Question

Own work.

A small cuboid box has a rectangular base of length 3x cm and width x cm, where x>0. The height is y cm, where y>0. The sum of the length, width and height is 20 cm. [9 marks]
Write down an expression for y in terms of x. [1 mark]


The volume of the box is V cm^3. Find an expression for V in terms of x.

Find dv/dx

Find the value of x for which V is a maximum ​

Answer 1

The expression for y in terms of x is y = 20 - 4x. The expression for the volume of the box is V = 3x^2y. The value of x for which V is a maximum is found by setting 6y - 9x = 0.

Step-by-step explanation:

To find an expression for y in terms of x, we can use the fact that the sum of the length, width, and height is 20 cm.

Given that the length is 3x cm, the width is x cm, and the height is y cm, we can write the equation:

3x + x + y = 20

Simplifying this equation gives us:

4x + y = 20

Thus, the expression for y in terms of x is y = 20 - 4x.

To find the expression for the volume of the box, we multiply the length, width, and height together:

V = (3x)(x)(y) = 3x^2y.

To find the value of x for which V is a maximum, we can take the derivative of V with respect to x and set it equal to 0:

dV/dx = 6xy - 9x^2 = 0

Factor out x to get:

x(6y - 9x) = 0

This equation has two solutions: x = 0 or 6y - 9x = 0. Since x must be greater than 0, the only valid solution is 6y - 9x = 0.

Answer 2

Refer to the attached images.