Answer:
length: 2sqrt(2)
width: 4
Step-by-step explanation:
Let the x-coordinate of the right lower vertex be x. Then the x-coordinate of the left lower vertex is -x. The distance between x and -x is 2x, so the lower side of the rectangle, that rests on the x-axis, measures 2x. The length of the rectangle is 2x.
We now use the equation of the parabola to find the y-coordinate of the upper vertices. y = 6 - 2^x. If you plug in x for x, naturally you get y = 6 - x^2, so the y-coordinates of the two upper vertices are 6 - x^2. Since the y-coordinates of the lower vertices is 0, the vertical sides of the rectangle have length 6 - x^2. The width of the rectangle is 6 - x^2.
We have a rectangle with length 2x and width 6 - x^2.
Now we can find the area of the rectangle.
area = length * width
A = 2x(6 - x^2)
A = 12x - 2x^3
To find a maximum value of x, we differentiate the expression for the area with respect to x, set the derivative equal to zero, and solve for x.
dA/dx = 12 - 6x^2
We set the derivative equal to zero and solve for x.
12 - 6x^2 = 0
2 - x^2 = 0
x^2 = 2
x = +/- sqrt(2)
length = 2x = 2 * sqrt(2) = 2sqrt(2)
width = 6 - x^2 = 6 - (sqrt(2))^2 = 6 - 2 = 4
Final answer:
The greatest possible area of a rectangle inscribed under the parabola y = 6 - x² is achieved with dimensions 2√3 (width) by 3 (height), obtained by optimizing the area function using calculus.
Explanation:
To find the dimensions of the rectangle with the greatest possible area that is inscribed with its base on the x-axis and its upper corners on the parabola y = 6 - x², we apply the optimization technique in calculus.
Let the x-coordinate of the rectangle's upper corners be x, which means that the width of the rectangle is 2x (since the parabola is symmetric with respect to the y-axis). The height of the rectangle is given by the y-coordinate on the parabola, which is 6 - x².
Therefore, the area A of the rectangle can be expressed as A = 2x(6 - x²). To find the maximum area, we take the derivative of A with respect to x and set it to zero, solving for x.
A'(x) = 2(6 - x²) - 4x²(x). Setting A'(x) = 0 gives us x = √3. The maximum area occurs when x = √3, so the width of the rectangle will be 2√3 and the height will be 6 - 3, which simplifies to 3.
Hence, the rectangle with the greatest possible area has dimensions of 2√3 by 3.
Use the Newton-Raphson method to find the root of the equation f(x) = In(3x) + 5x2, using an initial guess of x = 0.5 and a stopping criterion of &= 0.1%. Report your answer to 3 decimal places. What is the estimate for the root? What is the approximate relative error? %
Answer with explanation:
The equation which we have to solve by Newton-Raphson Method is,
f(x)=log (3 x) +5 x²
[tex]f'(x)=\frac{1}{3x}+10 x[/tex]
Initial Guess =0.5
Formula to find Iteration by Newton-Raphson method
[tex]x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}\\\\x_{1}=x_{0}-\frac{f(x_{0})}{f'(x_{0})}\\\\ x_{1}=0.5-\frac{\log(1.5)+1.25}{\frac{1}{1.5}+10 \times 0.5}\\\\x_{1}=0.5- \frac{0.1760+1.25}{0.67+5}\\\\x_{1}=0.5-\frac{1.426}{5.67}\\\\x_{1}=0.5-0.25149\\\\x_{1}=0.248[/tex]
[tex]x_{2}=0.248-\frac{\log(0.744)+0.30752}{\frac{1}{0.744}+10 \times 0.248}\\\\x_{2}=0.248- \frac{-0.128+0.30752}{1.35+2.48}\\\\x_{2}=0.248-\frac{0.17952}{3.83}\\\\x_{2}=0.248-0.0468\\\\x_{2}=0.2012[/tex]
[tex]x_{3}=0.2012-\frac{\log(0.6036)+0.2024072}{\frac{1}{0.6036}+10 \times 0.2012}\\\\x_{3}=0.2012- \frac{-0.2192+0.2025}{1.6567+2.012}\\\\x_{3}=0.2012-\frac{-0.0167}{3.6687}\\\\x_{3}=0.2012+0.0045\\\\x_{3}=0.2057[/tex]
[tex]x_{4}=0.2057-\frac{\log(0.6171)+0.21156}{\frac{1}{0.6171}+10 \times 0.2057}\\\\x_{4}=0.2057- \frac{-0.2096+0.21156}{1.6204+2.057}\\\\x_{4}=0.2057-\frac{0.0019}{3.6774}\\\\x_{4}=0.2057-0.0005\\\\x_{4}=0.2052[/tex]
So, root of the equation =0.205 (Approx)
Approximate relative error
[tex]=\frac{\text{Actual value}}{\text{Given Value}}\\\\=\frac{0.205}{0.5}\\\\=0.41[/tex]
Approximate relative error in terms of Percentage
=0.41 × 100
= 41 %
×
_+4=20
6
Solve the following equation. Then place the correct number in the box provided.
