Answer:
Method A.
Step-by-step explanation:
For solving this question we need to find out the z-scores for both methods,
Since, the z-score formula is,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where, [tex]\mu[/tex] is mean,
[tex]\sigma[/tex] is standard deviation,
Given,
For method A,
[tex]\mu = 33[/tex]
[tex]\sigma=8[/tex]
Thus, the z score for 34 is,
[tex]z_1=\frac{34-33}{8}=0.125[/tex]
While, for method B,
[tex]\mu = 37[/tex]
[tex]\sigma = 4[/tex]
Thus, the z score for 34 is,
[tex]z_2=\frac{34-37}{4}=-0.75[/tex],
Since, [tex]z_1 > z_2[/tex]
Hence, method A is preferred if the procedure must be completed within 34 minutes.
Comparison of two normal distribution can be done via intermediary standard normal distribution. The procedure to be preferred for getting the procedure completed within 34 minutes is: Method A
How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] )
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
Firstly, we need to figure out what the problem is asking, and a method which we can apply. Two normal distributions have to be compared. We can convert them to standard normal distribution for comparison. Then, we will get the p-value for 34 minutes(converted to standard normal variate's value) which will tell about the probability of obtaining time as 34 minutes(or under it)(this can be obtained with p value) in method A or B. The more the probability is there, the more chances for that method would be for getting completed within 34 minutes (compared to other method).
For method A:Let X = time taken for completion of surgical procedure by method A,
Then, by given data, we have: [tex]X \sim N(\mu = 33, \sigma = 8)[/tex]
The probability that X will fall within value 34 is [tex]P(X \leq 34)[/tex]
Converting this whole thing to standard normal distribution, we get the needed probability as:
[tex]P(X \leq 34) = P(Z = \dfrac{X - \mu}{\sigma} \leq \dfrac{34 - 33}{8} ) = P(Z \leq 0.15)[/tex]
From the z-tables, the p value for Z = 0.15 is 0.5596
Thus, we get:
[tex]P(X \leq 34) = P(Z \leq 0.15 ) \approx 0.5596[/tex]
For method B:Let Y = time taken for completion of surgical procedure by method B,
Then, by given data, we have: [tex]Y \sim N(\mu = 37, \sigma = 4)[/tex]
The probability that X will fall within value 34 is [tex]P(Y \leq 34)[/tex]
Converting this whole thing to standard normal distribution, we get the needed probability as:
[tex]P(Y \leq 34) = P(Z = \dfrac{Y - \mu}{\sigma} \leq \dfrac{34 - 37}{4} ) = P(Z \leq -0.25)[/tex]
From the z-tables, the p value for Z = -0.25 is 0.4013
Thus, we get:
[tex]P(Y \leq 34) = P(Z \leq -0.25 ) \approx 0.4013[/tex]
Thus, we see that:
P(method A will make surgical procedure last within 34 minutes) = 0.5596 > P(method B will make surgical procedure last within 34 minutes) = 0.4013
Thus, method A should be preferred, as there is higher chances for method A to get the surgery completed within 34 minutes than method B.
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Find the fixed points of f(x) = 2x - x3 and determine their stability.
Answer with explanation:
⇒ A point k is said to be fixed point of function, f(x) if
f(k)=k.
The given function is
f(x)=2 x - x³
To determine the fixed point
f(k)=2 k - k³=k
→2 k -k -k³=0
→k -k³=0
→k×(1-k²)=0
→k(k+1)(k-1)=0
→k=0 ∧ k+1=0∧k-1=0
→k=0∧ k= -1 ∧ k=1
So, the three fixed points are=0,1 and -1.
To Check Stability of fixed point
1.⇒ f'(x)=2-3 x²
|f'(0)|=|2×0-0³|=0
⇒x=0, is Superstable point.
2.⇒|f'(-1)|=2 -3×(-1)²
=2 -3
= -1
|f'(-1)| <1
⇒x= -1, is stable point.
3.⇒|f'(1)|=2 -3×(1)²
=2 -3
= -1
|f'(1)| <1
⇒x= 1, is also a stable point.
⇒⇒There are two points of Stability,which are, x=1 and , x=-1.
PLEASE HELP PRECALCULUS WILL MARK BRAINLIEST
see attachment'
Check the picture below.
Answer:
2
Step-by-step explanation:
Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = −xi − yj + z3k, S is the part of the cone z = x2 + y2 between the planes z = 1 and z = 2 with downward orientation.
The equation of the cone should be [tex]z=\sqrt{x^2+y^2}[/tex]. Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+u\,\vec k[/tex]
with [tex]1\le u\le2[/tex] and [tex]0\le v\le2\pi[/tex]. Take the normal vector to [tex]S[/tex] to be
[tex]\vec s_v\times\vec s_u=u\cos v\,\vec\imath+u\sin v\,\vec\jmath-u\,\vec k[/tex]
Then the integral of [tex]\vec F[/tex] across [tex]S[/tex] is
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_0^{2\pi}\int_1^2(-u\cos v\,\vec\imath-u\sin v\,\vec\jmath+u^3\,\vec k)\cdot(u\cos v\,\vec\imath+u\sin v\,\vec\jmath-u\,\vec k)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_0^{2\pi}\int_1^2(-u^2-u^4)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle-2\pi\int_1^2(u^2+u^4)\,\mathrm du\,\mathrm dv=\boxed{-\frac{256\pi}{15}}[/tex]
To evaluate the surface integral, we need to find the flux of the vector field F across the oriented surface S. Given that F(x, y, z) = −xi − yj + z3k, and S is the part of the cone z = x2 + y2 between the planes z = 1 and z = 2 with downward orientation, we can proceed as follows: First, find the unit normal vector to the surface. Next, calculate the dot product between the vector field F and the unit normal vector. Finally, integrate the dot product over the surface S using the downward orientation.
