The hypotenuse of a right triangle has one end at the origin and one end on the curve y = x 2 e −3x , with x ≥ 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis. Find the maximum area of such a triangle. At what x-value does it occur?

Step-by-step explanation:

1) Check if the function is differentiable on that interval. In this case, yes, because all polynomials are differentiable.

2) plug in the bounds of the interval to see if the y-values equal 0.

f(0)=0

f(5)=0

since the last 2 conditions are satisfied, DNE will not be an answer choice.

3)take derivative and make it equal to 0

f' (×) = 5- 2x

0 = 5- 2x

x = 5/2

4) at c = 5/2, f(x) satisfies rolle's theorem.

Rolle's Theorem can be applied to the function f(x) = x(5 - x) on the interval [0, 5], and the value of c guaranteed by the theorem is c = 2.5.

The function f(x) = x(5 - x) on the interval [0, 5] is continuous on the closed interval and differentiable on the open interval (0, 5). To check if Rolle's Theorem can be applied, we first need to verify that the function is continuous on [0, 5] and differentiable on (0, 5). Both of these conditions are satisfied by the given function.

To find the value(s) of c guaranteed by Rolle's Theorem, we need to find the values of x where the derivative of the function is zero. Let's find the derivative of f(x):

f'(x) = 5 - 2x

Setting f'(x) = 0 and solving for x:

5 - 2x = 0

2x = 5

x = 2.5

Therefore, Rolle's Theorem can be applied to the function f(x) = x(5 - x) on the interval [0, 5], and the value of c guaranteed by the theorem is c = 2.5.

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Answer:

The answer to your question is 18m + 2

Step-by-step explanation:

                                            5(3m - 2) + 3(m + 4)

Multiply 5 by 3m and -2 and 3 by m and 4

                                           15m - 10 + 3m + 12

Simplify like terms

                                          15m +3m - 10 + 12

Result                                 18m + 2

Answer:

Option b

Step-by-step explanation:

Given that a  researcher measures IQ and weight for a group of college students.

In general, we think that the weight has nothing to do with IQ of a person and hence not correlated.

But if we go deep, we find that after a certain weight, the person becomes lazy and inactive with a chance to have reduced IQ

Weight gain causes also health problems including less activity of both brain and body and hence there is a chance for less IQ

So we find that as weight increases iq decreases and when weight decreases, IQ increases.

Thus we can say that there is a negative correlation but not necessarily near to one.

Hence option b is right

The most likely correlation between IQ and weight among college students is near zero, indicating no meaningful relationship between these variables.

When it comes to the likelihood of obtaining a correlation between IQ and weight among college students, the correlation is expected to be c) a correlation near zero. This is because there is no theoretical basis or empirical evidence to suggest that these two variables are related in any systematic way. A correlation coefficient is a statistical measure that describes the strength and direction of a relationship between two variables. A coefficient close to 0 indicates a very weak or no correlation, whereas one closer to +1 or -1 indicates a strong positive or negative correlation, respectively. In our scenario, since weight and IQ are not assumed to be related, the correlation would likely be close to 0, suggesting no meaningful relationship.

Answer:

propability is the low of chance

Answer:

The probability of tail occurs =  [tex]\frac{1}{4}[/tex] = 0.25

and

The probability of heads occurs = [tex]\frac{3}{4}[/tex] = 0.75

Step-by-step explanation:

Given:

Let P be the tail as outcome in a toss.

Heads is three times as likely to come up as tails.

So, Probability of getting heads = 3P

The total probability is 1

So, P + 3P = 1

4P = 1

P = [tex]\frac{1}{4}[/tex] = 0.25

We denoted P as the probability of getting tail in a toss.

So probability of getting heads = 1 -  [tex]\frac{1}{4}[/tex] =  [tex]\frac{3}{4}[/tex]

Therefore, the probability of tail occurs =  [tex]\frac{1}{4}[/tex] = 0.25

and

The probability of heads occurs = [tex]\frac{3}{4}[/tex] = 0.75

Answer:

The correct answers are: Part A. A/π^1/2 = r and Part B. 4.15 centimeters.

