Problem Analysis:

The problem states that:

1. $a$ is directly proportional to $b^2$, i.e., $a = k_1 b^2$,
2. $a$ is also directly proportional to $\sqrt{c}$, i.e., $a = k_2 \sqrt{c}$.

This implies that $a$ is jointly proportional to $b^2$ and $\sqrt{c}$. Combining these relationships gives: $a = k b^2 \sqrt{c},$ where $k$ is a constant.

We are given:

• When $b = 3$, $c = 162$,
• We need to find $c$ when $b = 5$, rounded to 1 decimal place.

Step 1: Find $k$ using given data.

Substitute $b = 3$, $c = 162$ into $a = k b^2 \sqrt{c}$: $a = k (3^2) \sqrt{162}.$ $a = k (9) \sqrt{162}.$

Solve for $k$: $k = \frac{a}{9 \sqrt{162}}.$

Step 2: Use $k$ to find $c$ when $b = 5$.

Using the same equation, substitute $b = 5$: $a = k (5^2) \sqrt{c}.$ $a = k (25) \sqrt{c}.$

Rearrange to solve for $c$: $\sqrt{c} = \frac{a}{k \cdot 25}.$ $c = \left( \frac{a}{k \cdot 25} \right)^2.$

Let me calculate this step-by-step. math

Given values

b1 = 3 c1 = 162 b2 = 5

Compute k using the given values for b1 and c1

k = 1 / (b1**2 * math.sqrt(c1))

Calculate c when b = b2

c2 = (1 / (k * b2**2))**2

Round to 1 decimal place

c2_rounded = round(c2, 1) c2_roundedThe value of $c$ when $b = 5$ is 21.0.

Would you like a detailed explanation of the steps or have any specific questions?

Here are five related questions to consider:

1. How does joint proportionality affect the relationship between variables?
2. What happens to the proportionality constant $k$ if $a$, $b$, or $c$ changes?
3. How is the square root function critical in this problem?
4. How would the solution change if $a$ were inversely proportional to $\sqrt{c}$?
5. Could you use this method to solve for $b$ given $c$ and $a$?

Tip: Always verify units or values in real-world proportionality problems to ensure they make sense in context!