Find a bijection from (0,1] to (0,1)

Math: Analysis and proof

The average value of [tex]\(f(x)\) over \([-1, 3]\)[/tex] is 13. The function equals its average value at certain [tex]\(x\)[/tex] values.

To find the average value of the function [tex]\( f(x) = 4x^3 - 3x^2 \)[/tex] over the interval [tex]\([-1, 3]\)[/tex], we'll first calculate the definite integral of the function over that interval and then divide it by the length of the interval.

The formula for the average value of a function [tex]\( f(x) \)[/tex] over an interval [tex]\([a, b]\)[/tex] is given by:

[tex]\[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \][/tex]

For the function [tex]\( f(x) = 4x^3 - 3x^2 \)[/tex] over the interval [tex]\([-1, 3]\)[/tex], we have:

[tex]\[ \text{Average value} = \frac{1}{3-(-1)} \int_{-1}^{3} (4x^3 - 3x^2) \, dx \][/tex]

First, let's find the integral:

[tex]\[ \int (4x^3 - 3x^2) \, dx = \frac{4}{4}x^4 - \frac{3}{3}x^3 + C \]\[ = x^4 - x^3 + C \][/tex]

Now, we'll evaluate this integral from -1 to 3:

[tex]\[ \left[ x^4 - x^3 \right]_{-1}^{3} = (3^4 - 3^3) - ((-1)^4 - (-1)^3) \]\[ = (81 - 27) - (1 - (-1)) \]\[ = 54 - 2 \]\[ = 52 \][/tex]

So, the definite integral is 52.

Now, we'll find the average value:

[tex]\[ \text{Average value} = \frac{1}{3-(-1)} \times 52 \]\[ = \frac{52}{4} \]\[ = 13 \][/tex]

The average value of the function [tex]\( f(x) = 4x^3 - 3x^2 \)[/tex] over the interval [tex]\([-1, 3]\) is 13.[/tex]

To find the values of [tex]\( x \)[/tex] in the interval for which the function equals its average value, we set [tex]\( f(x) \)[/tex] equal to 13 and solve for [tex]\( x \):[/tex]

[tex]\[ 4x^3 - 3x^2 = 13 \][/tex]

This equation can be solved numerically. By using methods like graphing, Newton's method, or a numerical solver, we can find the roots of this equation within the interval [tex]\([-1, 3]\).[/tex] These roots will be the [tex]\( x \)[/tex] values where the function equals its average value.

The average value of [tex]\(f(x) = 4x^3 - 3x^2\)[/tex]over [tex]\([-1, 3]\) is 13.[/tex]

The values of x that make [tex]\(f(x) = 13\)[/tex] are approximately -0.771, 1.979, and 2.792.

To find the average value of the function [tex]\(f(x) = 4x^3 - 3x^2\)[/tex]over the interval [-1, 3], we can use the formula for the average value of a function over an interval [a, b] :

[tex]\[ A = \frac{1}{b - a} \int_{a}^{b} f(x) dx. \][/tex]

Determine the integral of f(x) :

  To find the integral of f(x), we first compute the antiderivative of[tex]\(4x^3 - 3x^2\):[/tex]

  [tex]\[ \int (4x^3 - 3x^2) dx = x^4 - x^3. \][/tex]

Evaluate the integral over [tex]\([-1, 3]\):[/tex]

  Now, let's find [tex]\(\int_{-1}^{3} (4x^3 - 3x^2) dx\):[/tex]

[tex]\[ \int_{-1}^{3} (4x^3 - 3x^2) dx = (x^4 - x^3) \Big|_{-1}^{3}. \][/tex]

  Evaluate the antiderivative at 3 and -1 :

  - When [tex]\(x = 3\), \(3^4 - 3^3 = 81 - 27 = 54\),[/tex]

  - When [tex]\(x = -1\), \((-1)^4 - (-1)^3 = 1 + 1 = 2\).[/tex]

  Thus, [tex]\[ \int_{-1}^{3} (4x^3 - 3x^2) dx = 54 - 2 = 52. \][/tex]

Find the average value over [-1, 3]:

  Using the result from step 2, the average value over [tex]\([-1, 3]\)[/tex]  is:

 [tex]\[ A = \frac{1}{3 - (-1)} \cdot 52 = \frac{1}{4} \cdot 52 = 13. \][/tex]

Therefore, the average value of [tex]\(f(x) = 4x^3 - 3x^2\)[/tex] over the interval [tex]\([-1, 3]\) is \(13\).[/tex]

Now, let's find the values of x in [-1, 3] for which f(x) = 13):

[tex]\[ 4x^3 - 3x^2 = 13. \][/tex]

Rearrange the equation:

[tex]\[ 4x^3 - 3x^2 - 13 = 0.[/tex]

This cubic equation is more complex to solve algebraically. The approximate solutions can be obtained numerically, using a graphing calculator, computational software, or by iterative methods.