Step-by-step explanation:

1. Let's review all the information provided for solving this question:

Area of the circle = π*r² where r is the circle's radius

2. Let's find the solution for r for part A and for part B:

Part A:

A = π*r²

A/π = r²

√A/π = r

A/π^1/2 = r

Part B:

If A = 54 centimeters² and π  = 22/7, what is the value of r in centimeters?

Using the result of part A and replacing with the real values, we have:

√A/π = r

√54/(22/7) = r

√54 * 7/22 = r

√378/22 = r

√17.1818 = r

4.1451 = r

4.15 = r (Rounding to two decimal places)

Answer:

620 minutes

Step-by-step explanation

In one day,Mata exercise 20minutes more than Greg

31 days so we have 20 *31=620

Answer:

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

For two lines to be perpendicular, the slope of one line is the negative reciprocal of the other line. The equation of the given line is

y = 2x - 2

Comparing with the slope intercept form,

Slope, m = 2

This means that the slope of the line that is perpendicular to it is -1/2

The given points are (-3, 5)

To determine c,

We will substitute m = -1/2, y = 5 and x = - 3 into the equation, y = mx + c

It becomes

5 = -1/2 × - 3 + c

5 = - 3/2 + c

c = 5 + 3/2

c = 13/2

The equation becomes

y = -x/2 + 13/2

Answer:

Output value for the following function is 4

Step-by-step explanation:

Given:

The input value are the value of x

[tex]y = 3x + 1[/tex]

Input value is 1

We substitute the value 1 for the input variable x in the given function.

[tex]y = 3x + 1[/tex]

[tex]y = 3(1) + 1[/tex]

[tex]y = 3 + 1[/tex]

[tex]y = 4[/tex]

The output value are the value of y

Therefore, for an input of 1, we have an output of 4.

Answer:

  3.  25/45

Step-by-step explanation:

[tex]\dfrac{5}{9}=\dfrac{5\cdot 5}{5\cdot 9}=\bf{\dfrac{25}{45}}[/tex]

Answer:

No

Step-by-step explanation:

N = 10,000

n= 800

p = .28

Sampling distribution drawn from specific population. So it is not safe to assume that sampling proportion of customers is approximately normal.

Final answer:

The sampling distribution of the sample proportion of customers planting a vegetable garden may not be approximately normal due to insufficient sample size, according to the central limit theorem.

Explanation:

No, it is not safe to assume the sampling distribution of the sample proportion of customers that plant a vegetable garden is approximately normal.

This situation involves calculating the normality of a sample proportion. The **central limit theorem** states that the sampling distribution of a sample proportion will be approximately normal if the sample size is large enough, specifically n * p >= 10 and n * (1-p) >= 10, where n is the sample size and p is the probability of success.

In this case, the sample size is 800 and the probability of success (customers planting a garden) is 28%, so 800 * 0.28 = 224, which is less than 10. Thus, the sampling distribution may not be normal.

Answer:

The answer should be C.

Answer:

c

Step-by-step explanation:

Answer:

(D) xy/(x + y)

Step-by-step explanation:

To find r in terms of x and y

Given,

Then,

1/r = 1/x + 1/y

1/r = (y + x)/xy

Find the reciprocal of both sides

r = xy/(x + y)

The right answer is option (D)  xy/(x + y)

Answer:

  f(t) = 30 +7t

Step-by-step explanation:

The fee f(t) in terms of months of membership t can be modeled as ...

  fee = startup fee + (monthly fee)×(number of months)

  f(t) = 30 + 7t

Answer: V = 0.09L feet^3

Step-by-step explanation:

The tubing has the shape of a cylinder. Formula for determining the volume of a cylinder is

Volume of cylinder = πr^2h

Where π = 22/7 or 3.14

r = the inner radius of the drainage tubing. It is given as 2 inches. We would convert the 2 inches to feets

If 12 inches = 1 foot,

2 inches will be 2/12 inches

h = height of the cylinder and it is replaced by L in feets. L is the length of the cut drainage tubing.

