the derivative of r(t) at t = 2 in the direction of r'(2) is 193.
We're asked to find the derivative of the function given by r(t) = (t,t³, 8t) using the product rule.
Recall that if we have two vector functions f(t) = (f1(t), f2(t), f3(t)) and g(t) = (g1(t), g2(t), g3(t)), then their product rule is given by:
(fg)'(t) = f(t)g'(t) + g(t)f'(t)
where the prime notation (') denotes differentiation with respect to t.
In our case, we have:
r(t) = (t, t³, 8t)
r'(t) = (1, 3t², 8)
We can use the product rule to find r''(t) as follows:
r''(t) = (r'(t))' = (1, 3t², 8)' = (0, 6t, 0)
Now, we can evaluate r''(2) by plugging in t = 2:
r''(2) = (0, 6(2), 0) = (0, 12, 0)
Therefore, the derivative of r'(t) at t = 2 is:
r''(2)·r(2) + r'(2)·r'(2) = (0, 12, 0)·(2, 1, 0) + (1, 4, 3)·(1, 3(2)², 8)
= 0 + (1, 12, 3)·(1, 12, 8)
= 1(1) + 12(12) + 3(8)
= 169 + 24
= 193
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to test this series for convergence ∑_(n=1 )^[infinity]▒n/√(n^3+4)
you could use the limit comparison test, comparing it to the series ∑_(n=1)^[infinity]▒1/n^p where p=
Using the limit comparison test by comparing it to the series ∑(n=1 to ∞) 1/n^p, for convergence where p is a positive constant if p > 1, the series ∑(n=1 to ∞) n/√(n^3 + 4) converges. Otherwise, it diverges.
Let's determine the value of p to compare the given series:
Taking the limit as n approaches infinity of the ratio between the terms of the two series:
lim(n→∞) (n/√(n^3 + 4)) / (1/n^p)
Simplifying the expression inside the limit:
lim(n→∞) (n/n^p) / √(n^3 + 4)
Taking the reciprocal of the denominator:
lim(n→∞) (n/n^p) * (1/√(n^3 + 4))
Now, let's simplify further by dividing both the numerator and denominator by n:
lim(n→∞) 1/n^(p-1) * (1/√(n^2 + 4/n))
Since the term 4/n approaches 0 as n approaches infinity, we have:
lim(n→∞) 1/n^(p-1) * (1/√n^2)
Simplifying inside the limit:
lim(n→∞) 1/n^(p-1) * (1/n)
Combining the terms:
lim(n→∞) 1/n^p
For the series to converge, the limit above must be finite and positive.
Let's analyze the cases for p:
If p > 1:
In this case, the limit is 0, indicating that the series ∑(n=1 to ∞) 1/n^p converges. Therefore, the given series ∑(n=1 to ∞) n/√(n^3 + 4) also converges.
If p ≤ 1:
In this case, the limit approaches infinity, indicating that the series ∑(n=1 to ∞) 1/n^p diverges. Therefore, the given series ∑(n=1 to ∞) n/√(n^3 + 4) also diverges.
In conclusion, if p > 1, the series ∑(n=1 to ∞) n/√(n^3 + 4) converges. Otherwise, it diverges
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Consider the following problem: The data set includes 107 body temperatures of healthy adult humans for which x=98.7°F and s = 0.72° F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans. What is the appropriate symbol to use for the answer?___ < δ < ______ < µ < ______ < p < ______ < z < ______ < n < ___
The appropriate symbols to use for the answer are: µ - z * (s / √n) < δ < µ + z * (s / √n)
To construct a confidence interval estimate for the mean body temperature of all healthy humans, we can use the symbol "µ" to represent the population mean.
A 99% confidence interval estimate for the mean body temperature can be represented as:
µ - z * (s / √n) < µ < µ + z * (s / √n)
In this expression:
"z" represents the critical value from the standard normal distribution corresponding to the desired confidence level (in this case, 99%).
"s" represents the sample standard deviation.
"n" represents the sample size.