Answer:
Step-by-step explanation:
X/6 +4 = 20
X/6 = 20 - 4
X/6 = 16
X = 16/6
How do you solve this system by substitution?
Answer is 'a'.(4;8;8)
All the details are provided in the attachment; the answer is marked with green colour.
In how many ways can 4 married couples attending a concert be seated in a row of 8 seats if: a) There are no restrictions? b) Each married couple is seated together?
Answer:
a) 40,320
b) 384
Step-by-step explanation:
Given,
The total number of seats = 8,
Also, these 8 seats are occupied by 4 married couples or 8 people,
a) Thus, if there is no restrictions of seating ( that is any person can seat with any person ),
Then, the total number of arrangement = 8 ! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
= 40320,
b) if each married couple is seated together,
Then, the 4 couples can seat in 4 pair of seats,
Also, in a pair of seats a couple can choose any of the two seats,
So, the total number of arrangement
[tex]=4! \times 2^4[/tex]
[tex]=24\times 16[/tex]
[tex]=384[/tex]
In 1898, L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.61. (a) What is the probability of more than 1 death in a corps in a year
Answer: Hence, Probability of more than 1 death in a corps in a year is 0.126.
Step-by-step explanation:
Since we have given that
Mean for a poisson distribution (λ) = 0.61
Number of years = 20 years
We need to find the probability of more than 1 death in a corps in a year.
P(X>1)=1-P(X=0)-P(X=1)
Here,
[tex]P(X=0)=\dfrac{e^{-0.61}(0.61)^0}{0!}=0.543\\\\and\\\\P(X=1)=\dfrac{e^{-0.61}(0.61)}{1}=0.331[/tex]
So,
P(X>1)=1-0.543-0.331=0.126
Hence, Probability of more than 1 death in a corps in a year is 0.126.
Using the Poisson distribution with a mean of 0.61, we calculate the probability of 0 or 1 death and subtract that from 1 to get the probability of more than 1 death in a Prussian cavalry corps in a year.
Explanation:Calculating Probability Using the Poisson DistributionBased on L. J. Bortkiewicz's study, the number of soldiers killed by horse kicks in the Prussian cavalry follows a Poisson distribution with a mean (λ) of 0.61. To calculate the probability of more than one death in a corps in a year, we use the Poisson probability formula:
P(X > k) = 1 - P(X ≤ k)
Where P(X > k) is the probability of having more than k events (in this case, deaths), and P(X ≤ k) is the probability of k or fewer events. In this scenario, k equals 1. So, we need to calculate the probability of 0 or 1 death and subtract from 1 to get the probability of more than 1 death.
Using the Poisson probability formula:
Calculate P(X = 0) and P(X = 1).Add P(X = 0) and P(X = 1) to get P(X ≤ 1).Subtract P(X ≤ 1) from 1 to obtain P(X > 1).Let's calculate:
P(X = 0) = (e^(-0.61))*(0.61^0)/(0!) = e^(-0.61)P(X = 1) = (e^(-0.61))*(0.61^1)/(1!) = 0.61*e^(-0.61)P(X ≤ 1) = P(X = 0) + P(X = 1)P(X > 1) = 1 - P(X ≤ 1)The resulting calculation will give us the probability of more than one death due to horse kicks in a Prussian cavalry corps within one year.
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Evaluate the Expression : (3+3)^2 / 10-4×3
Simplify 3 + 3 to 6
6^2/10 - 4 × 3
Simplify 4 × 3 to 12
6^2/10 - 12
Simplify 10 - 12 to -2
6^2/-2
Simplify 6^2 to 36
36/-2
Move the negative sign to the left
-36/2
Simplify 36/2 to 18
= -18
Factoring out the GCF
8m^2n^3-24m^2n^2 +4m^3 n
Answer:
[tex]\large\boxed{8m^2n^3-24m^2n^2+4m^3n=4m^2n(2n^2-6n+m)}[/tex]
Step-by-step explanation:
[tex]8m^2n^3-24m^2n^2+4m^3n\\\\8m^2n^3=\boxed{(2)}\boxed{(2)}(2)\boxed{(m)}\boxed{(m)}\boxed{(n)}(n)(n)\\\\24m^2n^2=\boxed{(2)}\boxed{(2)}(2)(3)\boxed{(m)}\boxed{(m)}\boxed{(n)}(n)\\\\4m^3n=\boxed{(2)}\boxed{(2)}\boxed{(m)}\boxed{(m)}(m)\boxed{(n)}\\\\8m^2n^3-24m^2n^2+4m^3n\\\\=\boxed{(2)}\boxed{(2)}\boxed{(m)}\boxed{(m)}\boxed{(n)}\bigg((2)(n)(n)-(2)(3)(n)+(m)\bigg)\\\\=4m^2n(2n^2-6n+m)[/tex]
2x + 1 < 5
Solve the following inequality. Then place the correct number in the box provided.