Using an approximation method, we get the following solutions (rounded to three decimal places):

1. [tex]\( x \approx -0.771 \),[/tex]

2. [tex]\( x \approx 1.979 \),[/tex]

3. [tex]\( x \approx 2.792 .[/tex]

These are the values of x in the interval [-1, 3] for which f(x) = 13).

Question :

Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. (Round your answer to three decimal places.) f(x) = 4x3 − 3x2, [−1, 3]

Answer:

no of success deemed to be usual 5

that is more than 5 success unusual

Step-by-step explanation:

Given data

number of successes  = less than μ − 2.5σ

number of successes  = greater than μ + 2.5σ

trials n  = 10

probability single trial = 0.2

to find out

is it unusual to have more than five successes

solution

we can say that

mean of the binomial, distribution that is

mean =  probability single trial × trials n

mean =  10 × .2

mean = 2

and standard deviation = √(mean× (1-probability))

standard deviation = √2× (1-0.2))

standard deviation = 1.2649

so no of successes are

= μ − 2.5σ                 and           = μ + 2.5σ

= 2 − 2.5(1.2649)      and           = 2 + 2.5(1.2649)

= -0.16225                and           = 5.16225

so now we say no of success deemed to be usual 5

that is more than 5 success unusual here

To determine whether it is unusual to have more than five successes in a binomial experiment with 10 trials and a probability of success of 0.2, we can use the mean and standard deviation of the binomial distribution. The probability of having six or more successes in this experiment is quite low (less than 0.05), so it would be considered unusual.

To determine whether it is unusual to have more than five successes in a binomial experiment with 10 trials and a probability of success of 0.2, we can use the mean and standard deviation of the binomial distribution. The mean is found by multiplying the number of trials by the probability of success (10 * 0.2 = 2), and the standard deviation is found using the formula √(npq) (where q = 1 - p). For this experiment, the standard deviation would be √(10 * 0.2 * 0.8) = 1.26.

To determine whether it is unusual to have more than five successes, we need to find the probability of having six or more successes. We can use the cumulative binomial distribution to find this probability. Using a calculator or a statistical software, we can find that the probability is approximately 0.003.

Since the probability is quite low (less than 0.05), we can consider it unusual to have more than five successes in this particular binomial experiment.

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The only way to get a term of degree 25 is by taking 3 copies of [tex]x^8[/tex], 1 copy of [tex]x[/tex], and 6 copies of 1. Then the coefficient of [tex]x^{25}[/tex] is

[tex]\dbinom{10}3\dbinom71\dbinom66=\dbinom{10}{3,1,6}=\dfrac{10!}{3!6!}=\boxed{840}[/tex]

Answer:

B. (8π + 12) in²

Step-by-step explanation:

1. Identify the formula for the area of both a triangle and a circle.

    Triangle: 1/2(b)(h)

       b = base

       h = height

    Circle: πr²

      r = radius

2. Start by finding the area of the circle, since we already have all the needed information for the variables in the equation.

     π(4)² → π(16) → 16π

3. Half the answer we just got as the area of the circle. We are doing this because we only have half a circle in the diagram, and we solved for the area of a full circle.

    (16π)/2 → 8π

4. Next find the base of the triangle, since this is the only information we do not yet have for the triangle. We will find this by doubling the 4, since 4 inches is only half the length of the base.

    4 × 2 = 8

5. Plug all the information of the triangle into the area of a triangle formula and solve.

    1/2(8)(3) → 1/2(24) → 12

6. Add both the area of the semi-circle and triangle together because they are one shape that we are finding the area for.

     8π + 12  

7. Label answer with units of measurement

    (8π + 12) in²            

ANSWER

The correct option is B

EXPLANATION

The composite figure is made up of a semicircle and an isosceles triangle.

The area of a semicircle is

[tex] \frac{1}{2}\pi {r}^{2} [/tex]

From the diagram, the radius is

[tex]r = 4 \: in[/tex]

When we substitute, area of the semicircle is

[tex] \frac{1}{2} \times \pi \times {4}^{2} [/tex]

[tex]\frac{1}{2} \times \pi \times 16[/tex]

[tex]8\pi \: \: {in}^{2} [/tex]

The area of the isosceles triangle is

[tex] \frac{1}{2} \times base \times height[/tex]

[tex] = \frac{1}{2} \times (4 + 4) \times 3[/tex]

[tex] = \frac{1}{2} \times 8 \times 3[/tex]

[tex] = 12 \: {in}^{2} [/tex]

We add the two areas to obtain the area of the composite figure to be:

[tex](8\pi + 12) {in}^{2} [/tex]

Answer:

Step-by-step explanation:

P\left ( defective item\right )=0.035

Using binomial distribution

Where p= probability of success

q=probability of failure

Here p=0.035

q=1-0.035=0.965

[tex]^nC_{r}P^{r}q^{n-r}[/tex]