Therefore, the appropriate symbols to use for the answer are:
µ - z * (s / √n) < δ < µ + z * (s / √n)
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show if m is a positive integer and a is an integer relatively prime to m such that ordma = m - 1, then m is prime.
Let us assume that m is not prime. This means that there exists a prime factor p of m such that p ≤ √m. Since a is relatively prime to m, it must also be relatively prime to p.
Now, let's consider the order of a modulo p. We know that ordpa divides p-1, since p is prime. However, since a and p are relatively prime, we also know that ordpa cannot be equal to p-1, since this would imply that a is a primitive root modulo p, which is impossible since p is a prime factor of m and therefore does not have any primitive roots modulo p.
So, ordpa must divide p-1, but it cannot be equal to p-1. Therefore, ordpa must be strictly less than m-1 (since m has p as a factor, which means that m-1 has p-1 as a factor). However, we know that ordma = m-1. This means that ordpa cannot be equal to ordma.
This is a contradiction, since we assumed that ordma = m-1 and that ordpa divides m-1. Therefore, our initial assumption that m is not prime must be false. Therefore, m must be prime.
In conclusion, if m is a positive integer and a is an integer relatively prime to m such that ordma = m-1, then m must be prime.
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4y = -2 help pls this is missing I will give pts!!
Answer:y=-4/2x
Step-by-step explanation:
HURRY MY TIMES RUNNING OUT
Answer:
C
Step-by-step explanation:
Input x 6 = output for each of these numbers
3x6 =18
6x6 =36
11x6 = 66
12x6 = 72
the other options are incorrect. A is divided by 4, B is times 4, and D is divided by 6.
6.5.3 if (x1, . . . , xn) is a sample from a pareto(α) distribution (see exercise 6.2.9), whereα > 0 is unknown, determine the fisher information.
The Fisher information for a Pareto(α) distribution is I(α) = nα² / (α - 1)².
To determine the Fisher information for a sample from a Pareto(α) distribution, follow these steps:
1. Recall the Pareto(α) probability density function (PDF): f(x) = αxᵃ⁺¹), where x ≥ 1 and α > 0.
2. Compute the log-likelihood function, L(α) = ln(f(x1,...,xn)) = ∑ ln(α) - (α+1)ln(xi) for i = 1 to n.
3. Differentiate L(α) with respect to α: dL/dα = ∑ (1/α) - ln(xi).
4. Differentiate dL/dα again: d²L/dα² = -∑ (1/α²).
5. The Fisher information is the negative expectation of the second derivative: I(α) = -E(d²L/dα²).
6. Apply the Pareto(α) distribution's expectation: I(α) = nα² / (α - 1)².
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By following the method, we think about communicating by reviewing the possible things (both general and specific) that might be said. Select one: O a. Free-form O b.Inverse. O c. Cyclical O d. Linear. O e. Topical
In the topical method, we focus on discussing different topics or subjects by considering various aspects and details related to them. This approach allows us to think about and communicate more effectively by addressing both general and specific points that might be relevant to the conversation.
The method of communication that involves reviewing possible things (both general and specific) that might be said. The correct answer is: e. Topical.
The method described in the question is a form of "topical" communication. This approach involves considering different topics or subjects that may need to be discussed and organizing thoughts and information around them. By reviewing possible things that may be said on each topic, one can prepare for a more effective and focused communication.
This method can be especially helpful in situations where there are multiple topics to cover or when discussing complex information that requires careful organization and planning.
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Kelly draws a rectangle. How many square corners does Kelly's rectangle have?
Choose the answer that makes the statement true. Kelly's rectangle has
Choose. Square corners
Kelly's rectangle has four square corners.
A rectangle is a quadrilateral with four sides and four angles. In a rectangle, opposite sides are equal in length, and all angles are right angles (90 degrees). A square is a special type of rectangle where all sides are equal in length
. Since a square is a type of rectangle, it also has four right angles, making all its corners square corners. Therefore, Kelly's rectangle, which is not specified as a square, may have different side lengths, but it will still have four right angles, resulting in four square corners.
