Question
The volume of a rectangular prism can be found by multiplying the base area, B, times the height.
If the volume of the prism is represented by 15x2 + x + 2 and the height is x2, which expression represents B, the area of the base?
15x + +
15x + +
15 + +
15 + +
If the volume of the prism is represented by 15x2 + x + 2 and the height is x2, which expression represents B, the area of the base?
15x + +
15x + +
15 + +
15 + +
Answers
Answered by
GPT-4o mini
To find the base area \( B \) of the rectangular prism, we can use the relationship between volume \( V \), base area \( B \), and height \( h \):
\[
V = B \cdot h
\]
We know that the volume \( V \) is given as \( 15x^2 + x + 2 \) and the height \( h \) is \( x^2 \). We can rearrange the equation to find the base area \( B \):
\[
B = \frac{V}{h}
\]
Substituting the given expressions for \( V \) and \( h \):
\[
B = \frac{15x^2 + x + 2}{x^2}
\]
Next, we can simplify this expression by dividing each term in the numerator by \( x^2 \):
\[
B = \frac{15x^2}{x^2} + \frac{x}{x^2} + \frac{2}{x^2}
\]
This simplifies to:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]
Therefore, the expression that represents the area of the base \( B \) is:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]
This matches the general structure of the provided options, which imply we have multiple ways of representing the expression. Therefore, the correct answer from the given options should be the one that aligns with \( 15 + \frac{1}{x} + \frac{2}{x^2} \). However, the formatting of the question seems inconsistent, as the answer options are incomplete. If the correct choices were given in a proper format, we could then match.
For clarity, the complete expression is:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]
\[
V = B \cdot h
\]
We know that the volume \( V \) is given as \( 15x^2 + x + 2 \) and the height \( h \) is \( x^2 \). We can rearrange the equation to find the base area \( B \):
\[
B = \frac{V}{h}
\]
Substituting the given expressions for \( V \) and \( h \):
\[
B = \frac{15x^2 + x + 2}{x^2}
\]
Next, we can simplify this expression by dividing each term in the numerator by \( x^2 \):
\[
B = \frac{15x^2}{x^2} + \frac{x}{x^2} + \frac{2}{x^2}
\]
This simplifies to:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]
Therefore, the expression that represents the area of the base \( B \) is:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]
This matches the general structure of the provided options, which imply we have multiple ways of representing the expression. Therefore, the correct answer from the given options should be the one that aligns with \( 15 + \frac{1}{x} + \frac{2}{x^2} \). However, the formatting of the question seems inconsistent, as the answer options are incomplete. If the correct choices were given in a proper format, we could then match.
For clarity, the complete expression is:
\[
B = 15 + \frac{1}{x} + \frac{2}{x^2}
\]