Math Problem Statement

If a is inversely proportional to 1/b and bis inversely proportional to c square then find the value of 9a + 5c, when b = 27. Given that when b = 3 then a = 4 and c = 6.

Solution

We are given two inverse proportionalities:

  1. aa is inversely proportional to 1b\frac{1}{b}, meaning a11/ba \propto \frac{1}{1/b}, which implies aba \propto b. So, we can express this as: a=k1ba = k_1 b where k1k_1 is a constant.

  2. bb is inversely proportional to c2c^2, meaning b1c2b \propto \frac{1}{c^2}. So, we can express this as: b=k2c2b = \frac{k_2}{c^2} where k2k_2 is another constant.

Step 1: Find the values of k1k_1 and k2k_2

We are given the following conditions:

  • When b=3b = 3, a=4a = 4 and c=6c = 6.

From a=k1ba = k_1 b:

Using a=4a = 4 and b=3b = 3: 4=k1×3k1=434 = k_1 \times 3 \quad \Rightarrow \quad k_1 = \frac{4}{3} Thus, a=43ba = \frac{4}{3} b.

From b=k2c2b = \frac{k_2}{c^2}:

Using b=3b = 3 and c=6c = 6: 3=k262k2=3×36=1083 = \frac{k_2}{6^2} \quad \Rightarrow \quad k_2 = 3 \times 36 = 108 Thus, b=108c2b = \frac{108}{c^2}.

Step 2: Find the value of aa and cc when b=27b = 27

Using the equation b=108c2b = \frac{108}{c^2}, substitute b=27b = 27: 27=108c2c2=10827=4c=227 = \frac{108}{c^2} \quad \Rightarrow \quad c^2 = \frac{108}{27} = 4 \quad \Rightarrow \quad c = 2

Now, use the equation a=43ba = \frac{4}{3} b and substitute b=27b = 27: a=43×27=36a = \frac{4}{3} \times 27 = 36

Step 3: Find the value of 9a+5c9a + 5c

Now that we have a=36a = 36 and c=2c = 2, calculate 9a+5c9a + 5c: 9a+5c=9×36+5×2=324+10=3349a + 5c = 9 \times 36 + 5 \times 2 = 324 + 10 = 334

Thus, the value of 9a+5c9a + 5c is 334\boxed{334}.


Would you like more details on any of these steps, or do you have further questions?

Here are 5 related questions:

  1. How do we derive the relationship for inverse proportionality?
  2. What happens if both quantities are inversely proportional to each other?
  3. How would the solution change if bb was proportional to c3c^3?
  4. Can we have multiple constants of proportionality in a single problem?
  5. How does inverse proportionality differ from direct proportionality?

Tip: In inverse proportionality, as one quantity increases, the other decreases at a rate dictated by the constant of proportionality.

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Math Problem Analysis

Mathematical Concepts

Inverse Proportionality
Algebra

Formulas

a = k1 * b
b = k2 / c^2
9a + 5c

Theorems

Inverse Proportionality Theorem

Suitable Grade Level

Grades 9-12