These corners are formed by the intersection of the sides at right angles, creating a shape with sharp, 90-degree angles. So, regardless of the specific dimensions of Kelly's rectangle, it will always have four square corners.
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We say XA is an indicator variable for event A: XA = 1 if A occurs, XA = 0 if A does not occur. If P(A) = 0.35, what is: • E(XA)? Var (XA)
The expected value of XA is 0.35, and the variance of XA is 0.2275.
To find the expected value of XA, we simply multiply the probability of A occurring (0.35) by 1 (the indicator variable when A occurs) and add the product of the probability of A not occurring (1 - 0.35 = 0.65) and 0 (the indicator variable when A does not occur). So, E(XA) = 0.35 * 1 + 0.65 * 0 = 0.35.
To find the variance of XA, we need to calculate the probability of each outcome (0 or 1) and its squared difference from the expected value. The variance formula for an indicator variable is Var(XA) = P(A)(1 - P(A)). Therefore, Var(XA) = 0.35 * (1 - 0.35) = 0.35 * 0.65 = 0.2275.
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The nth term test can be used to determine divergence for each of the following series except A arctann n=1 B 61 с n(n+3) = (n + 4) D Inn n=1
The nth term test, also known as the Test for Divergence, is a useful tool for determining the divergence of a given series. All of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.
In order to use this test, you should analyze the limit of the sequence's terms as n approaches infinity. If the limit is not zero, then the series diverges.
For each of the series provided, let's apply the nth term test:
A) arctan(n), n=1 to infinity:
The limit as n approaches infinity of arctan(n) is π/2, which is not zero. Therefore, the series diverges.
B) 61:
Since the series consists of a constant term, the limit as n approaches infinity is 61, which is not zero. Therefore, the series diverges.
C) n(n+3)/(n+4), n=1 to infinity:
As n approaches infinity, the limit of n(n+3)/(n+4) is 1, which is not zero. Therefore, the series diverges.
D) ln(n), n=1 to infinity:
The limit as n approaches infinity of ln(n) is infinity, which is not zero. Therefore, the series diverges.
In conclusion, all of the given series - A) arctan(n), B) 61, C) n(n+3)/(n+4), and D) ln(n) - diverge according to the nth term test.
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Evaluate the following integral using integration by parts. ∫ t^2 e^-17t dt Use the integration by parts formula so that the new integral is simpler than the original one. Choose the correct answer below. a. -2/17 t^2 e^-17t - ∫ (-1/17t^2 e^-17t) dt
b. -1/17 t^2 e^-17t - ∫ (-2/17t^2 e^-17t) dt
c. -1/17 t^2 e^-17t + ∫ (17t^2 e^-17t) dt
d. 1/17 t^2 e^17t - ∫ (2/17t e^17t) dt
Thus, the obtained function using the integration by parts: -1/17 t^2 e^-17t - ∫ (-2/17t^2 e^-17t) dt.
To evaluate the integral ∫ t^2 e^-17t dt using integration by parts, we will use the formula:
∫ u dv = uv - ∫ v du
where u and dv are functions of t that we choose appropriately. Let's choose:
u = t^2 (so that du/dt = 2t)
dv = e^-17t dt (so that v = (-1/17)e^-17t)
Using these choices, we can find du and v:
du = 2t dt
v = (-1/17)e^-17t
Now, we can apply the integration by parts formula:
∫ t^2 e^-17t dt = t^2 (-1/17)e^-17t - ∫ 2t (-1/17)e^-17t dt
Simplifying this expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t + (2/17) ∫ te^-17t dt
To evaluate the new integral ∫ te^-17t dt, we will use integration by parts again. This time, we will choose:
u = t (so that du/dt = 1)
dv = e^-17t dt (so that v = (-1/17)e^-17t)
Using these choices, we can find du and v:
du = dt
v = (-1/17)e^-17t
Now, we can apply the integration by parts formula again:
∫ te^-17t dt = t (-1/17)e^-17t - ∫ (-1/17)e^-17t dt
Simplifying this expression, we get:
∫ te^-17t dt = (-1/17) te^-17t + (1/289) e^-17t
Substituting this result back into our original expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t + (2/17) ((-1/17) te^-17t + (1/289) e^-17t))
Simplifying this expression, we get:
∫ t^2 e^-17t dt = (-1/17) t^2 e^-17t - (2/289) te^-17t - (2/4913) e^-17t
Therefore, the correct answer is (b): -1/17 t^2 e^-17t - ∫ (-2/17t^2 e^-17t) dt.