Answer:
[tex]\boxed{x<2}[/tex]
Step-by-step explanation:
You subtract by 1 from both sides of equation.
[tex]2x+1-1<5-1[/tex]
Simplify.
[tex]5-1=4[/tex]
[tex]2x<4[/tex]
Divide by 2 from both sides of equation.
[tex]\frac{2x}{2}<\frac{4}{2}[/tex]
Simplify, to find the answer.
[tex]4\div2=2[/tex]
X<2 is the correct answer.
Answer: [tex]x<2[/tex]
Step-by-step explanation:
Given the inequality [tex]2x + 1 < 5[/tex] you can follow this procedure to solve it:
The first step is to subtract 1 from both sides on the inequaltity.
[tex]2x + 1-(1) < 5-(1)\\\\2x < 4[/tex]
Now, the second and final step is to divide both sides of the inequality by 2. Therefore, you get this result:
[tex]\frac{2x}{2}<\frac{4}{2} \\\\(1)x<2\\\\x<2[/tex]
Quadrilateral ABCD has vertices A(-3, 4), B(1, 3), C(3, 6), and D(1, 6). Match each set of vertices of quadrilateral EFGH with the transformation that shows it is congruent to ABCD. E(-3, -4), F(1, -3), G(3, -6), and H(1, -6) a translation 7 units right E(-3, -1), F(1, -2), G(3, 1), and H(1, 1) a reflection across the y-axis E(3, 4), F(-1, 3), G(-3, 6), and H(-1, 6) a reflection across the x-axis E(4, 4), F(8, 3), G(10, 6), and H(8, 6)
Answer:
The set of vertices of quadrilateral EFGH with the transformation 7 units right is E(4 , 4) , F(8 , 3) , G(10 , 6) , and H(8 , 6)
The set of vertices of quadrilateral EFGH with a reflection across the y-axis is E(3 , 4) , F(-1 , 3) , G(-3 , 6) , and H(-1 , 6)
The set of vertices of quadrilateral EFGH with a reflection across the x-axis is E(-3 , -4) , F(1 , -3) , G(3 , -6) , and H(1 , -6)
Step-by-step explanation:
Lets revise some transformation
- If point (x , y) reflected across the x-axis
then Its image is (x , -y)
- If point (x , y) reflected across the y-axis
then Its image is (-x , y)
- If the point (x , y) translated horizontally to the right by h units
then its image is (x + h , y)
- If the point (x , y) translated horizontally to the left by h units
then its image is (x - h , y)
* Now lets solve the problem
- The vertices of the quadrilateral ABCD are:
A = (-3 , 4) , B = (1 , 3) , C = (3 , 6) , D = (1 , 6)
- The quadrilateral ABCD translated 7 units right to form
quadrilateral EFGH
- We add each x-coordinates in ABCD by 7
∵ A = (-3 , 4)
∴ E = (-3 + 7 , 4) = (4 , 4)
∵ B = (1 , 3)
∴ F = (1 + 7 , 3) = (8 , 3)
∵ C = (3 , 6)
∴ G = (3 + 7 , 6) = (10 , 6)
∵ D = (1 , 6)
∴ H = (1 + 7 , 6) = (8 , 6)
* The set of vertices of quadrilateral EFGH with the transformation
7 units right is E(4 , 4) , F(8 , 3) , G(10 , 6) , and H(8 , 6)
- The quadrilateral ABCD reflected across the y-axis to form
quadrilateral EFGH
- We change the sign of the x-coordinate
∵ A = (-3 , 4)
∴ E = (3 , 4)
∵ B = (1 , 3)
∴ F = (-1 , 3)
∵ C = (3 , 6)
∴ G = (-3 , 6)
∵ D = (1 , 6)
∴ H = (-1 , 6)
* The set of vertices of quadrilateral EFGH with a reflection across the
y-axis is E(3 , 4) , F(-1 , 3) , G(-3 , 6) , and H(-1 , 6)
- The quadrilateral ABCD reflected across the x-axis to form
quadrilateral EFGH
- We change the sign of the y-coordinate
∵ A = (-3 , 4)
∴ E = (-3 , -4)
∵ B = (1 , 3)
∴ F = (1 , -3)
∵ C = (3 , 6)
∴ G = (3 , -6)
∵ D = (1 , 6)
∴ H = (1 , -6)
* The set of vertices of quadrilateral EFGH with a reflection across the
x-axis is E(-3 , -4) , F(1 , -3) , G(3 , -6) , and H(1 , -6)
Answer:
here is the answer
Step-by-step explanation:
Find the solution of IVP for the differential equation (x+2)^2e^y when y(1) =0 y'= dy/dx=(x+2)^2.e^y; y(1)=0
Answer:
The solution is [tex]y\:=-\ln(-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3})[/tex]
Step-by-step explanation:
We need to find the solution of IVP for differential equation [tex]\frac{dy}{dx}=(x+2)^{2}e^{y}[/tex] when [tex]y(1)=0[/tex]
