Question

Solid a is a right, rectangular prism, in other words is a box. The length and width are double to form solid b, but the heigth of solid a and solid b are the same. How would each quantity in solid a be changed to find the quantity in solid b

Answer 1

The length and width dimensions of solid 'a' are multiplied by 2 while the height remains the same to get the representative quantity in solid, b

Explanation:

The given parameters of solid 'a' are;

The shape of solid, a = Right, rectangular prism

Let 'l' represent the length of solid 'a', let 'w' represent the width of solid 'a' and let 'h' represent the height of solid 'a'

We have;

The length of solid,
`l_b` = 2·l

The width of solid,
`w_b` = 2·w

The height of solid,
`h_b` = h

Given that we have;

The length of solid, b = 2 × The length of solid, a

The width of solid, b = 2 × The width of solid, a

The height of solid, b = The height of solid, a

The dimensions of length and width of each quantity in solid 'a' will be multiplied by 2 to find the dimension of a similar quantity in solid 'b'

The change in volume from solid 'a' to solid 'b' is given as follows;

The volume of solid, 'a', Vₐ = l × w × h = l·w·h

The volume of solid, 'b',
`V_b` = 2·l × 2·w ×h = 4·l·w·h

`V_b` = 4 × Vₐ

Therefore, the volume of each unit volume in solid 'a' is multiplied by 4 to get the volume of the image of the unit volume in solid 'b'

The cross sectional area of solid 'a', Aₐ = l × w

The cross sectional area of solid 'b',
`A_b` = 2·l × 2·w = 4·l·w

`A_b` = 4 × Aₐ

The cross sectional area of each unit of solid 'a' is multiplied by 4 to get the image of the unit cross sectional area in solid 'b'.

Therefore;

The change in the height and width of of solid 'b' is equal to a change in twice the height and width of solid 'a' at a given height, 'h'

The relationships are;

`l_b` = 2·l,
`w_b` = 2·w,
`h_b` = h