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I need help because need to bring my math grade
Mr. Anderson took Mrs. Anderson out
for a nice steak dinner. The food bill
came out to $89.25 before tax and tip.
If tax is 6% and tip is 15%, what is
the total cost?
Answer:
$108.80
Step-by-step explanation:
89.25x0.06 = $5.36 tax
89.25 + 5.36= 94.61
94.61 x 0.15 = 14.91 tip
94.61 + 14.91 = 108.80 total
A faster way: 89.25*1.06*1.15=108.80
using only the digits 0 and 1 how many different numbers consisting of 8 digits can be formed
The first digit must be 1. The remaining seven ones must be either 0 or 1.
Therefore, there can be formed [tex]2^7=128[/tex] different numbers.
give an indexed family of sets that is pairwise disjoint but the intersection over it is nonempty
The intersection over the indexed family {A_i} is nonempty, as there are no elements that belong to all three sets:
⋂(A_i) = A_1 ∩ A_2 ∩ A_3 = ∅ (empty set)
An indexed family of sets that is pairwise disjoint but has a nonempty intersection can be represented as follows:
Consider an indexed family of sets {A_i} where i belongs to the index set I, with I = {1, 2, 3}. Define the sets A_i as:
A_1 = {1, 2}
A_2 = {2, 3}
A_3 = {1, 3}
The sets are pairwise disjoint since no two sets share any common elements:
A_1 ∩ A_2 = {2} (not empty)
A_1 ∩ A_3 = {1} (not empty)
A_2 ∩ A_3 = {3} (not empty)
However, the intersection over the indexed family {A_i} is nonempty, as there are no elements that belong to all three sets:
⋂(A_i) = A_1 ∩ A_2 ∩ A_3 = ∅ (empty set)
In this example, the indexed family of sets is pairwise disjoint, but the intersection over the family is nonempty.
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Consider two independent random variables X and Y. X has a Uniform distribution on the interval (0, 3). The probability density function of Y is given by fY (y) = y^2/9 if 0 < y < 3; 0 otherwise (a) Calculate P(X / Y > 1). (b) Calculate P(X + Y > 2). (c) Calculate P(X * Y > 3)
Answer : ∫∫[Y > 3/X] (1/3) * (y^2/9) dx/dy.
(a) To calculate P(X/Y > 1), we need to find the probability that the ratio of X to Y is greater than 1.
The joint probability density function of X and Y, since they are independent, is given by f(X,Y) = fX(x) * fY(y).
Given that X has a Uniform distribution on (0, 3), the probability density function of X, fX(x), is:
fX(x) = 1/(3-0) = 1/3 for 0 < x < 3, and 0 otherwise.
The probability density function of Y, fY(y), is given as:
fY(y) = y^2/9 for 0 < y < 3, and 0 otherwise.
Now, we can calculate P(X/Y > 1) as follows:
P(X/Y > 1) = ∫∫[X/Y > 1] f(X,Y) dxdy
= ∫∫[X > Y] fX(x) * fY(y) dxdy
= ∫∫[X > Y] (1/3) * (y^2/9) dxdy
= ∫[0,3] ∫[0,x] (1/3) * (y^2/9) dydx
= (1/3) ∫[0,3] [(1/9) * (y^3/3)] evaluated from 0 to x dx
= (1/3) ∫[0,3] (x^3/27) dx
= (1/3) * [(1/108) * (x^4)] evaluated from 0 to 3
= (1/3) * [(1/108) * (3^4 - 0^4)]
= (1/3) * [(1/108) * 81]
= 1/4.
