If `a^3+b^3:a^3-b^3=133:117`, then find the value of a:b

Options

A. 0.085416666666667
B. 0.21111111111111
C. 0.20972222222222
D. none of these

Answer

To solve the problem, we need to find the ratio \( a:b \) given the ratio of \( a^3 + b^3 \) to \( a^3 - b^3 \).

**Step 1: Set up the equation based on the given ratio.**
We are given:
\[ \frac{a^3 + b^3}{a^3 - b^3} = \frac{133}{117} \]

**Step 2: Cross-multiply to eliminate the fraction.**
Cross-multiplying gives us:
\[ 117(a^3 + b^3) = 133(a^3 - b^3) \]

**Step 3: Expand both sides of the equation.**
Expanding both sides, we have:
\[ 117a^3 + 117b^3 = 133a^3 - 133b^3 \]

**Step 4: Rearrange the equation to group like terms.**
Now, let's bring all terms involving \( a^3 \) to one side and all terms involving \( b^3 \) to the other side:
\[ 117a^3 + 117b^3 + 133b^3 = 133a^3 \]
This simplifies to:
\[ 117a^3 + 250b^3 = 133a^3 \]

**Step 5: Move \( 133a^3 \) to the left side.**
Subtract \( 133a^3 \) from both sides:
\[ 117a^3 - 133a^3 + 250b^3 = 0 \]
This simplifies to:
\[ -16a^3 + 250b^3 = 0 \]

**Step 6: Rearrange the equation to isolate \( a^3 \) and \( b^3 \).**
Rearranging gives us:
\[ 16a^3 = 250b^3 \]

**Step 7: Divide both sides by \( b^3 \) to find the ratio of \( a^3 \) to \( b^3 \).**
Dividing both sides by \( b^3 \) results in:
\[ \frac{a^3}{b^3} = \frac{250}{16} \]

**Step 8: Simplify the fraction.**
We can simplify \( \frac{250}{16} \):
\[ \frac{250}{16} = \frac{125}{8} \]

**Step 9: Take the cube root of both sides to find the ratio \( a:b \).**
Taking the cube root gives:
\[ \frac{a}{b} = \frac{\sqrt[3]{125}}{\sqrt[3]{8}} = \frac{5}{2} \]

**Final Answer:**
Thus, the ratio \( a:b \) is:
\[ \boxed{5:2} \]

---