(i)for exactly 3 defective items i.e. r=3

P[tex]\left ( r=3\right )[/tex]=[tex]^{44}C_{3}[/tex][tex]\left ( 0.035\right )^{3}\left ( 0.965\right )^{44-3}[/tex]

P=[tex]\frac{44!}{41!3!}\times \left ( 0.035\right )^3\left ( 0.965\right )^{41}[/tex]

P=0.1317

(ii)No defective item i.e. r=0

P[tex]\left ( r=0\right )[/tex]=[tex]^{44}C_{0}[/tex][tex]\left ( 0.035\right )^{0}[/tex][tex]\left ( 0.965\right )^{44-0}[/tex]

P=[tex]\frac{44!}{44!0!}\times \left ( 0.035\right )^0\left ( 0.965\right )^{44}[/tex]

P=0.2085

(iii)At least 1 defective item

P=1-P(zero defective item)

P=1-[tex]^{44}C_{1}\left ( 0.035\right )^{1}\left ( 0.965\right )^{44-1}[/tex]

P=1-[tex]\frac{44!}{43!1!}\times \left ( 0.035\right )^1[/tex][tex]\left ( 0.965\right )^{43}[/tex]

P=0.6671

(a) The probability of exactly 3 defective items: approximately 0.1318
(b) The probability of no defective items: approximately 0.2085
(c) The probability of at least 1 defective item: approximately 0.7915

(a) Probability of Exactly 3 Defective Items
To find the probability of getting exactly 3 defective items in a run of 44, we will use the binomial probability formula:

[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]

where
- [tex]\( X \)[/tex] is the random variable representing the number of defective items,
- [tex]\( k \)[/tex] is the number of defective items we want to find the probability for (in this case, 3),
- [tex]\( \binom{n}{k} \)[/tex] is the number of combinations of [tex]\( n \)[/tex] items taken [tex]\( k \)[/tex] at a time.

So, substituting into the formula:
[tex]\[ P(X = 3) = \binom{44}{3} (0.035)^3 (1 - 0.035)^{41} \][/tex]

After calculating, we find:
[tex]\[ P(X = 3) \approx 0.13177807290504395 \][/tex]

Thus, the probability of getting exactly 3 defective items is approximately 0.1318.

(b) Probability of No Defective Items
To determine the probability of having no defective items, we calculate:

[tex]\[ P(X = 0) = \binom{44}{0} (0.035)^0 (1 - 0.035)^{44} \][/tex]

Here:
[tex]\[ \binom{44}{0} = 1 \\ (0.035)^0 = 1 \\ (1 - 0.035)^{44} \approx 0.20854596293662794[/tex]

Thus, the probability of having no defective items is approximately 0.2085.

(c) Probability of At Least 1 Defective Item
To find the probability of at least 1 defective item, it is easier to calculate the complement—the probability of having no defective items—and subtract it from 1:

[tex]\[ P(X \geq 1) = 1 - P(X = 0) \][/tex]

From part (b), we know [tex]\( P(X = 0) \)[/tex]:
[tex]\[ P(X \geq 1) = 1 - 0.20854596293662794 \approx 0.791454037063372 \][/tex]

Therefore, the probability of having at least 1 defective item is approximately 0.7915.

We have [tex]17\equiv-9\pmod{26}[/tex], so that [tex]x\equiv-1\pmod{26}[/tex], so [tex]x\equiv25\pmod{26}[/tex], and any solution of the form [tex]x=25+26n[/tex] satisfies the congruence, where [tex]n[/tex] is any integer.

The question involves solving a congruence equation. Given the equation 9x + 17 ≡ 0 (mod 26), we would typically isolate x to find the solution. However, the question seems to be missing an operator. Hence, an explicit answer cannot be provided.

The question involves solving a congruence. To solve the accord 9x ≡ 17 (mod 26), you must find an integer x such that 9x leaves a remainder of 17 when divided by 26.

This unity can be rewritten as 9x - 17 = 26k, where k is an integer. Our task is to solve for x given these parameters.

Given the nature of the question, I cannot provide a direct solution because it is missing an operator between 9x and 17. Assuming the operator is '+', the congruence will be 9x + 17 ≡ 0 (mod 26).

The steps to solve a congruence equation can vary, but generally, the goal is to isolate x on one side of the equation. However, it's easier to proceed with this congruence with explicit details.