Therefore, P(X/Y > 1) = 1/4.
(b) To calculate P(X + Y > 2), we need to find the probability that the sum of X and Y is greater than 2.
We can calculate this as follows:
P(X + Y > 2) = ∫∫[X + Y > 2] f(X,Y) dxdy
= ∫∫[X > 2 - Y] fX(x) * fY(y) dxdy
= ∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy.
To solve this integral, we can break it into two parts based on the range of Y:
For 0 < y < 2:
∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[0,2] ∫[2-y,3] (1/3) * (y^2/9) dxdy.
For 2 < y < 3:
∫∫[X > 2 - Y] (1/3) * (y^2/9) dxdy = ∫[2,3] ∫[0,3] (1/3) * (y^2/9) dxdy.
Calculating these integrals will give us the desired probability.
(c) To calculate P(X * Y > 3), we need to find the probability that the product of X and Y is greater than 3.
Similarly, we can set up the
integral:
P(X * Y > 3) = ∫∫[X * Y > 3] f(X,Y) dxdy
= ∫∫[Y > 3/X] fX(x) * fY(y) dxdy
= ∫∫[Y > 3/X] (1/3) * (y^2/9) dxdy.
We can then evaluate this integral over the appropriate ranges to find the desired probability.
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6. (20 points) the domain of a relation a is the set of integers. 2 is related to y under relation a it =u 2.
For any integer input x in the domain of relation a, if x is related to 2, then the output will be u2.
Based on the given information, we know that the domain of the relation a is the set of integers. Additionally, we know that 2 is related to y under relation a, with the output being u2.
Therefore, we can conclude that for any integer input x in the domain of relation a, if x is related to 2, then the output will be u2. However, we do not have enough information to determine the outputs for other inputs in the domain.
In other words, we know that the relation a contains at least one ordered pair (2, u2), but we do not know if there are any other ordered pairs in the relation.
The correct question should be :
In the given relation a, if an integer input x is related to 2, what is the corresponding output?
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the proportion of variation explained by the model is called the ____group of answer choices a. slope of the line b. sum of squares error c. coefficient of determination d. coefficient of correlation
The proportion of variation explained by the model is called the coefficient of determination, also denoted as R-squared.
It is a statistical measure that represents the percentage of the variance in the dependent variable that is explained by the independent variable(s) in the regression model. In other words, it measures the goodness of fit of the regression line to the observed data points. The coefficient of determination ranges from 0 to 1, where 0 indicates that the model does not explain any of the variance in the dependent variable, and 1 indicates that the model explains all of the variance in the dependent variable. The coefficient of determination is often used in regression analysis to evaluate the predictive power of the model and to compare the fit of different models.
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A new car is purchased for $16,500. The value of the car depreciates at 5.75% per year. What will the car be worth, to the nearest penny, after 5 years?
Answer:
Step-by-step explanation:
I think it would be 500
Answer:
The value of the car after 5 years is $12271.05
The present value of the car, PV = $16500
The rate of depreciation, r = 5.75%
r = 5.75/100
r = 0.0575
Step-by-step explanation:
find an inverse of a modulo m for the following pairs (whenever possible) a=your day of birth,m=your month of birth a=34,m=91
The inverse of 'a' modulo 'm' is not possible for the given pairs (your day and month of birth: a=34, m=91) because 'a' and 'm' are not relatively prime.
To find the inverse of 'a' modulo 'm', we need to determine a number 'x' such that (a * x) % m = 1. This means that 'x' is the multiplicative inverse of 'a' modulo 'm'. However, for an inverse to exist, 'a' and 'm' must be relatively prime, meaning they do not have any common factors other than 1. In the given pair (a=34, m=91), 'a' and 'm' share a common factor of 13. Therefore, an inverse does not exist.
