Chavez Harrington
05/13/2023 · Elementary School
\( 7 y \) is inversely proportional to the cube of \( x \), and \( y=1 \) when \( x=4 \). Find \( x \) when \( y=8 \)
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Tutor-Verified Answer
Step-by-step Solution
Since \( 7y \) is inversely proportional to the cube of \( x \), we can express this relationship mathematically as:
\[
7y = \frac{k}{x^3}
\]
where \( k \) is a constant.
Given that \( y = 1 \) when \( x = 4 \), we can substitute these values into the equation to find \( k \):
\[
7(1) = \frac{k}{4^3}
\]
Calculating \( 4^3 \):
\[
4^3 = 64
\]
Now substituting this back into the equation:
\[
7 = \frac{k}{64}
\]
To find \( k \), we multiply both sides by 64:
\[
k = 7 \times 64 = 448
\]
Now we have the relationship:
\[
7y = \frac{448}{x^3}
\]
Next, we need to find \( x \) when \( y = 8 \). Substituting \( y = 8 \) into the equation:
\[
7(8) = \frac{448}{x^3}
\]
Calculating \( 7(8) \):
\[
56 = \frac{448}{x^3}
\]
Now, we can solve for \( x^3 \) by multiplying both sides by \( x^3 \):
\[
56x^3 = 448
\]
Next, divide both sides by 56:
\[
x^3 = \frac{448}{56}
\]
Calculating \( \frac{448}{56} \):
\[
\frac{448}{56} = 8
\]
Thus, we have:
\[
x^3 = 8
\]
Taking the cube root of both sides gives:
\[
x = 2
\]
Therefore, the value of \( x \) when \( y = 8 \) is:
\[
\boxed{2}
\]
Quick Answer
The value of \( x \) when \( y = 8 \) is 2.
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