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[tex]\vec F(x,y,z)=\langle x^2,y^2,z^2\rangle\implies\mathrm{div}\vec F(x,y,z)=2x+2y+2z[/tex]

By the divergence theorem,

[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]

where [tex]R[/tex] the region with [tex]S[/tex] as its boundary. Convert to spherical coordinates, taking

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

Then the volume integral is

[tex]\displaystyle\iiint_R\mathrm{div}\vec F(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]

[tex]=2\displaystyle\int_0^{2\pi}\int_0^\pi\int_0^5(x+y+z)\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta[/tex]

[tex]=2\displaystyle\int_0^{2\pi}\int_0^\pi\int_0^5(\cos\theta\sin\varphi+\sin\theta\sin\varphi+\cos\varphi)\rho^3\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta=\boxed{0}[/tex]

In this exercise we have to use the divergent theorem to calculate the flow of the given equation, so we will find that:

[tex]\int\limits \int\limits \int\limits_R {divF(x,y,z)} \, dx dy dz= 0[/tex]

So from the given equation, we will find that:

[tex]\int\limits \int\limits_S {F} \, ds = \int\limits \int\limits \int\limits_R {div F(x, y, z) } \, dx dy dz[/tex]

where [tex]R[/tex] the region with [tex]S[/tex] as its boundary. Convert to spherical coordinates, taking:

[tex]\left[\begin{array}{c}x= \rho cos(\theta) sin(\phi) \\y= \rho sin(\theta) sin(\phi) \\z= \rho cos (\phi) \end{array}\right[/tex]

Then the volume integral is:

[tex]\int\limits \int\limits \int\limits_R {divF(x,y,z)} \, dxdydz\\= 2 \int\limits^{2\pi}_0 \int\limits^{\pi}_0 \int\limits^{5}_0 {(x+y+z)\rho ^2 sin(\phi) d(\rho) d(\phi) d(\theta)= 0[/tex]

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Answer:

D. 45.71 - 6m

Step-by-step explanation:

     Let m = the cost of a marker

Then 6m = the cost of six markers

Heather is paying for these from her savings account.

After deducting the cost of the markers, the amount in her account will be

45.71 - 6m

Answer:

The basic quadratic function is f(x) = x²

The equation of the graph is y = (x - 3)² - 1

Step-by-step explanation:

* Lets explain how to solve the problem

- The graph is a parabola which oped upward

∵ The function is represented by a parabola

∴ The graph is a quadratic function

∴ The basic quadratic function is f(x) = x²

- The vertex of the basic quadratic function is (0 , 0)

∵ From the graph the vertex of the parabola is (3 , -1)

∵ The x coordinate of the basic function change from 0 to 3

∴ The basic function translate 3 units to the right

- If the function f(x) translated horizontally to the right  by h units,

 then the new function g(x) = f(x - h)

∵ f(x) = x²

∴ The new function g(x) = (x - 3)²

∵ The y-coordinate of the basic function change from 0 to -1

∴ The basic function translate 1 unit down

- If the function f(x) translated vertically down  by k units, then the

 new function g(x) = f(x) - k

∵ g(x) = (x - 3)²

∴ The new function h(x) = (x - 3)² - 1

∵ h(x) = y

∴ The equation of the graph is y = (x - 3)² - 1

# Note: you can write the equation in general form by solve the

  bracket of power 2

∵ (x - 3)² - 1 = (x)(x) - (2)(3)(x) + (3)(3) - 1 = x² - 6x + 9 - 1 = x² - 6x + 8

∴ y = x² - 6x + 8

Answer:

The Final amount in the account after 5 years will be $108.26

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Equations.

Since we are talking about compounding interest we can use the Exponential Growth Formula to calculate the interest over the next couple of years. The formula is the following,

[tex]y = a*(1+r)^{t}[/tex]

Where:

We first need to calculate the 2% interest for the first year,

[tex]y = 100*(1+0.02)^{1}[/tex]

[tex]y = 100*1.02[/tex]

[tex]y = 102[/tex]

So after the first year the account will have $102. Now we can use the $102 to calculate the next 4 years of interest using the new interest rate of 1.5%

[tex]y = 102*(1+0.015)^{4}[/tex]

[tex]y = 102*(1.015)^{4}[/tex]

[tex]y = 102*1.0614[/tex]

[tex]y = 108.26[/tex]

The Final amount in the account after 5 years will be $108.26

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Answer: The predicted error is minimized.

Step-by-step explanation:

Ideally, residual analysis is used in a linear regression model to measure the appropriateness of the model by examining the residual plots on the graph.

And, residual referred as a difference between the noticed value of the dependent variable (y) and the estimated value (ŷ).  

Residual = Noticed value - Estimated value  

e = y - ŷ

Multiple regression analysis is used to make a linear model capable of giving predicting an output variable using two or more independent variables. Analysis of the residual is used to to test if the variation in the residuals is the same for all predicted values of y.

Therefore, residual analysis in multiple regression tests the variation in the residuals is the same for all predicted values of y.

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Answer: 0.1841

Step-by-step explanation:

Given: Mean : [tex]\mu=8.2[/tex]

Standard deviation : [tex]\sigma = 1.34[/tex]

The formula to calculate z-score is given by :_

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x= 9, we have

[tex]z=\dfrac{9-8.2}{1.34}\approx0.60[/tex]

For x= 10, we have

[tex]z=\dfrac{10-8.2}{1.34}\approx1.34[/tex]

The P-value = [tex]P(0.6<z<1.34)=P(z<1.34)-P(z<0.6)[/tex]

[tex]=0.9098773-0.7257469=0.1841304\approx0.1841[/tex]

Hence, the probability that at your next exam, you will have a stress level between 9 and 10 = 0.1841

Answer: The probability that the batch is accepted is 0.812.

