Answer to Question #138480 in Calculus for Michael

Question

Nathan is designing a box to keep his pet newt in. To make the box, he is going to start with a solid rectangle and cut squares with sides x cm in length from each corner. The dimensions of the solid rectangle are 51 cm by 45 cm.

a) Determine an equation for a function that models the volume of the box
b) Determine what dimensions can be used to create a box with a volume equal to 7750cm^3. Round final answers to one decimal place
c) Explain why not all of the solutions to the equation could be possible lengths of the squares that Nathan is going to cut out of the rectangle?

Show your work clearly for full marks

Answer 1

The volume of a rectangular parallelepiped is equal to the product of its length, width and height.

In our case, V= a*b*h, where

h=x

a = 51-2x

b = 45-2x

A)

V(x) = (51-2x)(45-2x)x

B)

(51-2x)(45-2x)x = 7750

4x³-192x²+ 2295x --7750 = 0

(4 x ³ - 152 x² + 775 Ñ…) - 40 Ñ…² + 1520 Ñ… - 7750 = 0

x (4 x² - 152 x + 775) + 10 * (- 4x² + 152 x - 775) = 0

(x - 10) (4x² - 152 x + 775) = 0

x₁ = 10

$x_2 = \frac{19+\sqrt{669}}{2}\approx31.9$

$x_3 = \frac{19-\sqrt{669}}{2}\approx6.1$

C)

since the distance is a positive number based on the expression b= 45 -2x

x = 31.9 is not a valid value

Answer:Nathan can cut out squares with that have sides of either 10 cm or 6.1cm