Question

1. the formula ( d = rt ) can be used to show the relationship between distance, rate and time. solve the formula for ( t ).

q. calvin is using the formula ( a = 3r^2 ) to estimate the area of a circle.
a. solve the formula for ( r ).

1. the used to rectang formul

Explanation:

Problem 6: Solve \( d = rt \) for \( t \)

Step 1: Isolate \( t \)

We have the formula \( d = rt \). To solve for \( t \), we divide both sides of the equation by \( r \) (assuming \( r
eq 0 \)).
\[
\frac{d}{r}=\frac{rt}{r}
\]

Step 2: Simplify

Simplifying the right - hand side, \( \frac{rt}{r}=t \) (since \( r
eq0 \), we can cancel out the \( r \) terms). So we get:
\[
t = \frac{d}{r}
\]

Step 1: Isolate \( r^{2} \)

We start with the formula \( A = 3r^{2} \). Divide both sides of the equation by 3:
\[
\frac{A}{3}=\frac{3r^{2}}{3}
\]

Step 2: Simplify and solve for \( r \)

Simplifying the right - hand side gives \( \frac{A}{3}=r^{2} \). Then, take the square root of both sides. Since \( r \) represents the radius (a non - negative quantity in the context of a circle), we have:
\[
r=\sqrt{\frac{A}{3}}=\frac{\sqrt{A}}{\sqrt{3}}=\frac{\sqrt{3A}}{3}
\]
(If we assume the correct formula for the area of a circle \( A = \pi r^{2} \), the steps would be:

Step 1: Isolate \( r^{2} \)

Divide both sides by \( \pi \): \( \frac{A}{\pi}=r^{2} \)

Step 2: Solve for \( r \)

Take the square root: \( r = \sqrt{\frac{A}{\pi}}=\frac{\sqrt{A\pi}}{\pi} \))

Answer:

\( t=\frac{d}{r} \)

Problem 9a: Solve \( A = 3r^{2} \) for \( r \) (assuming this is a typo and should be \( A=\pi r^{2} \) for the area of a circle, we will solve \( A = 3r^{2} \) as given)