The problem consists of two parts: (b) Determine the relationship between $x$ and $y$ given a table of values, and complete the sentence "y is inversely proportional to ...". (c) If $M$ is directly proportional to $L^3$, determine how many times larger $M$ becomes when $L$ is multiplied by 2.

1. Problem Description

The problem consists of two parts:

(b) Determine the relationship between $x$ and $y$ given a table of values, and complete the sentence "y is inversely proportional to ...".

(c) If $M$ is directly proportional to $L^3$, determine how many times larger $M$ becomes when $L$ is multiplied by

2.

2. Solution Steps

(b) We are given the following data:

When $x = 2$, $y = 25$.

When $x = 10$, $y = 1$.

Let's examine the product of $x$ and $y$ for each pair of values:

For the first pair, $x \cdot y = 2 \cdot 25 = 50$.

For the second pair, $x \cdot y = 10 \cdot 1 = 10$.

Since the product is not constant, $y$ is not inversely proportional to $x$.

Let's examine if $y$ is inversely proportional to $x^2$. Then $y = k / x^2$ or $x^2 y = k$.

For the first pair, $x^2 y = 2^2 \cdot 25 = 4 \cdot 25 = 100$.

For the second pair, $x^2 y = 10^2 \cdot 1 = 100 \cdot 1 = 100$.

Since the product $x^2y$ is constant, $y$ is inversely proportional to $x^2$.

(c) We are given that $M$ is directly proportional to $L^3$, so we can write

$M = kL^3$, where $k$ is a constant of proportionality.

If $L$ is multiplied by 2, let the new value of $L$ be $L' = 2L$. Then the new value of $M$, denoted by $M'$, is

$M' = k(L')^3 = k(2L)^3 = k(8L^3) = 8(kL^3) = 8M$.

So $M' = 8M$.

Therefore, $M$ becomes 8 times larger.

3. Final Answer

(b) $x^2$

(c) 8