Explanation:To evaluate the surface integral, we need to find the flux of the vector field F across the oriented surface S. Given that F(x, y, z) = −xi − yj + z3k, and S is the part of the cone z = x2 + y2 between the planes z = 1 and z = 2 with downward orientation, we can proceed as follows:
First, we need to find the unit normal vector to the surface. In this case, the unit normal vector is -∇(z - x^2 - y^2)/|∇(z - x^2 - y^2)|. By calculating the gradient and normalizing it, we get the unit normal vector as (2x, 2y, -1)/√(1 + 4x^2 + 4y^2).Next, we calculate the dot product between the vector field F and the unit normal vector. The dot product is -2x - 2y + z^3.Finally, we integrate the dot product over the surface S using the downward orientation. The integral is given by ∫∫S (-2x - 2y + z^3)dS.Learn more about Surface Integral here:https://brainly.com/question/32088117
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.3 3. True or False? For any integer m, 2m(3m + 2) is divisible by 4. Explain to get credit.
Answer with explanation:
We have to prove that, For any integer m, 2 m×(3 m + 2) is divisible by 4.
We will prove this result with the help of Mathematical Induction.
⇒For Positive Integers
For, m=1
L HS=2×1×(3×1+2)
=2×(5)
=10
It is not divisible by 4.
⇒For Negative Integers
For, m= -1
L HS=2×(-1)×[3×(-1)+2]
=-2×(-3+2)
= (-2)× (-1)
=2
It is not divisible by 4.
False Statement.
By convention, the independent variable is arrayed along the ____ in a scattergram. regression line calibration line vertical axis (the ordinate) horizontal axis (the abscissa)
Answer:
Horizontal axis
Step-by-step explanation:
By convention, the independent variables is arrayed along the Horizontal axis (abscissa) in a scattergram.
The stcattergram has two dimensions
The X (independent) variable is arrayed along the horizontal axis.The Y(dependent) variable is arrayed along the vertical axis.Each dot scattergram is case in data set.The dot is placed at the intersection cases scores on X and Y.You have $500,000 saved for retirement. Your account earns 4% interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 25 years?
Answer:
amount pull out each month is $4518.44
Step-by-step explanation:
principal (p) = $500000
rate (r) = 4% = 0.04 = 0.04/12 per month
time period = 25 years = 25 × 12 = 300 months
to find out
how much amount pull out each month
solution
we will calculate the amount by given formula i.e.
principal = amount ( 1 - [tex](1+r)^{t}[/tex] ) / r ....................1
now put the value amount rate time in equation 1
we get amount
500000 = amount ( 1 - [tex](1+0.04/12)^{300}[/tex] ) / 0.04/12
500000 = amount (2.711062 ) / 0.00333
amount = 1666.6666 * 2.711062
amount = 4518.44
amount pull out each month is $4518.44
Final answer:
To determine how much you can pull out each month, you can use the formula for the future value of an ordinary annuity. Plugging in the given values, you will be able to pull out approximately $1,408.19 each month if you want to be able to take withdrawals for 25 years.
Explanation:
To determine how much you can pull out each month, you can use the formula for the future value of an ordinary annuity. The formula is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV is the future value of the annuity
P is the monthly withdrawal amount
r is the monthly interest rate (4% divided by 12)
n is the number of months (25 years multiplied by 12)
Plugging in the values, we get:
FV = P * [(1 + 0.04/12)^(25*12) - 1] / (0.04/12)
To find the monthly withdrawal amount (P), we need to solve for P. Rearranging the formula:
P = FV * (0.04/12) / [(1 + 0.04/12)^(25*12) - 1]
Now substitute the values back in and calculate P:
P = $500,000 * (0.04/12) / [(1 + 0.04/12)^(25*12) - 1]
Simplifying the equation gives us:
P ≈ $1,408.19
Therefore, you will be able to pull out approximately $1,408.19 each month if you want to be able to take withdrawals for 25 years.
3. Find the inverse Laplace transform of F(s) = (-4s-9) / (s^2 + 25-8) f(t) =
[tex]f(s)\Longrightarrow L^{-1}=\{\frac{-4s-9}{s^2+25-8}\}[/tex]
First dismantle,
[tex]L^{-1}=\{-\frac{4s}{s^2+25-8}-\frac{9}{s^2+25-8}\}[/tex]
Now use the linearity property of Inverse Laplace Transform which states,
For functions [tex]f(s),g(s)[/tex] and constants [tex]a, b[/tex] rule applies,
[tex]L^{-1}=\{a\cdot f(s)+b\cdot g(s)\}=aL^{-1}\{f(s)\}+bL^{-1}\{f(s)\}[/tex]
Hence,
[tex]-4L^{-1}\{\frac{s}{s^2+25-8}\}-9L^{-1}\{\frac{1}{s^2+25-8}\}[/tex]
The first part simplifies to,
[tex]
L^{-1}\{\frac{s}{s^2+25-8}\} \\
\frac{d}{dt}(\frac{1}{\sqrt{17}}\sin(t\sqrt{17})) \\
\cos(t\sqrt{17})
[/tex]
The second part simplifies to,
[tex]
L^{-1}\{\frac{1}{s^2+25-8}\} \\
\frac{1}{\sqrt{17}}\sin(t\sqrt{17})
[/tex]
And we result with,
[tex]\boxed{-4\cos(t\sqrt{17})-\frac{9}{\sqrt{17}}\sin(t\sqrt{17})}[/tex]
Hope this helps.
