x)[/tex]
[tex]= (D + 1)(2e^x - 4xe^x - 2xe^x + 2x^2e^x - x^2e^x)[/tex]
[tex]= (D + 1)(2e^x - 6xe^x + 2x^2e^x)[/tex]
Therefore, the linear differential operator that annihilates [tex]e^{-x} + 2xe^x - x^2e^x is (D - 2)(D - 1)(D + 1).[/tex]
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the 2012 National health and nutrition examination survey reports a 95% confidence interval of 99.8 to 102.0 centimeters for the mean waist circumference of adult women in the United State. a) what is captured by the confidence interval? b) Express this confidence interval as a sequence written in the context of this problem c) what is the margin of error for this confidence interval? Express this interval in the format "estimate plus or minus margin of error" d) would a 99% confidence interval based on the same data be larger or smaller? Explain
Answers
The interval would be wider, with more plausible values, and the margin of error would be larger.
What is confidence interval ?
A confidence interval is a statistical range of values that is used to estimate an unknown population parameter, such as the mean or standard deviation of a distribution, based on a sample of data.
a) The confidence interval captures the plausible range of values for the population mean waist circumference of adult women in the United States. More specifically, it is the range of values that is likely to contain the true population mean with 95% confidence.
b) The confidence interval can be expressed as: "We are 95% confident that the true population mean waist circumference of adult women in the United States falls between 99.8 and 102.0 centimeters."
c) The margin of error for this confidence interval can be calculated by taking half the width of the interval. Therefore, the margin of error is (102.0 - 99.8) : 2 = 1.1. Thus, we can express the confidence interval as "The estimate is 100.9 centimeters plus or minus 1.1 centimeters."
d) A 99% confidence interval based on the same data would be larger than the 95% confidence interval. This is because a 99% confidence level requires a wider interval to capture the true population mean with 99% confidence.
Therefore, the interval would be wider, with more plausible values, and the margin of error would be larger.
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2. SIGNS A sign is in the shape of an ellipse. The eccentricity is 0.60 and the length is 48 inches.
a. Write an equation for the ellipse if the center of the sign is at the origin and the major axis is horizontal.
b. What is the maximum height of the sign?
Answers
a. The standard equation for an ellipse with center at the origin and major axis horizontal is:
x^2/a^2 + y^2/b^2 = 1
where a is the length of the semi-major axis and b is the length of the semi-minor axis. The eccentricity e is related to a and b by the equation:
e = √(a^2 - b^2)/a
We are given that the eccentricity e is 0.60 and the length of the major axis is 48 inches. Since the major axis is horizontal, a is half of the length of the major axis, so a = 24. We can solve for b using the equation for eccentricity:
0.6 = √(24^2 - b^2)/24
0.6 * 24 = √(24^2 - b^2)
14.4^2 = 24^2 - b^2
b^2 = 24^2 - 14.4^2
b ≈ 16.44
Therefore, the equation of the ellipse is:
x^2/24^2 + y^2/16.44^2 = 1
b. To find the maximum height of the sign, we need to find the length of the semi-minor axis, which is the distance from the center of the ellipse to the top or bottom edge of the sign. We can use the equation for the ellipse to solve for y when x = 0:
0^2/24^2 + y^2/16.44^2 = 1
y^2 = 16.44^2 - 16.44^2 * (0/24)^2
y ≈ 13.26
Therefore, the maximum height of the sign is approximately 26.52 inches (twice the length of the semi-minor axis).
I NEED HELP ON THIS ASAP!!
Answers
The answers are as follows
A: [tex]-2.3^{x-1} = a^{2} (r^{x-1} )\\[/tex]
∴[tex]y = a(-2.3)^{x}[/tex]
The constant ratio is [tex]-2,3[/tex] and the y-intercept is [tex](0,a)[/tex].
B: [tex]45.2^{x-1} = a(r^{x-1} )[/tex]
∴[tex]y = a(45.2)^{x}[/tex]
The constant ratio is [tex]45.2[/tex] and the y-intercept is [tex](0,a)[/tex].
C: [tex]1234-0.1^{x-1}[/tex] [tex]= a r^{x-1}[/tex]
∴ [tex]y = a(0.1)^{x} + 1234[/tex]
The constant ratio is [tex]0.1[/tex] and the y-intercept is[tex](0,a +1234)[/tex].