When 'a' and 'm' are not relatively prime, there is no integer 'x' that satisfies the equation (a * x) % m = 1. In this case, we cannot find the inverse of 'a' modulo 'm'. It is important to note that for an inverse to exist, 'm' must be a positive integer greater than 1, and 'a' must be a positive integer less than 'm'. In the given pair (34, 91), both conditions are met, but the lack of relative primality between 'a' and 'm' prevents the existence of an inverse.
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Simplify the expression Sqrt 39 (sqrt6+7)
The simplified expression √39(√6 + 7) is equal to √(39 * 6) + 7√(39 .
To simplify the expression √39(√6 + 7), we can follow these steps:
Step 1: Distribute the square-root (√) to both terms inside the parentheses:
√39 * √6 + √39 * 7
Step 2: Simplify the square roots separately:
√(39 * 6) + √(39 * 7)
Step 3: Calculate the products under the square roots:
√234 + √273
Step 4: Simplify the square roots further:
√(9 * 26) + √(9 * 3 * 3 * 3)
Step 5: Split the square root of a product into the product of the square roots:
√9 * √26 + √9 * √(3 * 3 * 3)
Step 6: Simplify the square roots of perfect squares:
3√26 + 3√(3 * 3 * 3)
Step 7: Multiply the numbers outside the square roots:
3√26 + 9√3
Note that the simplified form is obtained by simplifying the square roots as much as possible and combining like terms.
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Use the geometric series f(x) = 1/1 - x = sigma^infinity_k = 0 x^k, for |x| < 1. to find the power series representation for the following function (centered at 0). Give the interval of convergence of the new series. g(x) = x^3/1 - x Which of the following is the power series representation for g(x)? A. sigma^infinity_k = 0 x^3/x^k C. sigma^infinity_k = 0 1/1 - x^k + 3 B. sigma^infinity_k = 0 x^k + 3 D. sigma^infinity_k = 0 x^3k The interval of convergence of the new series is. (Simplify your answer. Type your answer in interval notation.)
B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).
To find the power series representation for g(x), we need to rewrite g(x) in terms of the given geometric series.
Notice that g(x) can be written as:
g(x) = x^3/1 - x = x^3 * (1/1-x)
We can now substitute the formula for the geometric series to get:
g(x) = x^3 * sigma^infinity_k = 0 x^k
= sigma^infinity_k = 0 (x^3 * x^k)
= sigma^infinity_k = 0 x^(k+3)
Therefore, the power series representation for g(x) is:
sigma^infinity_k = 0 x^(k+3)
The interval of convergence of this series is the same as that of the geometric series, which is |x| < 1.
In interval notation, this can be written as (-1, 1).
Therefore, the correct answer is B. sigma^infinity_k = 0 x^k + 3, and the interval of convergence is (-1, 1).
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Fine the perimeter of a rectangle 4m 4m
Answer:
16 m
Step-by-step explanation:
is a square, all sides congruent, we add up and we have the perimeter
Perimeter = 4 + 4 + 4 + 4 = 16 m
The result of the perimeter is 16 meters (m).
Step-by-step explanation:To solve, we must first know that the perimeters in this problem should only be added to each side, which is 4, where it gives a result of 16 meters (m).
¿What are the perimeters?First of all we must know that in geometry, the perimeter is the sum of all the sides. A perimeter is a closed path that encompasses, surrounds, or skirts a two-dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
With this we can say that the perimeters are those that are added from each side, so, what we need to do in this problem is just just add each side, each side is four, so we can add it by 4 since it asks us for that.
[tex] \bold{4 + 4 + 4 + 4 = \boxed{ \bold{16m}}}[/tex]
But we also have another step to solve this problem, which is just squaring it where it also gives us the same result, let's see:
[tex] \bold{2 {}^{4} = \boxed{ \bold{16 \: meters \: (m)}}}[/tex]
So, as we see, each resolution gives us the same result, therefore, the result of the perimeter is 16 meters (m).
A rectangle is 20cm long and 8cm wide. Find the diagonal of the rectangle.