If you have any additional questions please ask. I made process of solving as quick as possible therefore you might be left over with some uncertainty.
Hope this helps.
r3t40
divide
(3x^2 + 9x + 7) divide by (x+2)
Answer:
The remainder is: 3x+3
The quotient is: 1
Step-by-step explanation:
We need to divide
(3x^2 + 9x + 7) by (x+2)
The remainder is: 3x+3
The quotient is: 1
The solution is attached in the figure below.
Answer:
[tex]3x+3+\frac{1}{x+2}[/tex]
Step-by-step explanation:
We are to divide the polynomial [tex]3x^2 + 9x + 7[/tex] by [tex]x+2[/tex].
For that, we will first divide the leading coefficient of the numerator [tex]\frac{3x^2}{x}[/tex] by the divisor.
So we get the quotient: [tex]3x[/tex] and will multiply the divisor [tex]x+2[/tex] by [tex]3x[/tex] to get [tex]3x^2+6x[/tex].
Next, we will subtract [tex]3x^2+6x[/tex] from [tex]3x^2 + 9x + 7[/tex] to get the remainder [tex]3x+7[/tex].
Therefore, we get [tex]3x+\frac{3x+7}{x+2}[/tex].
Now again, dividing the leading coefficient of the numerator by the divisor [tex]\frac{3x}{x}[/tex] to get quotient [tex]3[/tex].
Then we will multiply [tex]x+2[/tex] by [tex]3[/tex] to get [tex]3x+6[/tex].
Then, we will subtract [tex]3x+6[/tex] from [tex]3x+7[/tex] to get the new remainder [tex]1[/tex].
Therefore, [tex]\frac{3x^2 + 9x + 7}{x+2}=3x+3+\frac{1}{x+2}[/tex]
4.D.15 Consider a student loan of $17,500 at a fixed APR of 6% for 25 years. a. Calculate the monthly payment. b. Determine the total amount paid over the term of the loan. c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest a. The monthly payment is $ (Do not round until the final answer. Then round to the nearest cent as needed.) ess ibrary
The monthly payment for a $17,500 loan at 6% APR over 25 years is about $113.36. The total amount paid over the duration of the loan is $34,008, of which about 51.46% goes towards the principal and 48.54% towards interest.
Explanation:To answer this question, we need to use the formula for calculating the monthly payment for a loan, which is P[r(1 + r)^n]/[(1 + r)^n - 1], where P is the principal loan amount, r is the monthly interest rate (annual rate divided by 12), and n is the number of payments (years times 12).
a. Monthly payment:
First, we have to calculate r: 6% APR implies a yearly interest rate of 6%/12 = 0.005 per month. Plugging P = $17,500, r = 0.005, and n = 25*12 = 300 into the formula, we get the monthly payment of approximately $113.36.
b. Total amount paid:
The total amount paid over the duration of the loan is simply the monthly payment times the total number of payments, so $113.36*300 = $34,008.
c. Percentages of principal and interest:
The percentage paid towards the principal is the original loan amount divided by the total payment amount times 100, so $17,500/$34,008*100 = approximately 51.46%. Therefore, the percentage paid for interest is 100 - 51.46 = 48.54%.
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The monthly payment for the loan is approximately $114.08. Over 25 years, the total amount paid will be about $34,224. About 51.11% of this amount goes toward the principal and 48.89% goes toward interest.
To solve this problem, we need to calculate the monthly payment, total amount paid, and the percentages of principal and interest paid for a student loan of $17,500 at a fixed APR of 6% for 25 years.
a. Calculate the monthly payment
We use the formula for the monthly payment on an amortizing loan:
→ [tex]M = P [r(1 + r)^n] / [(1 + r)^{n - 1}][/tex]
where:
→ P = loan principal ($17,500)
→ r = monthly interest rate (annual rate / 12)
= 6% / 12
= 0.005
→ n = total number of payments (years * 12)
= 25 * 12
= 300
Substituting the values into the formula:
→ [tex]M = 17500 [0.005(1 + 0.005)^{300}] / [(1 + 0.005)^{300 – 1}][/tex]
→ M = 17500 [0.005(4.2918707)] / [4.2918707 – 1]
→ M = 17500 [0.02145935] / [3.2918707]
→ M = 375.53965 / 3.2918707
→ M ≈ $114.08
b. Determine the total amount paid over the term of the loan
→ Total amount paid = monthly payment * total number of payments
→ Total amount paid = 114.08 * 300
≈ $34,224
c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest
→ Principal: $17,500
→ Interest: Total amount paid - Principal
→ Interest = 34,224 - 17,500
≈ $16,724
→ Percentage toward principal = (Principal / Total amount paid) * 100
≈ (17500 / 34224) * 100
≈ 51.11%
→ Percentage toward interest = (Interest / Total amount paid) * 100
≈ (16724 / 34224) * 100
≈ 48.89%
Kristie has taken five tests in science class. The average of all five of Kristie's test scores is 94. The average of her last three test scores is 92. What is the average of her first two test scores?
Answer: The average of the first two test scores is 97.
Step-by-step explanation: Given that Kristie has taken five tests in science class. The average of all five of Kristie's test scores is 94 and the average of her last three test scores is 92.
We are to find the average score of her first two tests.
Let a1, a2, a3, a4 and a5 be teh scores of Kristle in first , second, third, fourth and fifth tests respectively.