An expression for the volume of L feets of drainage tubing will be

V = πr^2L

= 3.14 × (2/12)^2 × L

= 3.14 × 4/144 × L

V = 0.087L feet^3

V = 0.09L feet^3

The solution is x = 4 and y = 2

We have the following system of two linear equations in two variables:

[tex]\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}y=2x-6\\y=-\frac{1}{2}x+4\end{array}\right.[/tex]

Subtract (2) from (1):

[tex]\begin{array}{c}(1)\\(2)\end{array}\left\{ \begin{array}{c}y=2x-6\\ -\left(y=-\frac{1}{2}x+4\right)\end{array}\right \\ \\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ \\ y-y=2x-6-(-\frac{1}{2}x+4) \\ \\ 0=2x-6+\frac{1}{2}x-4 \\ \\ Combine \ like \ terms: \\ \\ 2x+\frac{1}{2}x-6-4=0 \\ \\ 2.5x-10=0 \\ \\ 2.5x=10 \\ \\ x=\frac{10}{2.5} \\ \\ x=4[/tex]

Substituting the x-value into (1):

[tex]y=2(4)-6 \\ \\ y=8-6 \\ \\ y=2[/tex]

So the solution to this system is:

[tex]\boxed{x=4 \ and \ y=2}[/tex]

Methods for solving systems of linear equations:

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Answer:

The percentage of blue candies is equal to 28​%.

Step-by-step explanation:

Sample size = n = 1000

21​% of them are blue

So, No. of blue candies = [tex]21\% \times 100 =\frac{21}{100} \times 100=21[/tex]

Claim : The percentage of blue candies is equal to 28​%.

[tex]H_0:\mu = 0.28\\H_a:\mu \neq 0.28[/tex]

We will use one sample proportion test  

[tex]\widehat{p}=\frac{x}{n}[/tex]

[tex]\widehat{p}=\frac{21}{100}[/tex]

[tex]\widehat{p}=0.21[/tex]

Formula of test statistic =[tex]\frac{\widehat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

                                       =[tex]\frac{0.21-0.28}{\sqrt{\frac{0.28(1-0.28)}{100}}}[/tex]

                                       =−1.55

Now refer the p value from the z table

p value =0.0606

α =0.05

So, p value >  α

So, we failed to reject null hypothesis

So, the percentage of blue candies is equal to 28​%.

Given the problem, first, we denote the sample size, observed proportion of blue candies, claimed proportion of blue candies, and significance level as:

- n = 100
- p_observed = 0.21
- p_claimed = 0.28
- significance_level = 0.05

Our first step in the hypothesis testing process is to calculate the standard error. We do this using the formula:

 standard_error = sqrt((p_claimed*(1 - p_claimed))/n)

which gives us a standard error of approximately 0.0449. The standard error measures the variability or dispersion of our sample proportion from the claimed proportion.

Next, we will calculate the z-score, which measures the number of standard deviations an observation (or in this case, the sample proportion) is away from the mean, or the claimed proportion. We do this using the formula:

 z = (p_observed - p_claimed)/standard_error

which gives us a z-score of approximately -1.559. The negative sign indicates that the observed proportion is less than the hypothesized proportion.

Then, we need to calculate the p-value. The p-value is the probability of getting a sample as extreme, or more extreme, than the one we have, assuming the null hypothesis is true. In other words, it is the likelihood of observing our sample data if the candy company's claim of 28% blue candies is accurate.

As the observed proportion is less than the hypothesized proportion, we calculate the cumulative probability up to the z-score. Doing this gives us a p-value of approximately 0.0595.

Finally, we need to determine whether to accept or reject the null hypothesis based on the p-value and the significance level.

Here, we can see our p-value is slightly larger than the given significance level (0.0595 > 0.05), thus, we do not reject the null hypothesis. This means there is not enough evidence at the 5% significance level to reject the candy company's claim that 28% of their candies are blue.

In conclusion, given our sample and the given significance level, our analysis does not provide sufficient evidence to say with 95% confidence that the company's claim is false. Our data does not contradict the company's claimed proportion of 28%.

Answer:

The present value of 10,000 if  interest is paid at a rate of 6.2% compounded  weekly for 8 years is 6097.56

Explanation:

We know that compound interest is given by  

[tex]A=P\left(1+\frac{r}{n}\right)^{n t}[/tex]

Where ,  

Where A = final amount (which is given to be = 10000)

       P = Principal amount (which is the present amount which we have to find)

r  = interest rate = 6.2 = 0.062

n = no. of times interest applied per time period = it is given that the interest is applied weekly, so in one year there are 52 weeks so n = 52

t = time period = 8 years

Substituting the given values, we get

[tex]10000=\mathrm{P}\left(1+\frac{6.2}{52}\right)^{52\times 8}[/tex]

P = 6097.5

We get, P = 6097.56 which is the present value of a sum of money

Answer:

$24.20

Step-by-step explanation:

Multiply and add.

