Chapter 7: Problem 24

Table $7.3$ shows the values of $x$ and $y$. Given that $y$ is proportional to $x$ (a) find an equation connecting $y$ and $x$ (b) calculate the value of $y$ when $x=36$ (c) calculate the value of $x$ when $y=200$ $$ \begin{array}{lclllc} \hline x & 5 & 10 & 15 & 20 & 25 \\ y & 22.5 & 45 & 67.5 & 90 & 112.5 \\ \hline \end{array} $$

Short Answer

Expert verified

Answer: The values of y when x = 36 is 162, and the value of x when y = 200 is approximately 44.44.

Step by step solution

1. Understanding Proportionality

We are given that, $y$ is proportional to $x$. This means $y=kx$ where $k$ is the constant of proportionality.

2. Find the constant of proportionality ($k$) using the given values of $x$ and $y$

Utilize the given values of $x$ and $y$ from the table to find the value of $k$. For example, use the data from the first row, where $x = 5$ and $y = 22.5$. Plug these values into the equation $y = kx$: $22.5 = k(5)$ Now divide both sides of the equation by 5 to find out what $k$ is. $k = \dfrac{22.5}{5} = 4.5$

3. Write the equation connecting $y$ and $x$

Now that we've found the value of $k$, we can write the equation connecting $y$ and $x$: $y = 4.5x$

4. Calculate the value of $y$ when $x = 36$

To calculate the value of $y$ when $x=36$, substitute $x=36$ in the equation we found in step 3: $y = 4.5(36) = 162$ So, when $x = 36$, the value of $y$ is $162$.

5. Calculate the value of $x$ when $y=200$

Next, to calculate the value of $x$ when $y=200$, substitute $y=200$ in the equation we found in step 3: $200 = 4.5x$ Now, divide both sides of the equation by $4.5$ to find the value of $x$: $x = \dfrac{200}{4.5} \approx 44.44$ So, when $y = 200$, the value of $x$ is approximately $44.44$.

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Short Answer

Step by step solution

1. Understanding Proportionality

2. Find the constant of proportionality (\(k\)) using the given values of \(x\) and \(y\)

3. Write the equation connecting \(y\) and \(x\)

4. Calculate the value of \(y\) when \(x = 36\)

5. Calculate the value of \(x\) when \(y=200\)

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