D: [tex]-5(1/2)^{x-1}[/tex] is not a geometric sequence as there is no common ratio between consecutive terms.
The constant term is [tex]-5[/tex] and the y-intercept is [tex](0,-5)[/tex].
What is exponential function?An exponential function is a mathematical function in the form [tex]f(x) = a^{x}[/tex], where a is a positive constant called the base, and x is the variable. These functions have a constant ratio between consecutive outputs.
An explicit formula for a geometric sequence as an exponential function, we can write the nth term as[tex]a(r^{n-1})[/tex], where a is the first term and r is the common ratio.
This is equivalent to the general form of an exponential function, [tex]y = ab^{x}[/tex], where a is the initial value and b is the base. The constant ratio between consecutive terms is equal to the base of the exponential function.
Therefore,
A: [tex]-2.3^{x-1} = a^{2} (r^{x-1} )\\[/tex]
∴[tex]y = a(-2.3)^{x}[/tex]
The constant ratio is [tex]-2,3[/tex] and the y-intercept is [tex](0,a)[/tex].
B: [tex]45.2^{x-1} = a(r^{x-1} )[/tex]
∴[tex]y = a(45.2)^{x}[/tex]
The constant ratio is [tex]45.2[/tex] and the y-intercept is [tex](0,a)[/tex].
C: [tex]1234-0.1^{x-1}[/tex] [tex]= a r^{x-1}[/tex]
∴ [tex]y = a(0.1)^{x} + 1234[/tex]
The constant ratio is [tex]0.1[/tex] and the y-intercept is[tex](0,a +1234)[/tex].
D: [tex]-5(1/2)^{x-1}[/tex] is not a geometric sequence as there is no common ratio between consecutive terms.
The constant term is [tex]-5[/tex] and the y-intercept is [tex](0,-5)[/tex].
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(b) suppose = 148. what is the probability that x is at most 200? less than 200? at least 200? (round your answer to four decimal places.)
Answers
We need to determine the probabilities for the given scenarios. Probability is a branch of mathematics that deals with the study of random events or phenomena.
Here's the step-by-step explanation:
1. At most 200: This means x ≤ 200, including x = 200. Since x has 148 possible values, and all of them are less than or equal to 200, the probability, in this case, is 1 (all possible outcomes are included).
2. Less than 200: This means x < 200, not including x = 200. Since x has 148 possible values and all of them are less than 200, the probability is also 1 (all possible outcomes are included).
3. At least 200: This means x ≥ 200, including x = 200. Since there are no values of x in the given range that are greater than or equal to 200, the probability, in this case, is 0 (no possible outcomes are included).
So, the probabilities for the given scenarios are:
- At most 200: 1.0000 (rounded to four decimal places)
- Less than 200: 1.0000 (rounded to four decimal places)
- At least 200: 0.0000 (rounded to four decimal places)
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The height y (in feet) of a ball thrown by a child is y=−1/16x^2+6x+3 where x is the horizontal distance in feet from the point at which the ball is thrown. (a) How high is the ball when it leaves the child's hand? (Hint: Find y when x=0) 1- Your answer is y= 2- What is the maximum height of the ball? 3- How far from the child does the ball strike the ground?
Answers
(a) The ball is 3 feet high when it leaves the child's hand.
(b) The maximum height of the ball is 75 feet.
(c) The ball strikes the ground approximately 91.6 feet from the child
How to find the height of the ball?(a) To find the height of the ball moving in a projectile motion when it leaves the child's hand, we need to substitute x = 0 into the equation and solve for y:
[tex]y = -1/16(0)^2 + 6(0) + 3[/tex]
y = 3
Therefore, the ball is 3 feet high when it leaves the child's hand.
How to find the maximum height of the ball?(b) To find the maximum height of the ball, we need to find the vertex of the parabola defined by the equation. The x-coordinate of the vertex is given by:
x = -b / (2a)
where a = -1/16 and b = 6. Substituting these values, we get:
x = -6 / (2(-1/16)) = 48
The y-coordinate of the vertex is given by:
[tex]y = -1/16(48)^2 + 6(48) + 3 = 75[/tex]
Therefore, the maximum height of the ball is 75 feet.
How to find the distance of the ball?(c) To find how far from the child the ball strikes the ground, we need to find the value of x when y = 0 (since the ball will be at ground level when its height is 0).