Then, according to the given information, we have
[tex]\dfrac{a1+a2+a3+a4+a5}{5}=94\\\\\Rightarrow a1+a2+a3+a4+a5=94\times5\\\\\Rightarrow a1+a2+a3+a4+a5=470~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
and
[tex]\dfrac{a3+a4+a5}{3}=92\\\\\Rightarrow a3+a4+a5=92\times3\\\\\Rightarrow a3+a4+a5=276~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Subtracting equation (ii) from equation (i), we get
[tex](a1+a2+a3+a4+a5)-(a3+a4+a5)=470-276\\\\\Rightarrow a1+a2=194\\\\\Rightarrow \dfrac{a1+a2}{2}=\dfrac{194}{2}\\\\\Rightarrow \dfrac{a1+a2}{2}=97.[/tex]
Thus, the average of the first two test scores is 97.
Answer:
Thus, the average of the first two test scores is 97.
Step-by-step explanation:
Suppose consumers will demand 40 units of a product when the price is $12 per unit and 25 units when the price is $18 each. Find the demand equation assuming that it is linear. Find the price per unit when 30 units are demanded.
Answer: The price per unit is $48, when 30 units are demanded.
Step-by-step explanation:
Since we have given that
At price of $12 per unit, the number of units demanded = 40 units
At price of $18 per unit, the number of units demanded = 25 units.
So, the coordinates would be
(40,12) and (25,18)
As we know that x- axis denoted the quantity demanded.
y-axis denoted the price per unit.
So, the slope would be
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\m=\dfrac{18-12}{25-40}\\\\m=\dfrac{-6}{15}\\\\m=\dfrac{-2}{5}[/tex]
So, the equation would be
[tex]y-y_1=m(x-x_1)\\\\y-12=\dfrac{-2}{5}(x-40)\\\\5(y-12)=-2(x-40)\\\\5y-60=-2x+80\\\\5y+2x=80+60\\\\5y+2x=140[/tex]
So, if 30 units are demanded, the price per unit would be
[tex]5y=140+2x\\\\5y=140+2\times 30\\\\5y=140+60\\\\5y=240\\\\y=\dfrac{240}{5}\\\\y=\$48[/tex]
Hence, the price per unit is $48, when 30 units are demanded.
The linear demand equation is Qd = 100 - 5P. In this equation, if we want to find the price per unit when 30 units are demanded, substitute Qd with 30 to get P = 14.
Explanation:To find the linear demand equation, we can use the data given: consumers will buy 40 units of a product when the price is $12 per unit and 25 units when the price is $18 per unit. The general formula for a linear demand equation is Qd = a - bP, where Qd is the quantity demanded, P is the price per unit, a is the intercept, and b is the slope of the demand curve.
Let's use the two points (12, 40) and (18, 25) to formulate two equations with a and b as unknowns. We find that a = 100 and b = 5, therefore the demand equation is Qd = 100 - 5P. Now if we want to find the price per unit when 30 units are demanded, we just need to substitute Qd with 30 in the demand equation, hence P = 14.
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Several years ago, the mean height of women 20 years of age or older was 63.7 inches. Suppose that a random sample of 45 women who are 20 years of age or older today results in a mean height of 65.1 inches. (a) State the appropriate null and alternative hypotheses to assess whether women are taller today. (b) Suppose the P-value for this test is 0.16. Explain what this value represents. (c) Write a conclusion for this hypothesis test assuming an alphaequals0.10 level of significance.
Answer:
(a) State the appropriate null and alternative hypotheses to assess whether women are taller today.
Solution:
Definition of null hypothesis : The null hypothesis attempts to show that no variation exists between variables or that a single variable is no different than its mean. it is denoted
Alternative Hypothesis: In statistical hypothesis testing, the alternative hypothesis is a position that states something is happening, a new theory is true instead of an old one (null hypothesis).
We are given that The mean height of women 20 years of age or older was 63.7 inches.
So, null hypothesis : [tex]H_0: \mu=63.7[/tex]
Alternative Hypothesis : [tex]H_1: \mu>63.7[/tex]
b)The P-value for this test is 0.16.
Solution: The p-value represents the probability of getting a sample mean height of 65.1 inches.
c) Write a conclusion for this hypothesis test assuming an alpha equals 0.10 level of significance.
Solution:
[tex]\alpha = 0.10[/tex]
p- value = 0.16
[tex]p-value> \apha[/tex]
Since the p - value is high .
So, we will accept the null hypothesis
So, [tex]H_0: \mu=63.7[/tex]
Hence The mean height is 63.7 inches
The null hypothesis states that the mean height of women today is equal to the mean height several years ago. The P-value represents the probability of obtaining the observed sample mean if the null hypothesis is true. With an alpha level of 0.10, we fail to reject the null hypothesis, indicating no significant evidence to suggest that women today are taller.
Explanation:(a) Null hypothesis: The mean height of women 20 years of age or older today is equal to 63.7 inches. Alternative hypothesis: The mean height of women 20 years of age or older today is greater than 63.7 inches.
(b) The P-value represents the probability of obtaining a sample mean height of 65.1 inches or higher, given that the true mean height is 63.7 inches. A P-value of 0.16 indicates that there is a 16% chance of observing such a sample mean height even if the true mean height is 63.7 inches.
(c) Conclusion: Assuming an alpha level of 0.10, we fail to reject the null hypothesis. There is not enough evidence to conclude that women today are taller than before.
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Consider the region satisfying the inequalities.y ≤ e−x, y ≥ 0, x ≥ 0a) Find area of regionb) Find the volume of the solid generated by revolving the region about the x-axis.c) Find the volume of the solid generated by revolving the region about the y-axis.