Answer:

Step-by-step explanation:

Country 4 has the highest growth rate, as it has the largest exponent in its growth function.

The growth rate of each country is given by the coefficient in the exponent of the exponential equation. The greatest growth rate is for Country 4, which has a growth rate of 2.5% each year.

The growth rate of each country is represented by the coefficient in the exponent in each exponential equation. The coefficients represent the yearly percentage increase in population. Looking at the four models given, the coefficients are 0.001 for country 1, 0.016 for country 2, -0.005 for country 3, and 0.025 for country 4. Note that the coefficient for country 3 is negative, indicating that the population is actually decreasing each year. Hence, it can be concluded that Country 4 has the greatest growth rate, which is 0.025 or 2.5% each year (when the coefficient is expressed as a percentage).

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Answer:

162.5 meters

Step-by-step explanation:

With respect to the angle given, the side from Meg to Store (150m) is the side that is "opposite" to the angle.

The side from Store to Bank is the side that is "adjacent" to the angle.

So, we have Opposite side and want to know the Adjacent side.

Which trigonometric ratio relates "opposite" to "adjacent"??

Yes, it is Tan!

We write the trig equation and solve for the distance (letting it be x):

[tex]Tan(42.71)=\frac{Opposite}{Adjacent}=\frac{150}{x}\\x=\frac{150}{Tan(42.71)}\\x=162.49[/tex]

Rounding the answer to 1 decimal place, it is:

162.5 meters

Answer: Time it took her to get to the school is 0.6 hours

The distance of the school from their home is 27 miles

Step-by-step explanation:

Mrs Dang drove her daughter to school at the average speed of 45 miles per hour.

Let x miles = distance from the school to their home.

Distance = speed × time

Time = distance / speed

Time used in going to school will be

x/45

She returned home by the same route. This means that distance back home is also x miles.

She returned at an average speed of 30 miles per hour.

Time used in returning home from school will be x/30

x/45

If the trip took one half hour, then the time spent in going to school and the time spent in returning is 1 1/2 hours = 1.5 hours. Therefore

x/30 + x/45 = 1.5

(15x + 10x) /450 = 1.5

15x + 10x = 450 × 1.5 = 675

25x = 675

x = 675/25 = 27

Time it took her to get to the school will be x/45

= 27/45 = 0.6 hours

Answer:

[tex]-1.78>x\leq -1.46[/tex]

Step-by-step explanation:

1. Understanding the type of statement

We are given an OR statement. A certain x-value is a set of solution of the statement if it satisfies both of the inequalities.

Therefore, the solution of this statement is the Union of set of the solutions of both inequalities.

2. Finding the solutions to the two inequalities

[tex]-9x+2>18\\-9x>18-2\\-9x>16\\x<-\frac{16}{9}\\ \\x<-1.78[/tex]

Now Solving for other equation we get,

[tex]13x+15\leq-4\\13x\leq -4-15\\13x\leq -19\\x\leq -\frac{19}{13}\\ \\x\leq -1.46[/tex]

3. The solution is:

[tex]-1.78>x\leq -1.46[/tex]

Answer:

[tex]x \leq - 1.462[/tex]

Step-by-step explanation:

Let solve each inequation:

[tex]-9\cdot x + 2 > 18[/tex]

[tex]-16 > 9\cdot x[/tex]

[tex]9\cdot x < - 16[/tex]

[tex]x < - \frac{16}{9}[/tex]

[tex]x < -1.778[/tex]

[tex]13\cdot x + 15 \leq -4[/tex]

[tex]13\cdot x \leq -19[/tex]

[tex]x \leq -\frac{19}{13}[/tex]

[tex]x \leq -1.462[/tex]

The boolean operator OR means that proposition is true if at least one equation is true. Then, the domain that fulfill the proposition is:

[tex]x \leq - 1.462[/tex]

Answer:

1, 3, and 5

Step-by-step explanation:

Answer:

at y-intercept: (0,2) is the correct

Step-by-step explanation:

We have equation 1  

9x-6=3y

to find the x-intercept, substitute in 0 for y and solve for x

3(0)=9x+6

3(0)=9x+6

9x+6=0

Subtract 6 from both sides  

9x=−6

So x = -6/9 = -2/3

Now to find for y-intercept, substitute in 0 for x and solve for y

3y=9(0)+6

3y=0+6

3y=6

These are the x and y intercepts of the equation 3y=9x+6

x-intercept: (−2/3,0)

y-intercept: (0,2)

Answer:

Correct.

Step-by-step explanation:

Check if x = 0 and y = -2 fits the equation:

9(0) - 6 = -6

3(-2) = -6.

They do so the student is correct.

Answer:

20 peices

Step-by-step explanation:

To determine the temperature of an object after a specific time using the given equation T(t) = 69e−0.0174t + 79, you replace t with the desired time in minutes and calculate the result. In case of one and a half hours (90 minutes), the temperature can be found using T(90) = 69e−0.0174*90 + 79.

The subject of this question is mathematics, more specifically an application of exponential decay in the context of temperature change. The equation T(t) = 69e−0.0174t + 79 represents the temperature T of an object after time t in minutes. To find the temperature after one and a half hours, we need to convert this to minutes because the given equation uses time in minutes. One and a half hours is equivalent to 90 minutes. So we will plug t=90 into the equation:
T(90) = 69e−0.0174*90 + 79.

Around this value to the nearest degree will give us the desired temperature in degrees Fahrenheit after one and a half hours.

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The temperature of the object after one and a half hours is approximately [tex]\( {93^\circ} \)[/tex] Fahrenheit.

To find the temperature of the object after one and a half hours, we need to substitute  t = 90 minutes into the equation [tex]\( T(t) = 69e^{-0.0174t} + 79 \)[/tex], since there are 60 minutes in an hour and a half.

Let's calculate:

[tex]\[ T(90) = 69e^{-0.0174 \times 90} + 79 \]\[ T(90) = 69e^{-1.566} + 79 \][/tex]

Using a calculator:

[tex]\[ T(90) \approx 69 \times 0.208 + 79 \]\[ T(90) \approx 14.352 + 79 \]\[ T(90) \approx 93.352 \][/tex]

There will be 1.078x students next year and equation is number of students in next year = x + 7.8% of x

Given, There are "x" number of students at helms.  

The number of students increases by 7.8% each year which means if there "x" number of students in present year, then the number of students in next year will be x + 7.8% of x

Number of students in next year = number of students in present year + increased number of students.

[tex]\begin{array}{l}{\text { Number of students in next year }=x+7.8 \% \text { of } x} \\\\ {\text { Number of students in next year }=x\left(1+\frac{7.8}{100}\right)} \\\\ {\text { Number of students in next year }=x(1+0.078)=1.078 x}\end{array}[/tex]

Thus there will be 1.078x students in next year

Answer:

The greatest number of baskets can be 81 and the least number of baskets can be 9.

Step-by-step explanation:

We have a constraint on our actions, that every basket should have the same number of fruits in each basket.

To find the highest number of fruits in each basket we have to find the Highest Common Factor( HCF) of the number of apples , bananas and oranges.

HCF of 36, 27 and 18 is 9.

Therefore the number of fruits in each basket is 9.

Apples will require 4 baskets, bananas will require 3 baskets and oranges will require 2 baskets. Thus Nina is right and total 9 baskets will be required.

9 is the least of number of baskets required

If we have to maximize the number of baskets, then we have to place the least number of same fruits in a basket ie. 1.

Therefore, he maximum number of baskets required is 36+27+18=81 baskets.

Answer:

Step-by-step explanation:

inside the castle

Answer:

B

Step-by-step explanation:

To solve this, we use ratio.

Firstly, we need to know the number of hours traveled. The total number of hours traveled = x+y

Ratio of this used by high speed train = x/(x +y).

Total distance traveled before they meet = [x/(x + y)] × z

For low speed train = [y/(x + y)] × z.

The difference would be distance by high speed train - distance by low speed train.

= z [ (x - y)/x + y)]