Video Transcript
If a rectangular prism has a volume
of 40𝑥𝑦𝑧, a length of 𝑥, and a width of 10𝑦, find its height. Find the surface area of the
rectangular prism in its simplest form.
We begin by sketching the
rectangular prism with the given dimensions. We are told that the volume of the
prism is 40𝑥𝑦z. And we are asked to find its
height, which we’ll call ℎ. We recall that the volume of any
triangular prism can be calculated by multiplying the length by the width by the
height.
In this question, we have 𝑥
multiplied by 10𝑦 multiplied by ℎ is equal to 40𝑥𝑦𝑧. Dividing through by the common
factors of 𝑥 and 𝑦 on both sides, we have 10ℎ is equal to 40𝑧. We can then divide through by 10,
such that ℎ is equal to four 𝑧. The answer to the first part of
this question is that the height of the rectangular prism is four 𝑧, which we can
add to our diagram.
The second part of the question
asks us to find an expression for the surface area of the rectangular prism in its
simplest form. We recall that the surface area of
a rectangular prism is equal to twice the sum of the products of the pairs of side
lengths. This can be written as two
multiplied by 𝑙𝑤 plus 𝑙ℎ plus 𝑤ℎ, where 𝑙, 𝑤, and ℎ are the length, width, and
height of the prism, respectively.
Multiplying the length and width of
our rectangular prism, we have 𝑥 multiplied by 10𝑦, which is equal to 10𝑥𝑦. Next, multiplying 𝑥 and four 𝑧
gives us four 𝑥𝑧. Finally, 10𝑦 multiplied by four 𝑧
is equal to 40𝑦𝑧. Substituting these expressions into
our formula, we see that the surface area is equal to two multiplied by 10𝑥𝑦 plus
four 𝑥𝑧 plus 40𝑦𝑧. Distributing the two across our
parentheses, we have 20𝑥𝑦 plus eight 𝑥𝑧 plus 80𝑦𝑧. This is the surface area of the
rectangular prism in its simplest form.
We now have the two answers to this
question. The height of the rectangular prism
is four 𝑧, and its surface area is 20𝑥𝑦 plus 80𝑦𝑧 plus eight 𝑥𝑧.
