Missing Part of the Question
If n is directly proportional to the cube of m and n = 27 when m = 4
Answer:
i. [tex]n = \frac{27}{64}[/tex]
ii. [tex]n = \frac{20}{3}[/tex]
Explanation:
Given
Direct proportion between n and cube of m
This is represented as:
[tex]n\ \alpha\ m^3[/tex]
Convert proportion to equation
[tex]n = km^3[/tex]
Where k is the constant of variation;
Substitute 27 for n and 4 for m
[tex]27 = k * 4^3[/tex]
[tex]27 = k * 64[/tex]
[tex]27 = 64k[/tex]
Divide both sides by 64
[tex]k = \frac{27}{64}[/tex]
Solving for n when m = 2.
Recall that [tex]n = km^3[/tex]
Substitute [tex]\frac{27}{64}[/tex] for k and 2 for m
[tex]n = \frac{27}{64} * 2^3[/tex]
[tex]n = \frac{27}{64} * 8[/tex]
[tex]n = \frac{27 * 8}{64}[/tex]
[tex]n = \frac{27}{64}[/tex]
Solving for the value of m when n = 125
Recall that [tex]n = km^3[/tex]
Substitute 125 for n and [tex]\frac{27}{64}[/tex] for k
[tex]125 = \frac{27}{64} * n^3[/tex]
[tex]125 = \frac{27 * n^3}{64}[/tex]
[tex]125 = \frac{27 n^3}{64}[/tex]
Multiply both sides by 64
[tex]64 * 125 = \frac{27 n^3}{64} * 64[/tex]
[tex]64 * 125 = 27 n^3[/tex]
Divide both sides by 27
[tex]\frac{64 * 125}{27} = \frac{27n^3}{27}[/tex]
[tex]\frac{64 * 125}{27} =n^3[/tex]
Take Cube root of both sides
[tex]\sqrt[3]{\frac{64 * 125}{27}} = \sqrt[3]{n^3}[/tex]
[tex]\sqrt[3]{\frac{64 * 125}{27}} = n[/tex]
[tex]\frac{4 * 5}{3} = n[/tex]
[tex]\frac{20}{3} = n[/tex]
[tex]n = \frac{20}{3}[/tex]