Step-by-step explanation:

Since we have given that

Number of defective cars = 5

Total number of cars = 100

Probability of getting first car non defective = [tex]\dfrac{95}{100}[/tex]

probability of getting second car non defective = [tex]\dfrac{94}{99}[/tex]

Probability of getting third car non defective =  [tex]\dfrac{93}{98}[/tex]

Probability of getting fourth car non defective = [tex]\dfrac{92}{97}[/tex]

Using the multiplication rule, probability that the batch is accepted is given by

[tex]\dfrac{95}{100}\times \dfrac{94}{99}\times \dfrac{93}{98}\times \dfrac{92}{97}\\\\=0.812[/tex]

Hence, the probability that the batch is accepted is 0.812.

Answer:

98

Step-by-step explanation:

100-65=35

Turn 35 into a percent then a decimal, later multiply it by 280

280 X .35=98

Final answer:

The sale price of the down comforter after a 65% discount is $98. This is found by calculating the discount amount of $182 from the original price of $280, then subtracting it to find the final sale price.

Explanation:

To calculate the sale price of the down comforter after a reduction of 65%, we first find the discount amount and then subtract it from the original price. Here's how:

To calculate the discount amount:

The sale price of the down comforter after a 65% discount is $98.

Answer:

26

Step-by-step explanation:

Converting 11010 to base 10.

1*24=16

1*23=8

0*22=0

1*21=2

0*20=0

Adding all to get Ans=26_10

Step2 converting 26_10 to 10

The equation calculation formula for 26_10 number to 10 is like this below.

10|26  

10|2|6

10|2|2

Ans:26_10

Assuming the given number is in base 2, we have

[tex]11010_2=2^4+2^3+2^1=16+8+2=26_{10}[/tex]

Answer:

1/3

Step-by-step explanation:

Once you know there is a 3 in the hand, and only one card is drawn from the cards in the hand, then the probability of drawing the 3 is 1/3.

Answer:

P (chosen card is 3) = [tex]\frac{1}{3}[/tex]

Step-by-step explanation:

We are given that a 3​-card hand is dealt from a standard​ 52-card deck, and then one of the 3 cards is chosen at random.

Given that only one of the cards is a 3, we are to find the probability that the chosen card is a 3.

Since only one card is a 3 and one card is chosen so:

P (chosen card is 3) = 1 / 3

Step-by-step explanation:

Assuming that the 350 taking history and the 300 taking math each includes the 270 taking both history and math, then:

N(H or M) = N(H) + N(M) − N(H and M)

N = 350 + 300 − 270

N = 380

There are 380 first-year students taking history or mathematics.

A stem-and-leaf plot of the chemistry exam scores is created by dividing each score into a tens digit (the stem) and a ones digit (the leaf). We then group the data by stem and list the leaves for each stem, giving us a distribution of the exam scores.

To construct a stem-and-leaf plot, you can follow this process:

So, our stem-and-leaf plot would look like this:

5 | 0
7 | 6 7 9 9
8 | 1 2 3 6
9 | 9

In this plot, for example, '7 | 6 7 9 9' means there are scores of 76, 77, 79, and 79.

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Answer:163.54$

Step-by-step explanation:

Given data

Volume of Storage(V)=[tex]{10m^3}[/tex]

Length=2breadth

Let Length be L,Breadth be & height be H

therefore

10=LBH

Now substitutes the values

10=2[tex]{B^2}H[/tex]

5=[tex]{B^2}H[/tex]

Now cost for base is [tex]{C_1}=2{B^2}\times10[/tex]

Cost for side walls is[tex]{C_2}={2LH}\times6+2BH}\times6[/tex]

Now total cost(C)=[tex]{C_1}+{C_2}[/tex]

C=20[tex]{B^2}H[/tex]+[tex]{2LH}\times6[/tex]+[tex]2BH}\times6[/tex]

C=20[tex]{B^2}H[/tex]+24BH+[tex]12BH[/tex]

C=[tex]20{B^2}+36B\times\frac{5}{B^{2}}[/tex]

Now Differentiating With respect to Breadth to get minimum cost

[tex]\frac{\mathrm{d} C}{\mathrm{d} B}=0[/tex]

[tex]we\ get\ B=\sqrt[3]{4.5}=1.65m[/tex]

[tex]L=3.30m[/tex]

[tex]H=1.836m[/tex]

and mimimum cost C

[tex]{C=163.54\$}[/tex]

To find the cost of materials for the cheapest container, we need to minimize the total cost. The total cost function is C(x) = 20x^2 + 180x. However, there is no minimum cost for the container since it cannot have a negative width, resulting in a cost of $0.