[tex]\mathrm{First\:order\:separable\:Ordinary\:Differential\:Equation}[/tex]
[tex]\mathrm{A\:first\:order\:separable\:ODE\:has\:the\:form\:of}\:N\left(y\right)\cdot y'=M\left(x\right)[/tex]
[tex]\mathrm{Rewrite\:in\:the\:form\:of\:a\:first\:order\:separable\:ODE}[/tex]
[tex]\frac{1}{e^y}y'\:=\left(x+2\right)^2[/tex]
[tex]N\left(y\right)\cdot y'\:=M\left(x\right)[/tex]
[tex]N\left(y\right)=\frac{1}{e^y},\:\quad M\left(x\right)=\left(x+2\right)^2[/tex]
[tex]\mathrm{Solve\:}\:\frac{1}{e^y}y'\:=\left(x+2\right)^2[/tex]
[tex]\frac{1}{e^y}y'\:=x^{2}+4+4x[/tex]
Integrate both the sides with respect to dx
[tex]\int\frac{1}{e^y}y'dx\:=\intx^{2}dx+4\int dx+4\int x dx[/tex]
[tex]\int e^{-y}dy\:=\intx^{2}dx+4\int dx+4\int x dx[/tex]
[tex]-\frac{1}{e^{y}}\:=\frac{x^{3}}{3}+4x+4\frac{x^{2}}{2}+c_1[/tex]
[tex]-\frac{1}{e^{y}}\:=\frac{x^{3}}{3}+4x+2x^{2}+c_1[/tex]
Since, IVP is y(1)=0
put x=1 and y=0 in above equation
[tex]-\frac{1}{e^{0}}\:=\frac{1^{3}}{3}+4(1)+2(1)^{2}+c_1[/tex]
[tex]-1\:=\frac{1}{3}+4+2+c_1[/tex]
[tex]-1\:=\frac{19}{3}+c_1[/tex]
add both the sides by [tex]-\frac{19}{3}[/tex]
[tex]-1-\frac{19}{3}\:=\frac{19}{3}-\frac{19}{3}+c_1[/tex]
[tex]-1-\frac{19}{3}\:=c_1[/tex]
[tex]-\frac{22}{3}\:=c_1[/tex]
so,
[tex]-\frac{1}{e^{y}}\:=\frac{x^{3}}{3}+4x+2x^{2}-\frac{22}{3}[/tex]
Multiply both the sides by '-1'
[tex]\frac{1}{e^{y}}\:=-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3}[/tex]
[tex]e^{-y}\:=-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3}[/tex]
Take natural logarithm both the sides,
[tex]-y\:=\ln(-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3})[/tex]
Multiply both the sides by '-1'
[tex]y\:=-\ln(-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3})[/tex]
Therefore, the solution is [tex]y\:=-\ln(-\frac{x^{3}}{3}-4x-2x^{2}+\frac{22}{3})[/tex]
A car company claims that the mean gas mileage for its luxury sedan is at least 24 miles per gallon. A random sample of 7 cars has a mean gas mileage of 23 miles per gallon and a standard deviation of 1.1 miles per gallon. At α=0.05, can you support the company’s claim assuming the population is normally distributed?
Answer:
a) t sampling distribution because B the population is normal, and standard deviation is unknown
b) H0: mu <= 21
HA mu > 21
alpha = 0.05
t critical value at 4 df and alpha 0.05 is 2.132
The rejection region is t > 2.132
t = (xbar - µ)/(s/√n)
t = (19 - 21 )/(4/√5)
t = -2 / (4/2.2361)
t = -1.118
t does not fall into the rejection region, so we have insufficient evidence to reject the null hypothesis. The claim cannot be verified.
We tested the manufacturer's claim that the mean mpg is greater than 21, at alpha = 0.05. We used a one-tailed one-sample t-test (4 df). We placed the rejection region in the right tail of the t-distribution because we were only interested in the claim that the mileage was more than 21. The test result showed that the claim could not be validated. The sample mean was 19, which was less than the claim, so no calculations were needed to reject the null hypothesis. We were not able to find that the mean was statistically greater than 21.
Based on hypothesis testing in statistics, there isn't enough evidence to support the car company’s claim that the average gas mileage for its luxury sedan is at least 24 miles per gallon.
Explanation:This question involves the use of hypothesis testing in statistics. The null hypothesis for this test is that the mean gas mileage is at least 24 miles per gallon (μ >= 24), and the alternative hypothesis is that the mean gas mileage is less than 24 miles per gallon (μ < 24).
