Solve the linear programming problem. Minimize and maximize Upper P equals negative 20 x plus 30 y Subject to 2 x plus 3 y greater than or equals 30 2 x plus y less than or equals 26 negative 2 x plus 3 y less than or equals 30 x comma y greater than or equals 0

Answer:

a)  d = 4,712 cm

b) ∠ 108,52 ⁰

Step-by-step explanation:

The crcular path of the spider has radius = 3 cm

If the spider moves from the fa right side of the circle its start poin of the movement is P ( 3 , 0 ) (assuming the circle is at the center of the coordinate system. And the top of the web (going counterclockwise ) is the poin ( 0 , 3 )

So far the spider has traveled 1/4 of the circle

The lenght of the circle is

L = 2*π*r

The traveled distance for the spider (d) is

d = (1/4 )* 2*π*r       ⇒       d = (1/2)*3,1416*3      ⇒  d = 4,712 cm

And now spider go ahead a new distance of approximately 1 more cm

Therefore the spider went a total of 5.712 cm

Now we know that

πrad  =  180⁰    then   1 rad = 180/π         ⇒   1 rad  = 57⁰

rad = lenght of arc/radius  

so  5,712/3    =  1.904 rad

By rule of three

       1 rad           ⇒    57⁰

     1.904 rad     ⇒     ? x

x = 57 * 1.904      ⇒   108.52 ⁰

So the ∠ 108,52 ⁰

Answer:

By definition, the derivative of f(x) is

[tex]lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex]

Let's use the definition for [tex]f(x)=\frac{1}{x}[/tex]

[tex]lim_{h\rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{x-(x+h)}{x(x+h)}}{h}=\\lim_{h\rightarrow 0} \frac{\frac{(-1)h}{x^2+xh}}{h}=\\lim_{h\rightarrow 0} \frac{(-1)h}{h(x^2+xh)}=\\lim_{h\rightarrow 0} \frac{-1}{x^2+xh)}=\frac{-1}{x^2+x*0}=\frac{-1}{x^2}[/tex]

Then, [tex]f'(x)=\frac{-1}{x^2}[/tex]

Step-by-step explanation:

Let [tex]f(x)[/tex] be the number of shares in [tex]x[/tex] days.

Let the function [tex]f(x)=ab^{x}[/tex].

It is given that each friend shares with three friends the next day.

So,[tex]f(x)=3\times f(x-1)[/tex]

substituting [tex]f(x)=ab^{x}[/tex]

[tex]ab^{x}=3\times ab^{x-1}[/tex]

So,[tex]b=3[/tex]

Given that at day [tex]0[/tex],there are [tex]2[/tex] shares.

So,[tex]f(0)=2[/tex]

[tex]a3^{0}=2[/tex]

[tex]a=2[/tex]

So,[tex]f(x)=2\times 3^{x}[/tex]

Answer:

[tex]1117.01 \mathrm{ft}^{2} \text { is the watered are by the sprinkler. }[/tex]

Option: A

Step-by-step explanation:

A sprinkler sprays water over a distance of (r) = 40 feet

Rotates through an angle of (θ) =  80°

80° convert to radians

[tex]\text { Radians }=80^{\circ} \times\left(\frac{\pi}{180}\right)[/tex]

[tex]\text { Radians }=80^{\circ} \times 0.017453292[/tex]

θ in Radians = 1.396263402

We know that,

Area of sprinkler is [tex]\mathrm{A}=\frac{1}{2} \mathrm{r}^{2} \theta[/tex]

Substitute the given values,

[tex]A=\frac{1}{2} \times 40^{2} \times 1.396263402[/tex]

[tex]A=\frac{(1600 \times 1.396263402)}{2}[/tex]

[tex]\mathrm{A}=\frac{2234.021443}{2}[/tex]

[tex]\mathrm{A}=1117.01 \mathrm{ft}^{2}[/tex]

Area of sprinkler is [tex]1117.01 \mathrm{ft}^{2}[/tex]

Answer:

180000

Step-by-step explanation:

Answer:

It take 12 hours for 2 lawnmowers to cut 6 acres of grass

Step-by-step explanation:

We are given that It takes 4 lawnmowers to cut 2 acres of grass in 2 hours.

Let [tex]T_1 , M_1 and W_1[/tex] denotes the time taken , No. of land mowers and Amount of grass respectively in case 1 .

[tex]T_1=2 \\W_1=2\\M_1=4[/tex]

Now The amount of time varies directly with the amount of grass and inversely with the number of lawnmowers

Let [tex]T_2 , M_2 and W_2[/tex] denotes the time taken , No. of land mowers and Amount of grass respectively i case 2

[tex]T_1=? \\W_1=6\\M_1=2[/tex]

So, [tex]\frac{M_1 \times T_1}{W_1}=\frac{M_2 \times T_2}{W_2}[/tex]

Substitute the values

[tex]\frac{4 \times 2}{2}=\frac{2 \times ?}{6}[/tex]

[tex]\frac{4 \times 2 \times 6}{2 \times 2}=?[/tex]

[tex]12=?[/tex]

Hence it take 12 hours for 2 lawnmowers to cut 6 acres of grass

Answer:

D. 12

Step-by-step explanation:

hope this helps :)

Answer:

C

Step-by-step explanation:

f"(x) < 0, which means the function is concave down at all values of x.

For any such function, within the domain of a ≤ x ≤ b, the secant line S(x) is below the curve of f(x), and the tangent line T(x) is above the curve of f(x).

Here's an example:

desmos.com/calculator/fyektbi9yl

Answer:

The arrangement of the given equation in the slope - intercept form are

Step-by-step explanation:

Given:

x + y = 4 and

y - 2x = -5

Slope - intercept form :

[tex]y=mx+c[/tex]

Where,

m is the slope of the line.

c is the y-intercept.

When two points are given say ( x1 , y1 )  and ( x2 , y2)  we can remove slope by

Slope,

[tex]m=\frac{y_{2}- y_{1}}{x_{2}- x_{1}}[/tex]

Intercepts: Where the line cut X axis called X- intercept and where cut Y axis is called Y- intercept.

So, the Slope -intercept form of

x + y = 4   is  [tex]y=-x+4[/tex]

and

y - 2x = -5 is  [tex]y=2x-5[/tex]

Answer:

The answer to your question is the first rain lasted 40 hours and the second rain lasted 30 hours.

Step-by-step explanation:

Data

1st 15 ml/h = x

2nd 30 ml/h = y

Total time = 70 h and 1500 ml

1.- Write 2 equations

                                First rain + second rain = 70 h

                                             x + y = 70       (I)

                                15x + 30 y = 1500       (II)

2.- Solve equations by elimination

Multiply first equation by -15

                               -15x - 15y  = - 1050

                                15x + 30y =  1500

                                  0x + 15y = 450

                                           y = 450 / 15

                                           y = 30 h

                                   x + 30 = 70

                                  x = 70 - 30

                                  x = 40 h

3.- Conclude

The first rain lasted 40 h and the second rain lasted 30 h.

Final answer:

The first rainstorm lasted for 40 hours and the second rainstorm lasted for 30 hours. A system of equations method was used to find the duration of each storm.

Explanation:

Calculating Duration of Rainstorms

Let's denote the duration of the first storm as x hours and the second storm as y hours. The total amount of rain for the first storm would then be 15 mL/hour × x hours and for the second storm 30 mL/hour × y hours. Given that the total duration of the rain is 70 hours, we can set up the following equation: x + y = 70. Additionally, since the total volume of rain is 1500 mL, we have 15x + 30y = 1500.

We can solve this system of equations by multiplying the first equation by 15 to eliminate variable y: 15x + 15y = 1050 and then subtracting this from the second equation: (15x + 30y) - (15x + 15y) = 1500 - 1050, which simplifies to 15y = 450. Thus, y = 30 hours. Finally, we substitute y into x + y = 70 to find x: x = 70 - 30, so x = 40 hours.

Hence, the first storm lasted for 40 hours and the second storm lasted for 30 hours.

Answer:

2,692 cans of tomato soup were sold in January and July

Step-by-step explanation:

The number of cans of soup sold in January = 1346

Store sold same number of soup cans in July.

⇒The number of cans of soup sold in July = 1346

So, the total soup cans sold in January and July

=  Sum of soup cans sold in both months

=  1346 +  1346

=2,692

Hence, 2,692 cans of tomato soup were sold in January and July.

Answer:

70 standard boxes, 40 deluxe boxes

Step-by-step explanation:

standard: x

deluxe: y

x + y = 110

7x + 11y = 930

Multiply the first equation by 7 and subtract.

7x + 7y = 770

-(7x + 11y = 930)

You get -4y = -160

y = 40

Substitute y into the first equation.

x + 40 = 110

x = 70

Answer:

0.006

Step-by-step explanation:

This is joint probability and thus the probability of the given event is the product of:

(probability of drawing a 2 on the first draw)*(prob. of dring a 3 on the second draw), or

4         4          16

----- * ------ = ------------ = 0.006

52     52        (52)^2

Answer:

1/169

Step-by-step explanation:

if you mean a deck as in the 52 cards then the answer is the following.

there are 4 cards in a deck with the number 2 on it. 2 of spades, diamonds, hearts and clubs. so the probability would be 4/42 because there is 4 out of the 52 cards there are. next there is also 4 cards with three on it, so 4/52 again.you multiply these two to get the answer(but before you do i would recommend to simplify it to 1/13) to get 1/169

Answer:

0.05x

This is equivalent to 5% of the total amount of sales.

Step-by-step explanation:

Previous Weekly pay function for Owen; f (x) = 0.15x +325-------------------- (1)

Current   Weekly pay function for Owen; g(x) = 0.20x +325--------------------(2)

Change in Owen's Salary plan is equation (2) minus equation (1)

g(x) - f(x) = (0.20x+325)-(0.15x+325)

              =  0.05x

The new change in Owen's salary is 5% of the total amount of sales

The Chinese exports of automobiles and parts reached a value of approximately $12.3 billion around the year 2004.

Let's first assume that the value of the function f(x) equals the targeted exports, $12.3 billion. Therefore, we have the equation 1.8208e⁽°³³⁵⁷ˣ⁾= 12.3. We can solve this equation for x, which represents the number of years since 1998.

First, divide both sides of the equation by 1.8208 to isolate e⁽°³³⁵⁷ˣ⁾. You will get e⁽°³³⁵⁷ˣ⁾  = 6.7475 approximately. To get rid of the base e, we take the natural logarithm (ln) of both sides. This gives us .3387x = ln(6.7475).

Divide this by .3387 to solve for x. The solution approximates to x = 5.9. This means the exports reach $12.3 billion approximately 6 years after 1998, which would be around the year 2004.

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The year with $12.3 billion in Chinese auto exports is found by setting the provided exponential function equal to it and solving for x, which represents years past 1998. We apply logarithms to solve the equation. The final value of x, when added to 1998, gives us the requested year.

The goal is to find the year when the value of China's exports (represented by f(x)) reaches $12.3 billion. This can be achieved by setting f ( x ) = 1.8208 e .3387 x equal to $12.3 billion and solving for x using logarithmic properties.

Here are the steps:

Once you find the value of x, add this value to the base year 1998 to get the year when the exports reached $12.3 billion.

Note: You require a calculator with the capability to calculate natural logs to solve for x.

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Answer:

  see below

Step-by-step explanation:

The next step in copying the angle is to copy the width of it (the length of its chord) to the new location. First, you have to set the compass to that chord length, GF.

Answer:

2x^2 = 28 - x

-28+x -28+x

2x^2 + x - 28 = 0

ac=-56

m+p=-7,8

(2x-7)(2x+8)=0

2x-7+7=0+7

2x/2=7/2

x=3.5

2x+8-8=0-8

2x/2=-8/2

x= -4

Step-by-step explanation:

x=-4 or 3.5

Final answer:

The quadratic equation 2x² = 28 - x is solved by rearranging it to 2x² + x - 28 = 0, factoring by grouping, and finding the solutions x = 4 and x = -3.5. The solutions are verified by substituting them back into the original equation.

Explanation:

To solve the equation by factoring, we first need to rearrange the equation to set it to zero. The original equation given is 2x² = 28 - x. Moving all terms to one side gives us 2x² + x - 28 = 0. We can solve this quadratic equation by factoring:

First, look for two numbers that multiply to give ac (in this case, 2 * -28 = -56) and add to give b (in this case, 1).

These two numbers are 7 and -8 since 7 * -8 = -56 and 7 + (-8) = -1.

Rewrite the middle term using these two numbers: 2x² + 7x - 8x - 28 = 0.

Factor by grouping: (2x² + 7x) - (8x + 28) = 0.

Factor out the common terms: x(2x + 7) - 4(2x + 7) = 0.

Factor out the common binomial: (x - 4)(2x + 7) = 0.

Solving each factor separately gives us: x = 4 and x = -7/2 or -3.5.

The solutions we find are x = 4 and x = -3.5. To check if these solutions are correct, we can substitute them back into the original equation and verify if they satisfy the equation, thereby confirming they are correct.

Answer:

The probability is 0.74719

Step-by-step explanation:

Let's start defining the random variable X.

X : ''Number of children with hyperlipidemia out of 12 children''

X can be modeled as a binomial random variable.

X ~ Bi (n,p)

Where n is the sample size and p is the ''success probability''.

We defining as a success to find a child that has hyperlipidemia.

The probability function for X is :

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

Where nCx is the combinatorial number define as :

[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]

We are looking for [tex]P(X\geq 3)[/tex]

[tex]P(X\geq 3)=1-P(X\leq 2)[/tex]

[tex]P(X\geq 3)=1-[P(X=0)+P(X=1)+P(X=2)][/tex]

[tex]P(X\geq 3)=1-[(12C0)0.3^{0}0.7^{12}+(12C1)0.3^{1}0.7^{11}+(12C2)0.3^{2}0.7^{10}][/tex]

[tex]P(X\geq 3)=1-(0.7^{12}+0.07118+0.16779)=1-0.25281=0.74719[/tex]

There is a probability of 0.74719 that at least 3 children are hyperlipidemic.

To find the probability that at least 3 out of the 12 children are hyperlipidemic, we can use the binomial probability formula. The probability is 36.21% or 0.3621.

To find the probability that at least 3 out of the 12 children are hyperlipidemic, we need to use the binomial probability formula. The probability of success is 30% or 0.3 (since 30% of children are hyperlipidemic), and the probability of failure is 1 - 0.3 = 0.7.

The formula for the binomial probability is P(X >= k) = 1 - P(X < k), where X follows a binomial distribution with n trials (12 children in this case) and probability of success p (0.3).

To find P(X < k), we need to calculate the probabilities for X = 0, 1, and 2 children being hyperlipidemic and then sum them up.

Summing up these probabilities, we get P(X < 3) = 0.0687 + 0.2332 + 0.3361 = 0.6379

Finally, the probability of at least 3 children being hyperlipidemic is P(X >= 3) = 1 - P(X < 3) = 1 - 0.6379 = 0.3621 or 36.21%.

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Answer:

7

Step-by-step explanation:

Answer:

its 14 signs

Step-by-step explanation:

Answer:

56 and 57

Step-by-step explanation:

The numbers 56 and 57 are values that are very close it is better to estimate 56 and 57 then 6 and 56 and 57

The answer to the teacher's question is that approximately 95% of the sample lies within the range of 45 to 69. This conclusion is based on the features of the normal distribution

The study's results use elements of statistics, with the distribution in question being a normal distribution. A characteristic of a normal distribution is that approximately 95% of measurements will fall within two standard deviations, both above and below the mean. Given that the mean in this case is 57 and the standard deviation is 6, we can calculate the values between which approximately 95% of the sample falls.

Doing the math, we find that: 57 - (2*6) = 45 and 57 + (2*6) = 69. Thus, approximately 95% of the sample lies between the values of 45 and 69.

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Answer:

  (2646 +110.25π) cm² ≈ 2992.4 cm²

Step-by-step explanation:

The area of a sphere is 4 times the area of a circle with the same radius. Hence the area of a hemisphere will be 2 times the area of that circle. This means carving a hemispherical depression in the face of the cube will add an area that is equal to the area of the circular hole.

Of course the total surface area of a cube is 6 times the area of one square face. The area of a circle is ...

  A = πr² = π(d/2)² = (π/4)d²

The total surface area of the carved cube is ...

  S = 6·(21 cm)² + (π/4)·(21 cm)² = (441 cm²)(6 +π/4)

  S ≈ 2992.36 cm²

The total surface area of the remaining block is about 2992.4 cm².

Answer:

(2646 +110.25π) cm² ≈ 2992.4 cm²

Step-by-step explanation:

The problem is solved by using calculus to find the maximum of the area function for a rectangle. The dimensions that yield maximum area with 24 yards of fencing are a length of 6 yards and a width of 12 yards, resulting in an area of 72 square yards.

The problem involves maximizing the area of a rectangle with a fixed perimeter. Let's define the length of the rectangle as x and the width as y. Since you have 24 yards of fencing and you need to enclose three sides of the rectangle, the perimeter equation becomes 2x + y = 24, which can be rearranged as y = 24 - 2x.

The area of the rectangle can be represented by the equation A = xy, and substituting the y value from our perimeter equation, we get A = x(24 - 2x). To maximize this area, we take the derivative of A with respect to x and set it equal to zero, yielding the equation 24 - 4x = 0, or x = 6. Substituting x = 6 back into the y equation gives y = 12.

Therefore, the dimensions that maximize the area of the deck are a length of 6 yards and a width of 12 yards. Substituting these values back into the area equation gives a maximum area of 72 square yards.

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Answer:

(B) The correct interpretation of this interval is that 90% of the students in the population should have their scores improve by between 72.3 and 91.4 points.

Step-by-step explanation:

Confidence interval is the range the true values fall in under a given confidence level.

Confidence level states the probability that a random chosen sample performs the surveyed characteristic in the range of confidence interval. Thus,

90% confidence interval means that there is 90% probability that the statistic (in this case SAT score improvement) of a member of the population falls in the confidence interval.

The 95% confidence interval for the mean weight of all bags of oranges is calculated using the sample mean and the sample standard deviation. After calculation, we are 95% confident the mean weight is between 9.193 and 9.757 pounds.

To find the 95% confidence interval for the mean weight of all bags of oranges, we first need to calculate the sample mean and sample standard deviation. The sample mean, [tex]\overline{x}[/tex], is the sum of all sample weights divided by the number of samples. In this case, it is (9.3+9.7+9.2+9.7)/4 = 9.475 pounds.

The sample standard deviation, s, is the square root of the sum of the squared differences between each sample weight and the sample mean, divided by the number of samples minus one. The s value here is approximately 0.253 pounds.

For a 95% confidence interval with a sample size of 4, the z-score is 2.776 (obtained from a standard z-table). The margin of error is the z-score multiplied by the standard deviation divided by the square root of the sample size. This is approximately 0.282 pounds. So, the 95% confidence interval is (9.475-0.282, 9.475+0.282) = (9.193, 9.757) pounds.

So we can say that we are 95% confident that the mean weight of all bags of oranges is between 9.193 and 9.757 pounds.

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Answer:

- 0.2 cm/s

Step-by-step explanation:

Volume of a cylinder =  πr²l--------------------------------------------------------(1)

dV/dt =(dV /dr ) x (dr/dt) +  (dV /dl ) x (dl/dt) ---------------------------------(2)

dV/dr = 2 πrl

dV/dl  =  πr²

dr/dt   = Unknown

dl/dt   = 0.1 cm/s

dV/dt = 0

From equation (1), the length of the cylinder can be calculated when r = 5cm

V = πr²L

100 = π (5)²L

L =100/25π

  =4/π

To find the rate of radius change  (dr/dt) we substitute known values into equation (2):

0 = 2 π (5) (4/π) x  (dr/dt) + π(5)² x 0.1

0 =  40 (dr/dt)  + 2.5π

40 (dr/dt) = -2.5π

     dr/dt =   -2.5π/40

              =  -0.1963 cm/s

              ≈ - 0.2 cm/s

The negative sign shows that the radius of the cylinder of constant volume decrease at a rate twice the length.

Answer:

Step-by-step explanation:

First we have to find the worth of 50 Ib of both mixtures, so we multiply 50 by 2.72 since the owner wants to sell the mixture at $2.72 per pound

50 Multiplied by $2.72 equals $135

Then we divide that amount by 2 since we are considering two types of products, raisins and nuts

$135 divided by 2 equals $67.5

So we find the amount of raisins that is worth $67.5 and we know a pound of raisins cost $3.2

We divide $67.5 by $3.2 which will give 21.09 Ib (Amount for the raisins)

We divide $67.5 by $2.4 which will give 28.13 Ib (Amount for the nuts)

Final answer:

The store owner should mix 20 lb of raisins with 30 lb of nuts to make a 50 lb mixture that can be sold for $2.72 per pound. This is determined by solving a system of linear equations.

Explanation:

The owner of the store wants to mix raisins and nuts to sell a 50 lb mixture for $2.72 per pound. To find out the quantities of raisins and nuts he should use, we can set up a system of equations. Let R represent the amount of raisins in pounds and N represent the amount of nuts in pounds. The two equations area:

We can solve this system using the substitution or elimination method. Assuming we choose elimination, we can multiply the first equation by 2.40 to make the coefficient of N the same in both equations:

Subtracting these equations gives us:

Dividing both sides by 0.80 gives us the amount of raisins:

Using the first equation (R + N = 50), we can find the amount of nuts:

So the owner should mix 20 lb of raisins with 30 lb of nuts to make the mixture.

Answer:

D) (0.443, 0.497)

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The confidence interval for a proportion is given by this formula

[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

For the 99% confidence interval the value of [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.

[tex]z_{\alpha/2}=2.58[/tex]

And replacing into the confidence interval formula we got:

[tex]0.47 - 2.58 \sqrt{\frac{0.47(1-0.47)}{2294}}=0.443[/tex]

[tex]0.47 + 2.58 \sqrt{\frac{0.47(1-0.47)}{2294}}=0.497[/tex]

And the 99% confidence interval would be given (0.443;0.497).

We are confident that about 44.3% to 49.7% of registered voters approved of the job Barack Obama was doing as president.

Answer:

  A:  x = 20

  C:  x = -9

Step-by-step explanation:

You can solve the inequality and compare that with the offered choices, or you can try the choices in the inequality to see if it is true. Either approach works, and they take about the same effort.

Solving it:

Unfold it ...

  -17 ≤ x -7.5 ≤ 17

Add 7.5 ...

  -9.5 ≤ x ≤ 24.5

The numbers 20 and -9 are in this range: answer choices A and C.

_____

Trying the choices:

A:  |20 -7.5| = 12.5 ≤ 17 . . . . this works

B:  |-10 -7.5| = 17.5 . . . doesn't work

C:  |-9 -7.5| = 16.5 ≤ 17 . . . . .this works

D:  |27-7.5| = 19.5 . . . doesn't work

The choices that work are answer choices A and C.

Answer:

Minimum surface area =[tex]70.77 cm^2[/tex]

Step-by-step explanation:

We are given that

Width of container=x cm

Length of container=2x cm

Volume of container=[tex]54 cm^3[/tex]

We have to find the minimum surface areas that this container will have.

Volume of container=[tex]l\times b\times h[/tex]

[tex]x\times 2x\times h=54[/tex]

[tex]2x^2h=54[/tex]

[tex]h=\frac{54}{2x^2}=\frac{27}{x^2}[/tex]

Surface area of container=[tex]2(b+l)h+lb[/tex]

Because the container does not have lid

Surface area of container=[tex]S=2(2x+x)\times \frac{27}{x^2}+2x\times x[/tex]

[tex]S=\frac{162}{x}+2x^2[/tex]

Differentiate w.r.t x

[tex]\frac{dS}{dx}=-\frac{162}{x^2}+4x[/tex]

[tex]\frac{dx^n}{dx}=nx^{n-1}[/tex]

Substitute [tex]\frac{dS}{dx}=0[/tex]

[tex]-\frac{162}{x^2}+4x=0[/tex]

[tex]4x=\frac{162}{x^2}[/tex]

[tex]x^3=\frac{162}{4}=40.5[/tex]

[tex]x^3=40.5[/tex]

[tex]x=(40.5)^{\frac{1}{3}}[/tex]

[tex]x=3.4[/tex]

Again differentiate w.r.t x

[tex]\frac{d^2S}{dx^2}=\frac{324}{x^3}+4[/tex]

Substitute x=3.4

[tex]\frac{d^2S}{dx^2}=\frac{324}{(3.4)^3}+4=12.24>0[/tex]

Hence, function is minimum at x=3.4

Substitute x=3.4

Then, we get

Minimum surface area =[tex]\frac{162}{(3.4)}+2(3.4)^2=70.77 cm^2[/tex]

Answer: 3/13 is closer to 0 than 1/2 or 1.

- This is because 1/2 of 1/13 would be about 6.5/13.

- Since 3/13 is not close to 1/2, nor it will be to 1.

Answer: option B is the correct answer

Step-by-step explanation:

Candace's backyard has a flower garden that covers 40% of her backyard.

If the total area of her backyard is 450 square feet, it means that her garden covers 40% of 450 square feet.

To determine how many square feet is her flower garden, we will find the value of 40% of 450 square feet. It becomes

40/100 × 450= 0.4×460 = 180 square feet