Substituting y = 0 into the equation, we get:
[tex]0 = -1/16x^2 + 6x + 3[/tex]
Multiplying both sides by -16 to eliminate the fraction, we get:
[tex]x^2 - 96x - 48 = 0[/tex]
Using the quadratic formula, we can solve for x:
[tex]x = [96 \ ^+_- \sqrt{(96^2 - 4(-48))}]/2\\x = [96\ ^+_- \sqrt{(9408)}]/2\\x = 48 \ ^+_- \sqrt{(2352)[/tex]
x ≈ 4.4 or x ≈ 91.6
Since the ball is thrown from x = 0, we can discard the negative solution and conclude that the ball strikes the ground approximately 91.6 feet from the child.
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Two joggers run 6 miles south and then 5 miles east. What is the shortest distance they must travel to return to their starting point?
Answers
The shortest distance the joggers must travel to return to their starting point is 7.81 miles.
To find the shortest distance the joggers must travel to return to their starting point, we can use the Pythagorean theorem, as the southward and eastward distances form a right triangle. The theorem states that the square of the length of the hypotenuse (the shortest distance, in this case) is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
Here, a is the southward distance (6 miles), and b is the eastward distance (5 miles). We need to find c, the hypotenuse.
(6 miles)^2 + (5 miles)^2 = c^2
36 + 25 = c^2
61 = c^2
Now, take the square root of both sides to find c:
c = √61
c ≈ 7.81 miles
The two joggers form a right triangle with the starting point as the right angle. They have run 6 miles south and 5 miles east, so the legs of the right triangle have lengths 6 and 5.
To find the shortest distance they must travel to return to their starting point, we need to find the length of the hypotenuse of the right triangle. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs:
c^2 = a^2 + b^2
where c is the length of the hypotenuse, and a and b are the lengths of the legs.
Plugging in the values we have, we get:
c^2 = 6^2 + 5^2
c^2 = 36 + 25
c^2 = 61
Taking the square root of both sides, we get:
c = sqrt(61)
Therefore, the shortest distance the joggers must travel to return to their starting point is approximately 7.81 miles.
The discrete random variable X is the number of students that show up for Professor Smith's office hours on Monday afternoons. The table below shows the probability distribution for X. What is the expected value E(X) for this distribution?A. 1.2B. 1.0C. 1.5D. 2.0
Answers
The required answer is E(X) = 1.4
The expected value E(X) for this distribution can be calculated by multiplying each possible value of X by its probability, and then adding up these products. Using the table provided, we have:
E(X) = (0)(0.2) + (1)(0.3) + (2)(0.4) + (3)(0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4
Therefore, the closest option to our calculated expected value is option A, 1.2. However, none of the given options match exactly with our calculation.
To find the expected value E(X) of the discrete random variable X, we need the probability distribution table for X. However, the table is not provided in the question. Please provide the table with the probabilities for each possible value of X, and I will be happy to help you calculate the expected value E(X).
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Prove or disprove the identity:
[tex]\frac{(sin(t)+cos(t))^{2} }{sin(t)cos(t)} =2+csc(t)sec(t)[/tex]
Answers
The trigonometric identity [sint + cost]²/sin(t)cos(t) = 2 + csc(t)sec(t).
What are trigonometric identities?Trigonometric identities are mathematical equations that contain trigonometric ratios.
Since we have the trigonometric identity
[sint + cost]²/sin(t)cos(t) = 2 + csc(t)sec(t). We want to show that the left-hand-side L.H.S = right-hand-side R.H.S. We proceed as folows
Since we have L.H.S = [sint + cost]²/sin(t)cos(t), so expanding the numerator, we have that
[sint + cost]²/sin(t)cos(t), = [sin²t + 2sintcost + cos²(t)]/sin(t)cos(t)
Using the trigonometric identity sin²t + cos²t = 1, we have that
[sin²t + 2sintcost + cos²(t)]/sin(t)cos(t) = [sin²t + cos²(t) + 2sintcost]/sin(t)cos(t)
= [1 + 2sintcost]/sin(t)cos(t)
Dividing through by the denominator sin(t)cos(t) , we have that
[1 + 2sintcost]/sin(t)cos(t) = [1/sin(t)cos(t) + [2sintcost]/sin(t)cos(t)
= 1/sin(t) × `1/cos(t) + 2
= cosec(t)sec(t) + 2 [since cosec(t) = 1/sin(t) and sec(t) = 1/cos(t)]
= 2 + cosec(t)sec(t)
= R.H.S
Since L.H.S = R.H.S
So, the trigonometric identity [sint + cost]²/sin(t)cos(t) = 2 + csc(t)sec(t).