With a calculated sample mean of 23 miles per gallon and a sample standard deviation of 1.1 miles per gallon for 7 cars, we use the standard error formula SE = σ/√n = 1.1/√7 = 0.415 to calculate the standard error. The t value is then calculated as (X - μ) / SE = (23 - 24) / 0.415 = -2.41.
Using a t-distribution table, we find that the critical value for a one-tailed test with degrees of freedom = n - 1 = 6 and α = 0.05 is -1.943. Since our calculated t value (-2.41) is less than the critical value (-1.943), we reject the null hypothesis. Therefore, we cannot support the company’s claim that the mean gas mileage for its luxury sedan is at least 24 miles per gallon.
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1. If $18,000 is invested at 6% compounded monthly, what is the amount after 7 years?
Suppose more than kn marbles are distributed into n jars. Prove that one jar will contain at least ? k + 1 marbles.
Answer:yes
Step-by-step explanation:because there are more marbles than jars
Captain Ralph is in trouble near the sunny side of Mercury. The temperature of the ship's hull when he is at location (x, y, z) will be given by T (x, y, z) = e−x2 − 2y2 − 3z2, where x, y, and z are measured in meters. He is currently at (1, 1, 1). (a) In what direction should he proceed in order to decrease the temperature most rapidly?
The maximum rate of change occurs in the direction of the gradient vector at (1, 1, 1).
[tex]T(x,y,z)=e^{-x^2-2y^2-3z^2}\implies\nabla T(x,y,z)=\langle-2x,-4y,-6z\rangle e^{-x^2-2y^2-3z^2}[/tex]
At (1, 1, 1), this has a value of
[tex]\nabla T(1,1,1)=\langle-2,-4,-6\rangle e^{-6}[/tex]
so the captain should move in the direction of the vector [tex]\langle-1, -2, -3\rangle[/tex] (which is a vector pointing in the same direction but scaled down by a factor of [tex]2e^{-6}[/tex]).
The direction Captain Ralph should proceed in order to decrease the temperature most rapidly is towards the direction of the steepest temperature decrease gradient. This direction is given by the negative gradient of the temperature function.
In this case, the negative gradient of T(x, y, z) = e^(-x^2 - 2y^2 - 3z^2) at the point (1, 1, 1) would be (-2e^(-6), -4e^(-6), -6e^(-6)).
Therefore, Captain Ralph should proceed in the direction (-2e^(-6), -4e^(-6), -6e^(-6)) to decrease the temperature most rapidly at his current location.
8.) Out of 600 employees, only 270 are happy with their pay. What percent is this? (Please show your work.) *
For this case we can raise a rule of three:
600 employees -------------> 100%
270 employees -------------> x
Where the variable "x" represents the percentage of employees who are satisfied with their salary. So, we have:
[tex]x = \frac {270 * 100} {600}\\x = 45[/tex]
Thus, 45% of employees are satisfied with their salary.
Answer:
45%
ANSWER
[tex]45\%[/tex]
EXPLANATION
The total number of employees is 600.
The number of employees who are happy with their pay is 270.
The percentage of employees who are happy with their pay is the number who are happy with their pay divided by total number of employees times 100%
[tex] \frac{270}{600} \times 100\%[/tex]
This simplifies to
[tex]45\%[/tex]
A scientist mixes water (containing no salt) with a solution that contains 30% salt. She wants to obtain 180 ounces of a mixture that is 15% salt. How many ounces of water and how many ounces of the 30% salt solution should she use?
Answer:
water : 90 ounces
salt solution: 90 ounces
Step-by-step explanation:
Call w the amount of water and call the solution containing 30% salt.
We want to get 180 ounces of a mixture with 15% salt.
So:
The amount of mixture will be:
[tex]w + s = 180[/tex]
the amount of salt will be
[tex]0w + 0.3s = 180 * 0.15[/tex]
[tex]0.3s = 27[/tex]
[tex]s = 90\ ounces[/tex]
Now we substitute the value of s in the first equation and solve for w
[tex]w + 90 = 180[/tex]
[tex]w = 90\ ounces[/tex]
<c is opposite
please help me find the answer thank you :)
Answer:
A) [tex]\angle C \text{ is opposite } \overline{AB}[/tex]
Step-by-step explanation:
We need to name the side of the triangle that is across from angle C. A side of a triangle, if not otherwise given, is named based on the two points that form it. In this case, the side is formed by points B and C, so the side is called [tex]\overline{BC}[/tex].