Let's denote the width of the rectangular storage container as x. According to the given information, the length of the base is twice the width, so the length would be 2x. The height can be calculated by dividing the volume of the container by the area of the base. Therefore, the height would be 10/(x * 2x) = 5/(2x).

The cost of the base would be the area of the base multiplied by the cost per square meter, which is 10 * (x * 2x) = 20x^2. The cost of the sides would be the sum of each side multiplied by the cost per square meter, which is 6 * (2x * 5) + 6 * (x * 5) + 6 * (2x * 5) = 180x.

To find the cost of materials for the cheapest container, we need to minimize the total cost, which is the sum of the cost of the base and the cost of the sides. Therefore, the total cost function is C(x) = 20x^2 + 180x.

To find the minimum cost, we can take the derivative of C(x) with respect to x, set it equal to 0, and solve for x. The value of x that satisfies this equation will give us the width of the container that minimizes the cost.

C'(x) = 40x + 180 = 0

40x = -180

x = -4.5

Since the width cannot be negative, we disregard this solution.

Therefore, there is no minimum cost for the container since it cannot have a negative width. In this case, the cost of materials for the cheapest container would be $0.

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Answer:

17 bills

Step-by-step explanation:

There are 17 $50 bills is found in $890.

All you have to do is:

890 ÷ 50

However, that would equal 17.8, which is not a whole number. Therefore, there are only 17 $50 bills found in $890.

Answer:

JL=2 units

Step-by-step explanation:

we know that

If k is the midpoint in the line JL

then

JL=JK+KL

JK=KL

substitute the given values

2x-5=3x-8

Solve for x

3x-2x=-5+8

x=3

so

JK=2x-5=2(3)-5=1 units

KL=3x-8=3(3)-8=1 units

therefore

JL=JK+KL=1+1=2 units

Answer:

3/8 , 3/8

Step-by-step explanation:

Assumption: A boy is as likely as a girl

hence P(B)= P(G)= 1/2= 0.5

family has 3 children

find the probability of

A) two girls and a boy.

it can happen in following way

BGG, GGB or GBG

P(BGG)= P(GGB)= P(GBG) = [tex]\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}[/tex]

= 1/8

therefore , probability of two girl and a boy= [tex]\frac{1}{8} +\frac{1}{8} +\frac{1}{8}[/tex] = 3/8

B) At least One boy

it can happen is in

BGG, BBG, BBB

P(BGG)= P(BBG)= P(BBB) = [tex]\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}[/tex]

= 1/8

therefore , probability of at least one boy= [tex]\frac{1}{8} +\frac{1}{8} +\frac{1}{8}[/tex] = 3/8

Answer with explanation:

→Total number of people in the committee=6

→Number of votes in the system represented by q ,

      =6+ 4+4+ 3+ 2+ 1=20

a.→Value of q would represent a simple majority if , q>10 that is it would be greater than 50% of votes, means 11 votes.

b.→Value of q that would represent two-thirds majority

        [tex]=\frac{2}{3}\times 20\\\\=\frac{40}{3}\\\\=13.33[/tex]

Approximately 14 votes.

Final answer:

Allison would need to set aside approximately $21,338.60 in the investment now.

Explanation:

To calculate the amount Allison would have to set aside now in an investment paying 7% annual interest with daily compounding, we can use the formula for compound interest:

[tex]A = P(1 + r/n)^(nt)[/tex]

Where:

Plugging in the values, we get:

[tex]$1,90,000 = P(1 + 0.07/365)^(365 * 39)[/tex]

Simplifying the equation:

[tex]P = $1,90,000 / (1 + 0.07/365)^(365 * 39)[/tex]

Calculating this using a scientific calculator or software, Allison would need to set aside approximately $21,338.60 in the investment now.

Answer:

x (smallest to largest) = 0,45 ,55

y (smallest to largest) = 0,10,40

Step-by-step explanation:

(hours)        Dinghy Rowboat        Labor Available

Metal Work 2             3                       120

Painting         2              2                        110

Let x represent the number of dinghies the company can manufacture and y represent the number of rowboats.

So, total hours for metal work = [tex]2x+3y[/tex]

So, total hours for Painting = [tex]2x+2y[/tex]

So, equation becomes:

[tex]2x+3y\leq 120[/tex]

[tex]2x+2y\leq 110[/tex]

[tex]x\geq 0[/tex]

[tex]y\geq 0[/tex]

Plot the inequalities

Refer the attached figure

So, the vertices of the feasible region are (0,40),(45,10) and (55,0)

So, x values are 0 , 45 and 55

x represents the number of dinghies

So, x (smallest to largest) = 0,45 ,55

y values are 40,10,0

y represent the number of rowboats.

So, y (smallest to largest) = 0,10,40

Answer:  The equation, in "slope-intercept form" ; is:

________________________________________________

                   →   " y  =   x  +  4  " .

________________________________________________

Step-by-step explanation:

________________________________________________

Use the formula for linear equations;  written in "point-slope format" ;

which is:

      y −  y₁  = m( x −  x₁ )  ;

We are given the slope, "m" ;  has a value of:  "1 " ;  

           that is;  " m = 2 " .

________________________________________________

  We are given the coordinates to 1 (one) point on the line; in which the coordinates are in the form of :

    " ( x₁ , y₁ ) " ;  

→   that given point is:  "(10, 6)" ;  

            in which:  x₁ =  10 ;

                             y₁  =  6 .

→ Given:  The slope, "m" equals "1" ;  ________________________________________________

Let's plug our known values into the formula:

    →  "  y  −  y₁  = m( x − x₁ ) "  ;

_______________________________________________

    →   As follows:

    →   " y  −  10  = 1(x − 6) ;

______________________________________________

Now, focus on the "right-hand side of the equation" ;

    →    1(x  −  6) = ? ;   Simplify.

______________________________________________

Note the "distributive property" of multiplication:

            →  a(b + c) =  ab   +   ac ;

As such:  " 1(x  −  6) =  (1*x)  +  (1 * -6) " ;

                             =  1x   + (-6) ;

                             =   x  −  6 ;

[Note that:  " 1 x = 1 * x = x " ;

[Note that   " + (-6) "  =  " ( " − 6 " ) .] ;  

                 →  {since:  "Adding a negative" is the same as:

                                  "subtracting a positive."} ;

________________________________________________

          Now, let us bring down the "left-hand side of the equation" ; &

rewrite the entire equation; as follows:

________________________________________________

                           →   "  y  −  10  =   x  −  6 " ;  

________________________________________________

Note:  We want to rewrite the equation in "slope-intercept form" ;

           that is;  " y = mx + b "  ;

                 in which:  "y" ; stands alone as a single variable on the "left-hand side" of the equation;  with "no coefficients" [except for the "implied coefficient" of " 1 "} ;

                                 "m" is the coefficient of "x" ;

                                 and the "slope of the line" ;  

          Note that "m" may be a "fraction or decimal" ; and may be "positive or negative.

            If the slope is "1" ; (that is "1 over 1" ; or: "[tex]\frac1}{1}[/tex]" ;

then,  " m = 1 " ;  and we can write " 1x " as simply "x" ; since the implied coefficient is "1" ;  

               →  since " 1" , multiplied by any value {in our case, any value for "x"} , equals that same value.

________________________________________________

       "b"  refers to the "y-intercept" of the graph of the equation;  

           that is; the "y-value" of the point at which the graphed line of the equation crosses the "y-axis" ;  

           that is, the "y-value" of the coordinates of the point at which the graphed line of the equation crosses the "y-axis" ;

           that is, the ["y-value" of the]  y-intercept" .  

Note that the value of "b" may be positive or negative, and may be a decimal or fraction.

  If the value for "b" is negative, the equation can be written in the form:

          " y = mx - b " ;

    {since:  " y = mx + (-b) "  is a bit tedious .}

          If the y-intercept is "0" ; (i.e. the line crosses the y-axis at the origin, at point:  " (0,0) " ;  

then we simply write the equation as:  "y = mx " ;  

                                ________________________________________________

So;  we have:              →   " y  − 10  =  x  −  6 " ;  

________________________________________________

       →   We want to rewrite our equation in slope-intercept form,

that is;  " y = mx + b " ;  as explained above.

We can add "10" to each side of the equation ; to isolation the "y" on the "left-hand side" of the equation:

           →  " y  −  10  + 10  =  x  − 6  +  10 " ;

to get:

           →  " y  =  x  +  4 " ;

________________________________________________

          →  which is our answer.

________________________________________________

Note:  This answer:  " y = x + 4 " ;

                  →  is written in the "slope-intercept format";  

                     →   " y = mx +  b "  ;

  in which:  "y" is isolated as a single variable on the "left-hand side of the equation" ;

                   The slope of the equation is "1" ; or an implied value of "1" ;

    that is;   " m  = 1 " ;  

                   "b  =  4 " ;  

         →  {that is;  the "y-value" of the  "y-intercept" —  "(0, 4)" — of the graph of the equation is:  "4 ".} .

________________________________________________

Hope this answer is helpful!

          Best wishes to you in your academic pursuits

               —  and within the "Brainly" community!

________________________________________________      

[tex]x=7.5\overline{26}\\10x=75.\overline{26}\\1000x=7526.\overline{26}\\1000x-10x=7526.\overline{26}-75.\overline{26}\\990x=7451\\x=\dfrac{7451}{990}[/tex]

It can be expressed as a fraction with integer numerator and denominator, so it's a rational number.

Answer:

The probability that we shall make exactly n selections is [tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex].