Using the disk method, the volume is
[tex]\displaystyle\pi\int_0^\infty e^{-2x}\,\mathrm dx=\boxed{\frac\pi2}[/tex]
Alternatively, using the shell method, the volume is
[tex]\displaystyle2\pi\int_0^1y(-\ln y)\,\mathrm dy=\frac\pi2[/tex]
Revolving about the [tex]y[/tex]-axis:Using the shell method, the volume is
[tex]\displaystyle2\pi\int_0^\infty xe^{-x}\,\mathrm dx=\boxed{2\pi}[/tex]
Alternatively, using the disk method, the volume is
[tex]\displaystyle\pi\int_0^1(-\ln x)^2\,\mathrm dx=2\pi[/tex]
The area of the region is 1 square unit. The volume of the solid generated by revolving the region about the x-axis can be found by integrating π(y^2) dx from x = 0 to x = ∞.
Explanation:To find the area of the region, we need to find the intersection points between the two curves. In this case, the curves are y = e^(-x) and y = 0. Since y ≥ 0, the region will lie between the x-axis and the curve y = e^(-x). The intersection point is where y = 0, which occurs at x = 0. To find the area, we integrate y = e^(-x) from x = 0 to x = ∞:
A = ∫0∞ e^(-x) dx = [-e^(-x)]0∞ = -[e^0 - 0]
= -[1 - 0] = 1
The area of the region is 1 square unit.
To find the volume of the solid generated by revolving the region about the x-axis, we use the disk method. The radius of each disk is given by y = e^(-x), and the height of each disk is given by dx. The volume can be found by integrating π(y^2) dx from x = 0 to x = ∞:
V = π∫0∞ (e^(-x))^2 dx = π∫0∞ e^(-2x) dx
27-28 . Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection. 27. y = x, y = x 28. y = sinx, y = cos x, 0
Answer:
27.The angle between two given curves is [tex]0^{\circ}[/tex].
28.[tex]\theta=tan^{-1}(2\sqrt2)[/tex]
Step-by-step explanation:
27.We are given that two curves
y=x,y=x
We have to find the angle between the two curves
The angle between two curves is the angle between their tangent lines at the point of intersection
We know that the values of both curves at the point of intersection are equal
Let two given curves intersect at point [tex](x_1,y_1)[/tex]
Then [tex]y_1=x_1[/tex] because both curves are same
[tex]\frac{dy}{dx}=1[/tex]
[tex]m_1=1,m_2=1[/tex]
[tex]m_1(x_1)=1,m_2(x_1)=1[/tex]
Using formula of angle between two curves
[tex]tan\theta=\frac{m_1(x_0)-m_2(x_0)}{1+m_1(x_0)m_2(x_0)}[/tex]
[tex]tan\theta=\frac{1-1}{1+1}=\frac{0}{2}=0[/tex]
[tex]tan\theta=tan 0^{\circ}[/tex]
[tex]\theta=0^{\circ}[/tex]
Hence,the angle between two given curves is [tex]0^{\circ}[/tex].
28.y=sin x
y= cos x
By similar method we solve these two curves
Let two given curves intersect at point (x,y) then the values of both curves at the point are equal
Therefore, sin x = cos x
[tex]\frac{sin x}{cos x}=1[/tex]
[tex]tan x=1[/tex]
[tex]tan x=\frac{sin x}{cos x}[/tex]
[tex]tan x= tan \frac{\pi}{4}[/tex]
[tex]x=\frac{\pi}{4}[/tex]
Now, substitute the value of x then we get y
[tex]y= sin \frac{\pi}{4}=\frac{1}{\sqrt2}[/tex]
[tex] sin \frac{\pi}{4}=cos \frac{\pi}{4}=\frac{1}{\sqrt2}[/tex]
The values of both curves are same therefore, the point [tex](\frac{\pi}{4},\frac{1}{\sqrt2})[/tex] is the intersection point of two curves .
[tex]m_1=cos x[/tex]
At [tex] x=\frac{\pi}{4}[/tex]
[tex]m_1=\frac{1}{\sqrt2}[/tex]
and
[tex]m_2=-sin x=-\frac{1}{\sqrt2}[/tex]
Substitute the values in the above given formula
Then we get [tex]tan\theta =\frac{\frac{1}{\sqrt2}+\frac{1}{\sqrt2}}{1-\frac{1}{2}}[/tex]
[tex]tan\theta=\frac{\frac{2}{\sqrt 2}}{\frac{1}{2}}[/tex]
[tex]tan\theta=\frac{2}{\sqrt 2}\times 2[/tex]
[tex] tan\theta=\frac{4}{\sqrt2}=\frac{4}{\sqrt 2}\times\frac{\sqrt2}{\sqrt2}[/tex]
[tex]tan\theta=2\sqrt2[/tex]
[tex]\theta=tan^{-1}(2\sqrt2)[/tex]
Hence, the angle between two curves is [tex]tan^{-1}(2\sqrt2)[/tex].
I need the answer to this question.
1) Divide 704 days 11 hours by 29.
There are 24 hours to a day, so you can write this as
704 days, 11 hours = 704 + 11/24 days
Then dividing by 29 gives
704/29 + 11/696 days
We have
704 = 24*29 + 8
so that the time is equal to
24 + 8/29 + 11/696 days
24 + (192 + 11)/696 days
24 + 7/24 days
which in terms of days and hours is
24 days, 7 hours
Dagger Corporation uses direct labor-hours in its predetermined overhead rate. At the beginning of the year, the total estimated manufacturing overhead was $231,750. At the end of the year, actual direct labor-hours for the year were 17,500 hours, manufacturing overhead for the year was underapplied by $12,500, and the actual manufacturing overhead was $227,750. The predetermined overhead rate for the year must have been closest to:
1)$12.96
2)$12.30
3)$13.24
4)$11.43
Answer:
Predetermined overhead rate of the year = $12.3
Option 2 is correct.
Step-by-step explanation:
Let P = Predetermined overhead
Actual direct labor hours = 17,500
So, applied overhead = (17,500 *P)
Actual overhead = 227,750
Under applied overhead = 12,500
Applied Overhead = Actual overhead - Under applied overhead
Applied Overhead = 227,750 - 12500
Applied Overhead = 215,250
Using Formula:
215250 = (17,500 *P)
=> P = 215250/17500
P = 12.3
So, Predetermined overhead rate of the year = $12.3
Option 2 is correct.
\sum_{n=1}^{\infty } ((-1^n)/n)x^n
Find the interval of convergence.
Answer:
(-1,1).
Step-by-step explanation:
We need to calculate [tex]\lim_{n \to \infty}\frac{| a_{n+1}|}{| a_{n}|} = \frac{1}{R}[/tex] where R is the radius of convergence.
[tex]\lim_{n \to \infty}\frac{\frac{|(-1)^{n+1}|}{|n+1|}}{\frac{|(-1)^{n}|}{|n|}}[/tex]
[tex]\lim_{n \to \infty}\frac{\frac{1}{|n+1|}}{\frac{1}{|n|}}[/tex]
[tex]\lim_{n \to \infty}\frac{|n|}{|n+1|}[/tex]
Applying LHopital rule we obtaing that the limit is 1. So [tex]1=\frac{1}{R}[/tex] then R = 1.
As the serie is the form [tex](x+0)^{n}[/tex] we center the interval in 0. So the interval is (0-1,0+1) = (-1,1). We don't include the extrem values -1 and 1 because in those values the serie diverges.
If you are asked to provide a set of two or more numeric answers, separate them with commas. For example, to provide the year that Sputnik (the first satellite to be sent into orbit around the Earth) was launched and the year humans first walked on the Moon, you would enter 1957,1969 in the answer box.A rectangle has a length of 5.50 m and a width of 12.0 m. What are the perimeter and area of this rectangle?Enter the perimeter and area numerically separated by a comma. The perimeter should be given in meters and the area in square meters. Do not enter the units; they are provided to the right of the answer box.
Answer:
35, 66
Step-by-step explanation:
The perimeter is twice the sum of length and width:
2(5.50 m + 12.0 m) = 2(17.50 m) = 35.00 m
__
The area is the product of the length and width:
(5.50 m)(12.0 m) = 66.000 m^2
The perimeter of the rectangle is 35.0 m, and the area is 66.0 m², both calculated using standard geometry formulas and reported with three significant figures.
Explanation:To calculate the perimeter and area of a rectangle, we use the formulas: Perimeter = 2(length + width) and Area = length × width. In this case, the rectangle has a length of 5.50 m and a width of 12.0 m. Therefore, the perimeter is 2(5.50 m + 12.0 m) = 2(17.5 m) = 35.0 m. The area is 5.50 m × 12.0 m = 66.0 m².
It's important to express your answers with the correct number of significant figures and proper units. The length of 5.50 m has three significant figures, and the width of 12.0 m has three significant figures as well. Thus, our final answers for both perimeter and area should also be reported to three significant figures: 35.0 m for the perimeter and 66.0 m² for the area.
PLEASE HELP PRECALCULUS WILL MARK BRAINLIEST -SEE ATTACHMENT-
Answer:
Maximum is 8 while the minimum is -8.
Step-by-step explanation:
If we consider y=cos(x), the maximum is 1 and the minimum is -1.
This is the parent function of y=8cos(x) which has been vertically stretched by a factor of 8. So now the maximum of y=8cos(x) is 8 while the minimum of y=8cos(x) is -8.
How many 3 digit pass codes can be made from the digits 0 to 9 if the first number is not allowed to be a 0?
Answer:
total 900 pass codes can be made.
Step-by-step explanation:
Given situation is 3 digit pass codes are required made from the digits 0 to 9.
But the first number is not allowed to be a 0.
So for the third place we have 10 choices ( 0,1,2,3,4,5,6,7,8,9 )
For the second place we have 10 choices ( 0,1,2,3,4,5,6,7,8,9 )
But for the first place we have 9 choices ( 1,2,3,4,5,6,7,8,9 )
( 9 × 10 × 10 ) = 900
Therefore, total 900 pass codes can be made from the digits 0 to 9.
PLEASE HELP PRECALCULUS WILL MARK BRAINLIEST
Answer:
[tex]\dfrac{21\pi}{10},\ -\dfrac{19\pi}{10}[/tex]
Step-by-step explanation:
Any of the angles (in radians) π/10 +2kπ (k any integer) will be co-terminal with π/10. The angles listed in the answer above have k=1, k=-1.
_____
Comment on the last answer choice
The answer choices π/10+360° and π/10-360° amount to the same thing as the answer shown above, but use mixed measures. 1 degree is π/180 radians, so 360° is 2π radians. Then π/10+360° is fully equivalent to 21π/10 radians.
Samâs Auto Shop services and repairs a particular brand of foreign automobile. Sam uses oil filters throughout the year. The shop operates fifty-two weeks per year and weekly demand is 150 filters. Sam estimates that it costs $20 to place an order and his annual holding cost rate is $3 per oil filter. Currently, Sam orders in quantities of 650 filters. Calculate the total annual costs associated with Samâs current ordering policy
Total annual costs = $
the total annual costs associated with Sam's current ordering policy amount to $1,215.