An owner of a key rings manufacturing company found that the profit earned (in thousands of dollars) per day by selling n number of key rings is given by , where n is the number of key rings in thousands. Find the number of key rings sold on a particular day when the total profit is $5,000.
n^2-2n-3
Answer:
The number of key rings sold on that day is 4000 key rings
Step-by-step explanation:
* Lets explain the information in the problem
- The profit earned (in thousands of dollars) per day by selling n number
of key rings is given by the function P(n) = n² - 2n - 3, where n is the
number of key rings in thousands and P is the profit in thousands
for one day
- On a particular day the total profit is $5,000
∵ 5000 = 5 in thousands
∵ The function P(n) is the profit of n key ring in thousands
∴ P(n) = 5
- Lets solve the function to find the number of key rings
∵ P(n) = n² - 2n - 3
∴ 5 = n² - 2n - 3 ⇒ subtract 5 from both sides
∴ 0 = n² - 2n - 8 ⇒ factorize it
∵ n² = n × n ⇒ 1st terms in the 2 brackets
∵ -8 = -4 × 2 ⇒ 2nd terms in the 2 brackets
∵ n × -4 = -4n ⇒ nears
∵ n × 2 = 2n ⇒ extremes
∵ -4n + 2n = -2n ⇒ the middle term
∴ (n - 4)(n + 2) = 0 ⇒ equate each bracket by 0 to find n
∴ n - 4 = 0 ⇒ add 4 to both sides
∴ n = 4 key ring in thousands = 4000 key rings
- OR
∴ n + 2 = 0 ⇒ subtract 2 from both sides
∴ n = -2 ⇒ we will refused this value because number of key rings
must be positive
∴ The number of key rings sold on that day is 4000 key rings
To find the number of key rings sold on a particular day when the total profit is $5,000, we need to solve the given equation for n.
Explanation:The owner of a key rings manufacturing company found that the profit earned (in thousands of dollars) per day by selling n number of key rings is given by the equation P(n) = n^2-2n-3. To find the number of key rings sold on a particular day when the total profit is $5,000, we need to solve the equation P(n) = 5000 for n.
Step 1: Set the equation equal to 5000: n^2-2n-3 = 5000.
Step 2: Rearrange the equation and set it equal to zero: n^2-2n-5003 = 0.
Step 3: Solve the quadratic equation using factoring, completing the square, or the quadratic formula to find the values of n.
Step 4: The solutions will give us the possible values of n, representing the number of key rings sold on the particular day when the total profit is $5,000.
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A sample of size n=10n=10 is drawn from a population. The data is shown below.
138.8
116.4
97
114.2
138.8
100.4
128.8
138.8
101.9
117.6
What is the range of this data set?
range =
What is the standard deviation of this data set? (Remember, it is a sample.) Please report the answer with appropriate rounding, rounding to two more decimal places than the original data. Please, please, please do not calculate the value by hand.
stdev =
Answer:
The range of this data set is 41.8
The standard deviation of the data set is 16.42
Step-by-step explanation:
* Lets read the information and use it to solve the problem
- There is a sample of size n = 10, is drawn from a population
- The data are: 97 , 100.4 , 101.9 , 114.2 , 116.4 , 117.6 , 128.8 , 138.8 ,
138.8 , 138.8
- The range is the difference between the largest number and
the smallest number
∵ The largest number is 138.8
∵ The smallest number is 97
∴ The range = 138.8 - 97 = 41.8
* The range of this data set is 41.8
- Lets explain how to find the standard deviation
# Step 1: find the mean of the data set
∵ The mean = the sum of the data ÷ the number of the data
∵ The data set is 97 , 100.4 , 101.9 , 114.2 , 116.4 , 117.6 , 128.8 , 138.8 ,
138.8 , 138.8
∵ Their sum = 97 + 100.4 + 101.9 + 114.2 + 116.4 + 117.6 + 128.8 + 138.8 +
138.8 + 138.8 = 1192.7
∵ n = 10
∴ The mean = 1192.7 ÷ 10 = 119.27
# Step 2: subtract the mean from each data and square the answer
∴ (97 - 119.27)² = 495.95
∴ (100.4 - 119.27)² = 356.08
∴ (101.9 - 119.27)² = 301.72
∴ (114.2 - 119.27)² = 25.70
∴ (116.4 - 119.27)² = 8.24
∴ (117.6 - 119.27)² = 2.79
∴ (128.8 - 119.27)² = 90.82
∴ (138.8 - 119.27)² = 381.42
∴ (138.8 - 119.27)² = 381.42
∴ (138.8 - 119.27)² = 381.42
# Step 3: find the mean of these squared difference
∵ A Sample: divide by n - 1 when calculating standard deviation of
a sample
∵ The mean = the sum of the data ÷ (the number of the data - 1)
∵ The sum = 495.95 + 356.08 + 301.72 + 25.70 + 8.24 + 2.79 + 90.82 +
381.42 + 381.42 + 381.42 = 2425.56
∴ The mean = 2425.56 ÷ (10 - 1) = 269.51
# Step 4: the standard deviation is the square root of this mean
∴ The standard deviation = √(269.51) = 16.416658 ≅ 16.42
* The standard deviation of the data set is 16.42
The sum of Eli’s age and Cecil’s age is 60. Six years ago, Eli was three times old as Cecil. Find Eli’s age now
Answer:
Eli's age = 42 years
Step-by-step explanation:
Let x be Eli's age and y be Cecil's age
So,
According to the statement given
x+y=60 eqn 1
Eli's age 6 years ago = x-6
Cecil's age 6 years ago = y-6
So according to the given statement
x-6 = 3(y-6)
x-6 = 3y - 18
x-3y = -18+6
x-3y= -12 eqn 2
Subtracting eqn 2 from eqn 1
x+y - (x-3y) = 60 - (-12)
x+y-x+3y = 60+12
4y = 72
y = 18
Cecil's age = 18 years
Putting y = 18 in eqn 1
x+18=60
x = 60-18
x = 42
Eli's age = 42 years ..
