Question
A sandbox in the shape of a rectangular prism is 1/2 ft. high and needs to contain at least 42 ft.^3 of sand. How would you write an inequality to express the possible area of the base, b, of the sandbox?
42 ____ 1/2b
42 ____ 1/2b
Answers
GPT-4o mini
To express the relationship between the volume of the sandbox and the area of its base, we can use the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Base area} \times \text{Height}
\]
Given that the height of the sandbox is \( \frac{1}{2} \) ft and the volume needs to be at least 42 ft³, we can express this as:
\[
\text{Volume} \geq 42 \text{ ft}^3
\]
Substituting the volume formula into this inequality gives:
\[
\frac{1}{2}b \geq 42
\]
To isolate \( b \), you can multiply both sides by 2:
\[
b \geq 84
\]
Thus, the inequality representing the possible area of the base \( b \) of the sandbox is:
\[
42 \leq \frac{1}{2}b
\]
or, equivalently, to express it explicitly with \( b \):
\[
b \geq 84
\]
\[
\text{Volume} = \text{Base area} \times \text{Height}
\]
Given that the height of the sandbox is \( \frac{1}{2} \) ft and the volume needs to be at least 42 ft³, we can express this as:
\[
\text{Volume} \geq 42 \text{ ft}^3
\]
Substituting the volume formula into this inequality gives:
\[
\frac{1}{2}b \geq 42
\]
To isolate \( b \), you can multiply both sides by 2:
\[
b \geq 84
\]
Thus, the inequality representing the possible area of the base \( b \) of the sandbox is:
\[
42 \leq \frac{1}{2}b
\]
or, equivalently, to express it explicitly with \( b \):
\[
b \geq 84
\]