The student has provided the necessary information to calculate the total annual costs associated with Sam's current ordering policy at his auto shop. Sam orders 650 oil filters at a time and the shop uses 150 filters per week. The key components involved in this calculation are the order cost, annual demand, holding cost, and the size of each order.
Total Annual Costs CalculationThe total annual demand (D) is the weekly demand (d) multiplied by the number of weeks per year:
D = d × 52
D = 150 filters/week × 52 weeks/year
D = 7,800 filters/year
The order cost (S) is given as $20 per order and the annual holding cost per unit (H) is $3 per filter. The size of each order (Q) is 650 filters.
The total annual ordering cost (AOC) can be calculated as the annual demand divided by the order size, multiplied by the order cost:
AOC = (D/Q) × S
AOC = (7,800/650) × $20
AOC = 12 × $20
AOC = $240
The total annual holding cost (AHC) can be calculated as the average inventory level (which is Q/2 for consistent orders) multiplied by the holding cost per unit:
AHC = (Q/2) × H
AHC = (650/2) × $3
AHC = 325 × $3
AHC = $975
The total annual costs (TAC) are the sum of the total annual ordering cost and the total annual holding cost:
TAC = AOC + AHC
TAC = $240 + $975
TAC = $1,215
Therefore, the total annual costs associated with Sam's current ordering policy amount to $1,215.
Suppose that 15% of people dont show up for a flight, and suppose that their decisions are independent. how many tickets can you sell for a plane with 144 seats and be 99% sure that not too many people will show up.
The book says to do this by using the normal distribution function and that the answer is selling 157 tickets.
Answer: 157 tickets
Explanation:
The people not showing up for the flight can be treated as from Binomial distribution.
The binomial distribution B(n, p) is approximately close to the normal i.e. N(np, np(1 − p)) for large 'n' and for 'p' and neither too close to 0 nor 1 .
Now, Let us assume 'n' = n
and we are given
p=0.15
So now B(n,0.15n) follows Normal distribution
u=n
[tex]\sigma^{2}[/tex] = 0.15n
We have to calculate P(X<144) with 99% accuracy
P(X<144) = P(Z<z)
where;
z= [tex](144-\bar{X})\div \sigma[/tex]
z score for 99% is 2.33
i.e.
[tex](144-\bar{X})\div \sigma = (144-n)/\sqrt{np} = 2.33\\\ (144-n)^{2} = \ np*2.33^{2}\\\ 20736 +n^{2} - 288n = 0.15n*5.43\\\ n^{2} - 288.81n + 20736=0[/tex]
solving this we will get one root nearly equal to 157 and other root as 133
Hence the answer is 157.
Approximate the number below using a calculator. Round your answer to three decimal places. el.3 a) 3.534 b) 3.669 c) 3.969 d) 2.040
Answer:
a. 3.53 b. 3.61 c. 3.91 d. 2.04
Step-by-step explanation:
a. 3.534 take 4 out
b. 3.669 nine is higher then 5 so, take six makes out of one
c. 3.969 nine is higher then 5 so, take six makes out of one
d. 2.040 take zero out
above that is the answer.... hope i helped this....
Find the equation of the following line and graph. Through (3,-10) perpendicular to 5x-y=9
bearing in mind that perpendicular lines have negative reciprocal slopes, let's find the slope of 5x - y = 9 then.
[tex]\bf 5x-y=9\implies -y=-5x+9\implies y=\stackrel{\stackrel{m}{\downarrow }}{5}x-9\leftarrow \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{5\implies \cfrac{5}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{1}{5}}\qquad \stackrel{negative~reciprocal}{-\cfrac{1}{5}}}[/tex]
so then, we're really looking for the equation of a line whose slope is -1/5 and runs through (3,-10).
[tex]\bf (\stackrel{x_1}{3}~,~\stackrel{y_1}{-10})~\hspace{10em} slope = m\implies -\cfrac{1}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-10)=-\cfrac{1}{5}(x-3)\implies y+10=-\cfrac{1}{5}x+\cfrac{3}{5} \\\\\\ y=-\cfrac{1}{5}x+\cfrac{3}{5}-10\implies y=-\cfrac{1}{5}x+\cfrac{53}{5}[/tex]
and it looks like the one in the picture below.
define the variables, write a system if equations corresponding to the problem, and solve the problem. 2. A group of four golfers pays $150 to play a round of golf. Of these four, one is a member of the club and three are nonmembers. Another group of golfers consists of two members and one nonmember and pays a total of $75. What is the cost for a member to play a round of golf, and what is the cost for a nonmember?
Answer: Cost of member = $15 and Cost of non member = $45
Step-by-step explanation:
a) Define variables.
Let x be the cost of member.
Let y be the cost of non member.
b) Write a system of equations:
According to question, we get that
x+3y=$150--------------(1)
2x+y=$75---------------(2)
Now, we need to calculate the cost for a member and a non member.
Using graphing method, we get that (15,45) is the solution set.
Hence, cost of member = $15 and Cost of non member = $45
Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a red card for the second card drawn, if the first card, drawn without replacement, was a diamond? Express your answer as a fraction or a decimal number rounded to four decimal places.
If the first card drawn is a diamond, the total number of cards reduces to 51 for the second draw. Out of these, 25 are red (13 hearts + 12 diamonds). So, the probability of drawing a red card on the second draw, if the first drawn card was a diamond, is 25/51 approximated to 0.4902.