Light bulbs of a certain type are advertised as having an averagelifetime of 750 hours. The price of these bulbs is very favorable,so a potential customer has decided to go ahead with a purchasearrangement unless it can be conclusively demonstrated thatthe true average lifetime is smaller than what is advertised. Arandom sample of 50 bulbs was selected, the lifetime of each bulbdetermined, and the appropriate hypothesis were tested usingMINITAB, resulting in the accompanying output.Variable N Mean St Dev SEMean ZP -Valuelifetime 50 738.44 38.20 5.40-2.14 0.016 What conclusion would be appropriate for a significance level of.05 ? .A significance level of .01 ?. What significance level wouldyou recommend ?
answer 40
Step-by-step explanation:
because you added all together to make one
Solve |2x - 6| > 10
{x|x < -8 or x > 2}
{x|x < -2 or x > 8}
{x|-2 < x < 8}
ANSWER
{x|x < -2 or x > 8}
EXPLANATION
The given absolute inequality is
[tex] |2x - 6| \: > \: 10[/tex]
By the definition of absolute value,
[tex] (2x - 6)\: > \: 10 \: or \: \: - (2x - 6)\: > \: 10[/tex]
Multiply through the second inequality by -1 and reverse the inequality sign
[tex]2x - 6\: > \: 10 \: or \: \: 2x - 6\: < \: - 10[/tex]
[tex]2x \: > \: 10 + 6\: or \: \: 2x \: < \: - 10 + 6[/tex]
Simplify
[tex]2x \: > \: 16\: or \: \: 2x \: < \: -4[/tex]
Divide through by 2
[tex]x \: > \: 8\: or \: \: x \: < \: -2[/tex]
Answer:
{x|x < -2 or x > 8}
Step-by-step explanation:
|2x - 6| > 10
We split the inequality into two functions, one positive and one negative. The negative one flips the inequality. since this is greater than, this is an or problem
2x-6 >10 or 2x-6 < -10
Add 6 to each side
2x-6+6 > 10+6 2x-6+6 < -10+6
2x > 16 2x < -4
Divide by 2
2x/2 > 16/2 2x/2 < -4/2
x >8 or x < -2
Please show me steps to find 1/3 + m = 13/15
Hello!
Answer:
[tex]\boxed{m=\frac{8}{15}}[/tex]
Step-by-step explanation:
First, you switch sides.
[tex]m+\frac{1}{3}=\frac{13}{15}[/tex]
Then, you subtract by 1/3 from both sides.
[tex]m+\frac{1}{3}-\frac{1}{3}=\frac{13}{15}-\frac{1}{3}[/tex]
Simplify and solve.
[tex]\frac{13}{15}=\frac{8}{15}[/tex]
Therefore, [tex]\boxed{\frac{8}{15}}[/tex], which is our final answer.
I hope this helps you!
Have a nice day! :)
A skateboarder went a kilometer in 3 minutes with the wind, and returned in four minutes against the wind.
How fast could he skateboard one kilometer if there was no wind?
Please explain your answer! Please!
Step-by-step answer:
Answer to problems of this kind is the reciprocal of the harmonic mean of the time required.
We need to find the average of the speeds, not the average of the time.
The respective speeds are 1/3 and 1/4.
The average of the speeds is therefore (1/3+1/4)/2 = 7/24 (harmonic mean of the time taken).
The time required is therefore the reciprocal of the unit speed,
T = 1/(7/24) = 24/7 = 3 3/7 minutes, or approximately 3.43 minutes.
According to an article in Newsweek, the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100: 114 (46.7% girls). Suppose you don’t believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 60 girls and 90 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7?
Based on the study results, the percentage of girls born in China could range from 32.98% to 47.02%.
Explanation:To determine if the percentage of girls born in China is 46.7%, we can calculate the confidence interval for the proportion of girls in the population using a binomial distribution. Based on the study, out of 150 births, 60 were girls and 90 were boys.