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Susan went to the supermarket to buy some itemsShe bought 5 pounds of meat at $14 per pound and a packets of sodas at $9 per pack which equation can be used to determine the total amount y that Susan paid A y=9x+70 B y=14x+9
C y=9x-14 D Y=70x+9
PLEASE HELPP
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The equation that can be used to determine the total amount y that Susan paid is [tex]y=14x+9[/tex]. The Option B is correct.
What equation can be used to calculate Susan's total purchase cost?To calculate the total amount paid by Susan at the supermarket, we need to add the cost of all the items she bought.
From the given information, we know that:
She bought 5 pounds of meat at $14 per pound She bought pack of sodas at $9 per pack.We can use the equation y = $14x + $9, where y represents the total amount paid by Susan and x represents the pounds of meat which can be multiplied.
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show how you can factor n = pq given the quantity (p − 1)(q − 1).
Answers
If we are given the quantity (p-1)(q-1), we can use it to factor n=pq. This is because of the relationship between Euler's totient function and the prime factors of n.
Specifically, if we know (p-1)(q-1), we can calculate the value of Euler's totient function for n as follows:
φ(n) = (p-1)(q-1)
We can then use this value to find the prime factors of n. One way to do this is to use the fact that for any prime factor p of n, we have:
n = p^k * m, where m is not divisible by p.
Then, we can use Euler's totient function to calculate the value of φ(n) as:
φ(n) = (p-1) * p^(k-1) * φ(m)
By substituting the value we calculated earlier for φ(n), we can solve for p and q:
φ(n) = (p-1)(q-1) = pq - (p+q) + 1
Solving for p and q using the quadratic formula gives:
p = (p+q) / 2 + sqrt((p+q)^2 / 4 - n)
q = (p+q) / 2 - sqrt((p+q)^2 / 4 - n)
Therefore, if we are given the quantity (p-1)(q-1), we can use it to calculate the value of Euler's totient function for n, and then use that to find the prime factors p and q of n.
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Find the limit of the sequence using L'Hôpital's Rule. an = (In(n))^2/Зn (Use symbolic notation and fractions where needed. Enter DNE if the sequence diverges.) lim n->[infinity] an =
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The limit of the sequence an = [(ln(n))²]/(3n) using L'Hôpital's Rule is 0.
We can apply L'Hôpital's Rule to find the limit of the given sequence:
an = [(ln(n))²]/(3n)
Taking the derivative of the numerator and denominator with respect to n:
an = [2 ln(n) * (1/n)] / 3
Simplifying:
an = (2/3) * (ln(n)/n)
Now taking the limit as n approaches infinity:
lim n->∞ an = lim n->∞ (2/3) * (ln(n)/n)
We can again apply L'Hôpital's Rule:
lim n->∞ (2/3) * (ln(n)/n) = lim n->∞ (2/3) * (1/n) = 0
Therefore, the limit of the sequence is 0.
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Find the angle of elevation of the sun from the ground when a tree that is 15 yard tall casts a shadow of 24 yards long. Round to the nearest degree.
A 38
B 39
C 63
D 32
E 51
Answers
Answer:
Set your calculator to degree mode.
Please draw the figure to confirm my answer.
[tex] \tan( \alpha ) = \frac{15}{24} [/tex]
[tex] \alpha = {tan}^{ - 1} \frac{5}{8} = 32 \: degrees[/tex]
So the angle of elevation is 32°.
D is the correct answer.
Which graph shows the solution to y > x – 8?
Answers
Answer: Your answer is B.
Step-by-step explanation: y > x means y is bigger than x and also -8 means that you subtract 8 from x so its B
From the set {22, 14, 12}, use substitution to determine which value of x makes the equation true. 2x = 24
Answers
Answer:
x = 12
Step-by-step explanation:
{22, 14, 12}
2x = 24
2(22) = 44
2(14) = 28
2(12) = 24
Is a trapezoid a quadrilateral or parallelogram or both? Explain.
Answers
Answer: both
Step-by-step explanation: