# If the garden is to be 1250 square feet, and the fence along the driveway costs \$6 per foot while on the other three sides it costs only \$2 per foot, find the dimensions that will minimize the cost.

So, the minimum cost is \$400.

### Area of the rectangle:

The area of a rectangle is the region occupied by a rectangle within its four sides or boundaries.

And the formula is,

Given that,

Area of the garden=1250 square feet.

Let, the length be and the breadth be then,

The total cost of the fence is,

Now, differentiating the obtained equation we get,

Therefore the length is 25 ft

Now, calculating the minimum cost,

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Dimensions of rectangular garden:

x = 25 feet   ( sides along the driveway)

y = 50 feet

Step-by-step explanation:

Rectangular area is:

A(r)  = x*y           (1)

if we call x one the driveway side the cost of that side will be

6*x

The cost of the other side parallel to driveway side is 2*x and cost of the others two sides are 4*y

Total costs:  C = 6*x + 2*x  * 4*y     (2)

From equation (1)

A(r)  = 1250 = x*y      ⇒⇒   y = 1250/ x

Plugging that value in equation (2) we get costs as a function of x

that is:

C(x) = 6*x + 2*x +  4* 1250/x

Taking derivatives on both sides of the equation

C´(x)  = 6 + 2 - 5000/x²

C´(x)  = 8 - 5000 /x²

C´(x) = 0       ⇒       8 - 5000 /x² = 0

8*x² -5000 = 0

x² = 5000/8

x² = 625

x = 25 feet

and    y = 1250/ 25

y = 50 ft

C(min) = 50*2*2 + 6*25 + 2*25

C(min) = 200 + 200

C(min) = 400 \$

## Related Questions

Solve for the given variable: 4. 24 - 9x = -3x

5. 4(5x + 2) + 11 = 18x + 3

6. 3x - 8x = -27 – 13

4. x=4

5. x=-8

6. x=8

Step-by-step explanation:

It may take one extra step to get to the solution, but this method always works.

1. find the variable term that is smallest or most negative. Subtract all the terms on that side of the equation from both sides of the equation.

2. collect terms

3. divide the equation by the coefficient of the variable

4. add the opposite of the constant

___

4. The most negative variable term is -9x, which is on the left side. Subtracting (24-9x) from both sides of the equation, we have ...

0 = -3x -24 +9x

0 = -24 +6x

0 = -4 +x . . . . . divide by 6

4 = x . . . . . . . . add the opposite of -4

__

5. The smallest variable term is 18x, on the right. (The variable term on the left is 20x.)

4(5x +2) +11 -18x -3 = 0 . . . subtract the right side

2x +16 = 0 . . . collect terms

x +8 = 0 . . . . . . divide by 2

x = -8 . . . . . . . . add -8

__

6. All variables are on the left side, so we can just collect terms and divide by the coefficient of the variable.

-5x = -40 . . . collect terms

x = 8 . . . . . divide by -5

If you were to literally follow the steps above, you would recognize that -5x is less than 0x (the x-term on the right side of the equation), so you would subtract the left side, giving ...

0 = 5x -40

0 = x -8 . . . . . divide by 5

8 = x . . . . . . . . add 8

_____

Comment on this solution technique

You will often be told to solve these equations by separating the variable terms from the constant terms. This method actually puts the variable terms and constant terms together (and zero on the other side of the equal sign). The constant is separated from the variable as the last step of this solution process, rather than as one of the first steps. By doing this, we don't have to worry about which variable term or which constant term we're going to mess with.

The only reason for choosing the variable term with the smallest (least) coefficient in the first step is to ensure that the resulting variable coefficient is positive. This tends to reduce errors later on. You can also use that same strategy when solving the equation following the "separate constant terms and variable terms" approach.

What is the circumference of a circle with a diameter of 14 cm? Approximate using pi equals 22 over 7.22 cm
44 cm
154 cm
616 cm

Step-by-step explanation:

154 cm

Step-by-step explanation:

What is the slope of the linear table below?

A. 2/3

Step-by-step explanation:

Plot the points, so you visualize rise/run.

When you start at the point (-3,8) it rises 2 then moves 3 to the right to get to the point (0,10). Then it rises 2 and moves 3 to the right to the point (3,12).

Find x. Round to the nearest tenth if needed.

x = 16

Step-by-step explanation:

Use that the addition of all internal angles of a triangle must add up to 180, and the fact that the two given triangles are similar:

51 + 65 + 4 x = 180

combine and solve for "x"

116 + 4 x = 180

4 x = 180 - 116

4 x = 64

x = 64 / 4

x = 16

x = 16

Step-by-step explanation:

A triangle is 180 degrees. Since two of the angles are congruent to each other, the third angle must be congruent as well. This means that the third measure for both triangles is 4x. Knowing all of this, we can solve for x:

65 + 51 + 4x = 180

116 + 4x = 180

4x = 64

x = 16

What is the answer to this equation?? (-3)^2=( )( )=

I did (-3)^2= cause I'm not sure if you meant to put ( ) ( ) in it. If you did well, sorry. I got (-3)^2=9.

For any graph of a hyperbola, which of the following statements describes the locations of the foci, vertices, and center with respect to one another ? Select all that apply .A- the foci are closer to the center than vertices

B- the foci are further from the center than the vertices

C- the center is located at the midpoint of the two foci

D- the center is located at the midpoint of the two vertices