Arthur wants to make a raised rectangular frame, shown below, to grow basil plants. The basil plants are transplanted, with their soil, from 2-in.-wide pots into the frame. The diagram at the right below shows a top-down view of the frame. Each circle represents a transplanted basil plant with its soil. Arthur will add more soil to the frame until the soil is 3x in. deep. A cuboid with a height of 3x inches. A rectangle with six circles of two inches diameter placed at equal distance in two rows consisting of three circles each. The distance between each circle and the sides of the rectangle is marked as x inches. Write a polynomial function V to represent the volume of soil in the frame in terms of x. Explain.

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  V = 36x^3 +102x^2 +(72 -18π)x cubic inches

Step-by-step explanation:

The frame contains 6 pots of diameter 2 inches, so their total area is ...

  A = 6(πr^2) = 6(π(1 in)^2) = 6π in^2

The dimensions of the frame, in inches are ...

  length: 4x +6

  width: 3x +4

The area of the pots will subtract from the area of the frame to give the area covered by soil. The other dimension of the frame is ...

  height: 3x

So the volume of soil in the frame is ...

  V = (LW - 6π)(3x)

  V = ((4x +6)(3x +4) -6π)(3x) = 3x(12x^2 +34x +24 -6π)

  V = 36x^3 +102x^2 +(72 -18π)x . . . . cubic inches

Option B is correct

The minimum number of pens the company must sell to make a profit is, 174.

Explanation:

Let x be the number of pens  and y be the cost of the pens.

To find the cost of the equation.

It is given that cost , y , of manufacturing the pens is a quadratic function i.,e

                ......[1]

and  y-intercept of 120 which means that for x=0 , y=120 and Vertex = (250 , 370).

Put x = 0  and y =120  in [1]

120 = 0+0+c

⇒  c= 120.

Since, a quadratic function has axis of symmetry.

The axis of symmetry is given by:

                ......[2]

Substitute the value of x = 250 in [2];

or

                        ......[3]

Substitute the value of x=250, y =370, c =120 and b = -500 a in [1];

 or

or

or

1 = -250 a

⇒

We put the value of a in [3]

So,

b =-500 a=

Simplify:

b =2

Therefore, the cost  price of the pens is:  

And the selling  of the pens is 2x [ as company sell pens $ 2 each]

To find the minimum number of pens the company must sell to make a profit:

profit = selling price - cost price

Since to make minimum profit ; profit =0

then;

or

Simplify:

⇒  or

Simplify:

x =173.205081

or

x = 174 (approx)

Therefore, the minimum number of pens the company must sell to make a profit is, 174