Calculate the sample proportion of girls: p = 60/150 = 0.4Calculate the standard error: SE = sqrt((p*(1-p))/n) = sqrt((0.4*(1-0.4))/150) ≈ 0.0357Calculate the margin of error: ME = z*(SE) = 1.96*(0.0357) ≈ 0.0702Calculate the confidence interval: CI = p ± ME = 0.4 ± 0.0702 = (0.3298, 0.4702)The confidence interval suggests that the true proportion of girls born in China could range from 32.98% to 47.02%. Since the reported figure of 46.7% falls within this interval, it is plausible based on the study results.
Choose the equation of the graph shown.
y=-2cosx
y=-2sinx
y=2cosx
y= 2sinx
Answer:
y = -2 sin x
Step-by-step explanation:
As a basic,
y = cos x has a value of 1 at x = 0, and
y = sin x has a value of 0 at x = 0
Note: the value 1 can change to 2, 3, 4, etc. if the amplitude increases
Looking at the graph at x = 0, we see the y-value is 0, so definitely this is a sin graph. We can eliminate the cos choices.
So is it y = 2 sin x or y = -2 sin x??
If the graph goes downward from 0 (at x = 0), it is reflected of original, so that would be y = - sinx.
Since the graph decreases (goes downward) from x = 0, it is definitely the graph of negative sin. So y = - 2 sin x
Find the directional derivative of the function at the given point in the direction of the vector v. f(x, y, z) = xe^y + ye^z + ze^x, (0, 0, 0), v = 6, 3, −3
Final answer:
The directional derivative of the function f(x, y, z) = [tex]xe^y + ye^z + ze^x[/tex] at the point (0, 0, 0) in the direction of the vector v = 6, 3, −3 is 0.
Explanation:
To find the directional derivative of the function f(x, y, z) = [tex]xe^y + ye^z + ze^x[/tex] at the point (0, 0, 0) in the direction of the vector v = 6, 3, −3, we first need to find the gradient of f. The gradient of f, denoted as ∇f, is a vector of partial derivatives with respect to each variable. We calculate the partial derivatives as follows:
∂f/∂x = [tex]e^y + ze^x[/tex]∂f/∂y = [tex]xe^y + e^z[/tex]∂f/∂z = [tex]ye^z + xe^x[/tex]At the point (0, 0, 0), the gradient ∇f is (0 + 0, 0 + 1, 0 + 1) = (0, 1, 1).
Next, we need to normalize the given vector v. The normalization process involves dividing v by its magnitude to obtain a unit vector u in the direction of v. The magnitude of v is √(6² + 3² + (-3)²) = √(36 + 9 + 9) = √54. Therefore, the unit vector u is (6/√54, 3/√54, -3/√54).
Finally, the directional derivative of f at (0, 0, 0) in the direction of v is the dot product of ∇f and u, which is (0, 1, 1) ⋅ (6/√54, 3/√54, -3/√54) = 0*6/√54 + 1*3/√54 + 1*(-3)/√54 = 0.
verify sin^4x-sin^2x=cos^4x-cos^2x is an identity
ANSWER
See below
EXPLANATION
We want to verify that,
[tex] { \sin ^{4} x} - { \sin^{2} x} = { \cos ^{4} x} - { \cos^{2} x}[/tex]
To verify this identity, we can take the left hand side simplify it to get the right hand side or vice versa.
[tex]{ \sin ^{4} x} - { \sin^{2} x} =( { \sin ^{2} x} )^{2} - { \sin^{2} x}[/tex]
[tex]{ \sin ^{4} x} - { \sin^{2} x} ={ \sin ^{2} x}({ \sin ^{2} x} - 1)[/tex]
[tex]{ \sin ^{4} x} - { \sin^{2} x} ={ \sin ^{2} x} \times - (1 - { \sin ^{2} x})[/tex]
[tex]{ \sin ^{4} x} - { \sin^{2} x} =({1 - \cos^{2} x} )\times - ({ \cos^{2} x})[/tex]
[tex]{ \sin ^{4} x} - { \sin^{2} x} =({ \cos^{2} x} - 1 )\times ({ \cos^{2} x})[/tex]
We now expand the right hand side to get:
[tex] { \sin ^{4} x} - { \sin^{2} x} = { \cos ^{4} x} - { \cos^{2} x}[/tex]
Plz help out!! honest answer!
Answer:
2nd choice: Counterclockwise rotation about the origin by 180 degrees followed by a reflection about the y-axis
Step-by-step explanation:
A simple reflection across the x-axis will do.
A rotation of 180 degrees about the origin is equivalent to a reflection across both axes. Then a reflection back across the y-axis leaves the net effect being the desired reflection across the x-axis.
Consider the function f(x)=2x^3+24x^2−54x+9,−9≤x≤2 This function has an absolute minimum value equal to? and an absolute maximum value equal to ?
If you evaluate it, it's −9≤x≤2 and trying to find the absolute maximum/minimum of it then you'll get nothing due to it being an improper fraction of some sorts.. And there's still nothing when trying to find it all together.. Sorry that I wasn't that much help.