Answer:
21.5 cm
Step-by-step explanation:
a² + b² = c²
20² + 8² = c²
400 + 64 = c²
464 = c²
c = √464
c = 21.5
Answer: 21.5 cm
find the area enclosed by the polar curve r=12sinθ. write the exact answer. do not round.
To find the area enclosed by the polar curve r = 12sinθ, we can use the formula for the area of a polar curve: A = 1/2 * ∫(r^2)dθ. For r = 12sinθ, the integral limits are from 0 to π because the curve covers a full period of the sine function.
Let's evaluate the integral using angle identity:
A = 1/2 * ∫(r^2)dθ
A = 1/2 * ∫((12sinθ)^2)dθ, with θ from 0 to π
A = 1/2 * ∫(144sin^2θ)dθ
Now, we can use the double angle identity sin^2θ = (1 - cos(2θ))/2:
A = 1/2 * ∫(144(1 - cos(2θ))/2)dθ
A = 72 * ∫(1 - cos(2θ))dθ, with θ from 0 to π
Now, we can integrate:
A = 72 * [θ - 1/2 * sin(2θ)] from 0 to π
A = 72 * [π - 0 - (1/2 * sin(2π) - 1/2 * sin(0))]
A = 72 * π
The exact area enclosed by the polar curve r = 12sinθ is 72π square units.
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A parking garage has 230 cars in it when it opens at 8 ( = 0). On the interval 0 ≤ ≤ 10, cars enter the parking garage at the rate ′ () = 58 cos(0.1635 − 0.642) cars per hour and cars leave the parking garage at the rate ′ () = 65 sin(0.281) + 7.1 cars per hour (a) How many cars enter the parking garage over the interval = 0 to = 10 hours? (b) Find ′′(5). Using correct units, explaining the meaning of this value in context of the problem. (c) Find the number of cars in the parking garage at time = 10. Show the work that leads to your answer.
Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.
(a) To find the number of cars entering the parking garage over the interval 0 ≤ t ≤ 10, we need to integrate the rate of cars entering the garage with respect to time. ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars.
(b) To find ′′(5), we need to differentiate the rate of cars leaving the garage with respect to time twice. ′′(t) = -65cos(0.281) and ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour. This value represents the rate of change of the rate of cars leaving the garage at t = 5.
(c) To find the number of cars in the parking garage at time t = 10, we need to subtract the total number of cars leaving the garage from the total number of cars entering the garage from t = 0 to t = 10. This gives approximately 559 cars in the garage at t = 10.
Therefore, (a) ∫58cos(0.1635t - 0.642)dt from 0 to 10 gives approximately 822.6 cars, (b) ′′(5) = -65cos(0.281) which is approximately -62.4 cars per hour per hour, (c) Approximately 559 cars in the garage at t = 10.
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a couple plan to have three children. there are eight possible arrangements of girls and boys. for example, ggb means the first two children are girls and the third child is a boy. all eight arrangements are (approximately) equally likely. write down all eight arrangements of the sexes of three children. what is the probability of any one of these arrangements? enter your answer to three decimal places.
The probability of any one of the eight arrangements is 0.125, or 12.5% when rounded to three decimal places.
There are eight possible arrangements of boys and girls when a couple plans to have three children. They are:
BBB (all boys)
BBG (two boys, one girl)
BGB (one boy, two girls)
BGG (one boy, two girls)
GBB (two boys, one girl)
GBG (one boy, two girls)
GGB (two boys, one girl)
GGG (all girls)
The probability of any one of these arrangements can be calculated using the formula for probability:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
In this case, since all eight arrangements are equally likely, the probability of any one of them is:
Probability = 1 / 8
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Panchito discovered a refreshing beverage by mixing a 5% cranberry juice with a 90% orange juice. How much of each should he mix together to make 150 ml of a 22% cranberry-orange juice blend
Panchito should mix 30 ml of cranberry juice with 120 ml of orange juice to make 150 ml of a 22% cranberry-orange juice blend.