Explanation:The subject of this question is probability in mathematics, specifically related to a scenario that involves sampling without replacement from a deck of playing cards. Firstly, let us familiarize ourselves with the composition of a standard deck of cards. It consists of 52 cards divided into four suits: clubs, diamonds, hearts, and spades. Clubs and spades are black cards, while diamonds and hearts are red cards. Each suit has 13 cards.
If the first card drawn is a diamond, the total count of cards in the deck reduces to 51 (because we are drawing without replacement), and, since a diamond card has been withdrawn, the count of remaining red cards is 25 (13 hearts and 12 remaining diamonds). Thus, to find the probability of drawing a red card on the second draw, we simply count the remaining red cards and divide by the remaining total cards, giving us a probability of 25/51 or approximately 0.4902 when rounded to four decimal places.
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The probability of choosing a red card for the second card drawn, if the first card was a diamond, is 0.4902.
Determine the total number of cards in a standard deck: A standard deck has 52 cards in total.So, the calculated probabilities are:
- The probability of drawing a diamond first: 0.25
- The probability of drawing a red card second, given that the first card drawn was a diamond: 0.4902
Therefore, the probability of choosing a red card for the second card drawn, if the first card was a diamond, is approximately 0.4902.
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = x3, y = 8, x = 0; about x = 9
Answer:
200π cubic units.
Step-by-step explanation:
Use the general method of integrating the area of the surface generated by an arbitrary cross section of the region taken parallel to the axis of revolution.
Here the axis x = 9 is parallel to the y-axis.
The height of one cylindrical shell = 8 - x^3.
The radius = 9 - x.
2
The volume generated = 2π∫ (8 - x^3) (9 - x) dx
0
= 2π ∫ ( 72 - 8x - 9x^3 + x^4) dx
2
= 2 π [ 72x - 4x^2 - 9x^4/4 + x^5 / 4 ]
0
= 2 π ( 144 - 16 - 144/4 + 32/4)
= 2 π * 100
= 200π.
The range of for y = 4/5 sin x for pi [tex]\leq[/tex] x [tex]\leq[/tex] 3pi/2 is
Choices:
4/5 [tex]\leq[/tex] y [tex]\leq[/tex] 1
-1 [tex]\leq[/tex] y[tex]\leq[/tex] 4/5
-4/5[tex]\leq[/tex] y [tex]\leq[/tex] 0
-4/5 [tex]\leq[/tex] y[tex]\leq[/tex] 4/5
Answer:
see attachment
Step-by-step explanation:
The range of [tex]\(y = \frac{4}{5}\sin x\)[/tex] for [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex] is [tex]\(-\frac{4}{5} \leq y \leq 0\)[/tex] ( Option C).
To find the range of [tex]\(y = \frac{4}{5}\sin x\)[/tex] for [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex], we need to determine the minimum and maximum values of sin x in the given interval and then scale them using [tex]\(\frac{4}{5}\)[/tex].
In the interval [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex], the sine function is negative since it corresponds to the third and fourth quadrants on the unit circle. The minimum value of sin x in this interval is -1, and the maximum value is 0.
Now, scale these values using [tex]\(\frac{4}{5}\)[/tex]:
[tex]\(-1 \times \frac{4}{5} = -\frac{4}{5}\) (minimum)\\\\\(0 \times \frac{4}{5} = 0\) (maximum)[/tex]
Therefore, the range of [tex]\(y = \frac{4}{5}\sin x\) for \(\pi \leq x \leq \frac{3\pi}{2}\)[/tex] is [tex]\(-\frac{4}{5} \leq y \leq 0\)[/tex]. The correct choice is:
-4/5 <= y <= 0
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An instructor at a major research university occasionally teaches summer session and notices that that there are often students repeating the class. Out of curiosity, she designs a random sample of students enrolled in summer sessions and counts the number repeating a class. She counts 105 students in the sample, of which 19 are repeating the class. She decides a confidence interval provides a good estimate of the proportion of students repeating a class. She wants a 95% confidence interval with a margin of error at most ????=0.025m=0.025 . She has no idea what the true proportion could be. How large a sample should she take? 250 1537 1500 400
Answer: 1537
Step-by-step explanation:
Given : Margin of error : [tex]E=0.025[/tex]
Significance level : [tex]\alpha=1-0.95=0.05[/tex]
Critical value : [tex]z_{\alpha/2}=z_{0.025}=1.96[/tex]
The formula to calculate the sample size if prior estimate pf population proportion does not exist :-
[tex]n=0.25(\dfrac{z_{\alpha/2}}{E})^2\\\\\Rightarrow\ n=0.25(\dfrac{1.96}{0.025})^2\\\\\Rightarrow\ n=1536.64\approx1537[/tex]
Hence, she should take a sample with minimum size of 1537 .
Final answer:
To estimate the proportion of students repeating a class with a 95% confidence level and a margin of error of 0.025, the instructor would need a sample size of at least 1537 students.
Explanation:
To calculate the sample size needed for constructing a 95% confidence interval for the proportion of students who are repeating a class with a specified margin of error, we can use the formula for sample size in a proportion:
n = (Z² × p × (1-p)) / E²
Where:
Substituting the values, we get:
Thus, the calculation is:
n = (1.96² × 0.5 × (1 - 0.5)) / 0.025²
n = (3.8416 × 0.25) / 0.000625
n = 0.9604 / 0.000625
n = 1536.64
As we cannot have a fraction of a person, we would round up to the next whole number.
Therefore, the instructor would need to sample at least 1537 students.