Answer :
Let's solve the problem step-by-step.
### Part (a): Write the relationship as a function [tex]\( r = f(t) \)[/tex].
Given the equation [tex]\( 3r + 4t = 24 \)[/tex], we need to express [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex].
1. Start by isolating [tex]\( r \)[/tex] on one side of the equation:
[tex]\[
3r + 4t = 24
\][/tex]
2. Subtract [tex]\( 4t \)[/tex] from both sides:
[tex]\[
3r = 24 - 4t
\][/tex]
3. Divide every term by 3 to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{24 - 4t}{3}
\][/tex]
So, the function [tex]\( r = f(t) \)[/tex] is:
[tex]\[
r = f(t) = \frac{24 - 4t}{3}
\][/tex]
### Part (b): Evaluate [tex]\( f(-3) \)[/tex].
To find [tex]\( f(-3) \)[/tex], we substitute [tex]\( t = -3 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
1. Substitute [tex]\( -3 \)[/tex] for [tex]\( t \)[/tex]:
[tex]\[
r = f(-3) = \frac{24 - 4(-3)}{3}
\][/tex]
2. Calculate inside the parentheses:
[tex]\[
24 + 12 = 36
\][/tex]
3. Divide by 3:
[tex]\[
r = \frac{36}{3} = 12
\][/tex]
Thus, [tex]\( f(-3) = 12 \)[/tex].
### Part (c): Solve [tex]\( f(t) = 16 \)[/tex].
To find [tex]\( t \)[/tex] for which [tex]\( f(t) = 16 \)[/tex], set the function equal to 16:
1. Set the equation:
[tex]\[
\frac{24 - 4t}{3} = 16
\][/tex]
2. Multiply both sides by 3 to eliminate the fraction:
[tex]\[
24 - 4t = 48
\][/tex]
3. Subtract 24 from both sides to isolate terms with [tex]\( t \)[/tex]:
[tex]\[
-4t = 24
\][/tex]
4. Divide by [tex]\(-4\)[/tex] to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{24}{-4} = -6
\][/tex]
Therefore, [tex]\( t = -6 \)[/tex].
These are the solutions to each part:
- [tex]\( r = f(t) = \frac{24 - 4t}{3} \)[/tex]
- [tex]\( f(-3) = 12 \)[/tex]
- [tex]\( t = -6 \)[/tex] when [tex]\( f(t) = 16 \)[/tex]
### Part (a): Write the relationship as a function [tex]\( r = f(t) \)[/tex].
Given the equation [tex]\( 3r + 4t = 24 \)[/tex], we need to express [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex].
1. Start by isolating [tex]\( r \)[/tex] on one side of the equation:
[tex]\[
3r + 4t = 24
\][/tex]
2. Subtract [tex]\( 4t \)[/tex] from both sides:
[tex]\[
3r = 24 - 4t
\][/tex]
3. Divide every term by 3 to solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{24 - 4t}{3}
\][/tex]
So, the function [tex]\( r = f(t) \)[/tex] is:
[tex]\[
r = f(t) = \frac{24 - 4t}{3}
\][/tex]
### Part (b): Evaluate [tex]\( f(-3) \)[/tex].
To find [tex]\( f(-3) \)[/tex], we substitute [tex]\( t = -3 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
1. Substitute [tex]\( -3 \)[/tex] for [tex]\( t \)[/tex]:
[tex]\[
r = f(-3) = \frac{24 - 4(-3)}{3}
\][/tex]
2. Calculate inside the parentheses:
[tex]\[
24 + 12 = 36
\][/tex]
3. Divide by 3:
[tex]\[
r = \frac{36}{3} = 12
\][/tex]
Thus, [tex]\( f(-3) = 12 \)[/tex].
### Part (c): Solve [tex]\( f(t) = 16 \)[/tex].
To find [tex]\( t \)[/tex] for which [tex]\( f(t) = 16 \)[/tex], set the function equal to 16:
1. Set the equation:
[tex]\[
\frac{24 - 4t}{3} = 16
\][/tex]
2. Multiply both sides by 3 to eliminate the fraction:
[tex]\[
24 - 4t = 48
\][/tex]
3. Subtract 24 from both sides to isolate terms with [tex]\( t \)[/tex]:
[tex]\[
-4t = 24
\][/tex]
4. Divide by [tex]\(-4\)[/tex] to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{24}{-4} = -6
\][/tex]
Therefore, [tex]\( t = -6 \)[/tex].
These are the solutions to each part:
- [tex]\( r = f(t) = \frac{24 - 4t}{3} \)[/tex]
- [tex]\( f(-3) = 12 \)[/tex]
- [tex]\( t = -6 \)[/tex] when [tex]\( f(t) = 16 \)[/tex]