To determine the amounts of cranberry juice and orange juice that Panchito should mix, we can set up a system of equations based on the given information. Let's assume Panchito mixes x ml of cranberry juice and y ml of orange juice.
The total volume of the mixture is 150 ml: x + y = 150.
The percentage of cranberry juice in the mixture is 22%: (0.05x) / 150 = 0.22.
Simplifying the second equation, we get:
0.05x = 0.22 * 150
0.05x = 33
x = 33 / 0.05
x = 660 ml
Substituting this value back into the first equation, we can solve for y:
660 + y = 150
y = 150 - 660
y = -510 ml
Since the solution for y is negative, it is not feasible. This indicates that there is no way to create a 22% cranberry-orange juice blend using a 5% cranberry juice and a 90% orange juice.
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The cost of one pound of bananas is greater than $0. 41 and less than $0. 50. Sarah pays $3. 40 for x pounds of bananas. Which inequality represents the range of possible pounds purchased? 0. 41 < 0. 41 less than StartFraction 3. 40 over x EndFraction less than 0. 50. < 0. 50 0. 41 < 0. 41 less than StartFraction x over 3. 40 EndFraction less than 0. 50. < 0. 50 0. 41 < 3. 40x < 0. 50 0. 41 < 3. 40 x < 0. 50.
A) is correct answer. The inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.
The inequality that represents the range of possible pounds purchased is as follows:
0.41 < (3.40/x) < 0.50.
Let's discuss the given problem step-by-step.
Sarah pays $3.40 for x pounds of bananas.
The cost of one pound of bananas is greater than $0.41 and less than $0.50.
Therefore, the cost of x pounds of bananas can be written as:
3.40 < x(0.50) and 3.40 > x(0.41)
⇒ 0.41x < 3.40 < 0.50x
⇒ 0.41 < (3.40/x) < 0.50
Hence, the inequality that represents the range of possible pounds purchased is 0.41 < (3.40/x) < 0.50.
The answer is option A.
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The base of each triangle measures 2 centimeters and the perimeter of each triangle is 10 centimeters. What is the approximate total area of the plastic triangles on the spinner? 3. 9 square centimeters 6. 7 square centimeters 7. 7 square centimeters 13. 4 square centimeters.
The answer is option 13. 4 square centimeters.
Let's first find the length of the sides of each triangle. Since the perimeter of each triangle is 10 centimeters, and each triangle has 3 sides of equal length, the length of each side of the triangles is given by;
Side length = Perimeter ÷ Number of sides
= 10 ÷ 3= 3.33 (rounded to 2 decimal places)
The base of each triangle measures 2 centimeters, and the length of the side is 3.33 centimeters.
We can use the Pythagorean theorem to find the height of the triangles. Using Pythagorean theorem,
a² + b² = c²where a = 1, b = h and c = 3.33
From the formula above, we can find that:
h² = c² - a²
= 3.33² - 1²
≈ 10.77h
≈ √10.77
≈ 3.28
The area of each triangle is given by the formula;
Area = 1/2 x base x height
= 1/2 x 2 x 3.28
= 3.28 square centimeters (rounded to 2 decimal places)
Since there are 4 triangles, the total area of the plastic triangles on the spinner is approximately:
Total area = 4 x 3.28
= 13.12 square centimeters (rounded to 2 decimal places)
Therefore, the answer is option 13. 4 square centimeters.
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Write a function when a baseball is thrown into the air with an upward velocity of 30 ft/s.
this function assumes that the baseball is thrown from ground level, and it does not take into account any external factors that may affect the trajectory of the ball (such as air resistance, wind, or spin).
Assuming that air resistance can be ignored, the height (in feet) of a baseball thrown upward with an initial velocity of 30 ft/s at time t (in seconds) can be modeled by the function:
h(t) = 30t - 16t^2
This function represents the position of the baseball above the ground, and it is a quadratic equation with a downward-facing parabolic shape. The initial velocity of 30 ft/s corresponds to the coefficient of the linear term, and the coefficient of the quadratic term (-16) is half the acceleration due to gravity (32 ft/s^2).
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