Step-by-step explanation:

It is given that an urn contains 4 white and 4 black balls and we randomly choose 4 balls. If 2 of them are white and 2 are black, we stop.

The total number of ways to select exactly 2 white and 2 black balls.

[tex]^4C_2\times ^4C_2=\frac{4!}{2!(4-2)!}\times \frac{4!}{2!(4-2)!}=6\times 6=36[/tex]

The total number of ways to select 4 balls from 8 balls is

[tex]^8C_4=\frac{8!}{4!(8-4)!}=\frac{8\times 7\times 6\times 5\times 4!}{4\times 3\times 2\times 1\times !4!}=70[/tex]

The probability of selecting exactly 2 white and 2 black balls is

[tex]p=\frac{36}{70}=\frac{18}{35}[/tex]

The probability of not selecting exactly 2 white and 2 black balls is

[tex]q=1-p=1-\frac{18}{35}=\frac{17}{35}[/tex]

If we not get exactly 2 white and 2 black balls, then we replace the balls in the urn and again randomly select 4 balls.

The probability that we shall make exactly n selections is

[tex]P(X = n)=(q)^{n-1}p[/tex]

[tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex]

Therefore the probability that we shall make exactly n selections is [tex]P(X = n)=(\frac{17}{35})^{n-1}\frac{18}{35}[/tex].

Answer:  0.0971

Step-by-step explanation:

Given : Sample size : [tex]n=26[/tex]

The percent of people are afraid of flying [tex]=43\%[/tex]

Thus the proportion of people are afraid of flying [tex]P=0.43[/tex]

We know that the formula to find the standard deviation of the distribution of sample proportions is given by :-

[tex]\text{S.E.}=\sqrt{\dfrac{P(1-P)}{n}}\\\\\Rightarrow\text{S.E.}=\sqrt{\dfrac{0.43(1-0.43)}{26}}\\\\\Rightarrow\ \text{S.E.}=0.0970923430396\approx0.0971[/tex]

Hence, the standard deviation of the distribution of sample proportions = 0.0971

The standard error for the distribution of sample proportions is approximately 0.0966 (rounded to four decimal places).

The standard error (SE) for a sample proportion can be calculated using the following formula:

SE = √[p(1 - p) / n]

Where:

p is the population proportion (0.43, or 43% expressed as a decimal).

n is the sample size (26).

Let's calculate the standard error:

SE = √[0.43 * (1 - 0.43) / 26]

SE = √[0.43 * 0.57 / 26]

SE = √(0.009351923076923077)

SE ≈ 0.096645

So, the standard error for the distribution of sample proportions is approximately 0.0966 (rounded to four decimal places).

Learn more about Standard Error here:

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Answer:

239,580 ways of seating

Step-by-step explanation:

11 students will be divided into 2 groups. One group of 7 people and one group of 4 people. So first we need to find the number of ways of dividing 11 students into these 2 groups.

First group is of 7 people. We have to select 7 people out of 11. The order of selection does not matter so this is a combination problem. Selecting 7 people from 11 can be expressed as 11C7.

Formula for combination is:

[tex]^{n}C_{r}=\frac{n!}{r!(n-r)!}[/tex]

For the given case this would be:

[tex]^{11}C_{7}=\frac{11!}{7! \times 4!}=330[/tex]

So, there are 330 ways of selecting a group of 7 from 11 students. When these 7 students are selected the remaining 4 will go to the other group. So, we can say there are 330 ways to divide the 11 students in groups of 7 and 4. Note that if you start with group of 4 students, the answer will still the same because 11C4 is also equal to 330.

Next we have to arrange 7 students on a round table. The number of possible arrangements would be = (7 - 1)! = 6! = 720

Similarly, to arrange 4 people on a round table, the number of possible arrangements would be = (4 - 1)! = 3! = 6

Since, for each selection of the 330 groups, there are 720 + 6 possible seating arrangements, so the total number of possible seating arrangements would be:

330 ( 720 + 6) = 239,580 ways

Thus, there are 239,580 ways of seating 11 students.

There are 86400 ways the students can be seated in the dining hall.

There are two circular tables in the dining hall, one can seat 7 people and the other can seat 4 people. The students need to be seated in a way that they can be accommodated on these two tables.

The number of ways the students can be seated is:

1) Assign the even-numbered students to the table that can seat 7 people. There are 6 even-numbered students.

2) Assign the odd-numbered students to the table that can seat 4 people. There are 5 odd-numbered students.

3) Calculate the number of ways these students can be arranged on their respective tables. For the table with 7 seats, there are 6 students to be seated, so the number of ways is 6!. For the table with 4 seats, there are 5 students to be seated, so the number of ways is 5!.

4) Multiply the number of ways for each table to get the total number of ways to seat the students: 6! * 5! = 720 * 120 = 86400.

Therefore, there are 86400 ways the students can be seated.

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