Answer :
Sure, let's work through this step-by-step!
### Part a: Write the relationship as a function [tex]\( r = f(t) \)[/tex]
We start with the given equation:
[tex]\[ 7r + 8t = 7 \][/tex]
Our goal is to express [tex]\( r \)[/tex] as a function of [tex]\( t \)[/tex]. To do this, we'll solve for [tex]\( r \)[/tex]:
1. Subtract [tex]\( 8t \)[/tex] from both sides of the equation:
[tex]\[ 7r = 7 - 8t \][/tex]
2. Divide both sides by 7 to isolate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{7 - 8t}{7} \][/tex]
Therefore, the function [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex] is:
[tex]\[ f(t) = \frac{7 - 8t}{7} \][/tex]
### Part b: Evaluate [tex]\( f(-7) \)[/tex]
To find [tex]\( f(-7) \)[/tex], we'll substitute [tex]\( t = -7 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
[tex]\[ f(-7) = \frac{7 - 8(-7)}{7} \][/tex]
Calculate inside the parentheses first:
[tex]\[ 7 - 8(-7) = 7 + 56 = 63 \][/tex]
Now, divide by 7:
[tex]\[ f(-7) = \frac{63}{7} = 9 \][/tex]
So, [tex]\( f(-7) = 9 \)[/tex].
### Part c: Solve [tex]\( f(t) = 49 \)[/tex]
We need to find the value of [tex]\( t \)[/tex] such that [tex]\( f(t) = 49 \)[/tex]. We'll set up the equation:
[tex]\[ \frac{7 - 8t}{7} = 49 \][/tex]
To clear the fraction, multiply both sides by 7:
[tex]\[ 7 - 8t = 343 \][/tex]
Next, solve for [tex]\( t \)[/tex]:
1. Subtract 7 from both sides:
[tex]\[ -8t = 343 - 7 \][/tex]
[tex]\[ -8t = 336 \][/tex]
2. Divide by -8:
[tex]\[ t = \frac{336}{-8} \][/tex]
[tex]\[ t = -42 \][/tex]
So, the value of [tex]\( t \)[/tex] that makes [tex]\( f(t) = 49 \)[/tex] is [tex]\( t = -42 \)[/tex].
### Summary
a. [tex]\( f(t) = \frac{7 - 8t}{7} \)[/tex]
b. [tex]\( f(-7) = 9 \)[/tex]
c. [tex]\( t = -42 \)[/tex]
I hope this helps! Let me know if you have any more questions.
### Part a: Write the relationship as a function [tex]\( r = f(t) \)[/tex]
We start with the given equation:
[tex]\[ 7r + 8t = 7 \][/tex]
Our goal is to express [tex]\( r \)[/tex] as a function of [tex]\( t \)[/tex]. To do this, we'll solve for [tex]\( r \)[/tex]:
1. Subtract [tex]\( 8t \)[/tex] from both sides of the equation:
[tex]\[ 7r = 7 - 8t \][/tex]
2. Divide both sides by 7 to isolate [tex]\( r \)[/tex]:
[tex]\[ r = \frac{7 - 8t}{7} \][/tex]
Therefore, the function [tex]\( r \)[/tex] in terms of [tex]\( t \)[/tex] is:
[tex]\[ f(t) = \frac{7 - 8t}{7} \][/tex]
### Part b: Evaluate [tex]\( f(-7) \)[/tex]
To find [tex]\( f(-7) \)[/tex], we'll substitute [tex]\( t = -7 \)[/tex] into the function [tex]\( f(t) \)[/tex]:
[tex]\[ f(-7) = \frac{7 - 8(-7)}{7} \][/tex]
Calculate inside the parentheses first:
[tex]\[ 7 - 8(-7) = 7 + 56 = 63 \][/tex]
Now, divide by 7:
[tex]\[ f(-7) = \frac{63}{7} = 9 \][/tex]
So, [tex]\( f(-7) = 9 \)[/tex].
### Part c: Solve [tex]\( f(t) = 49 \)[/tex]
We need to find the value of [tex]\( t \)[/tex] such that [tex]\( f(t) = 49 \)[/tex]. We'll set up the equation:
[tex]\[ \frac{7 - 8t}{7} = 49 \][/tex]
To clear the fraction, multiply both sides by 7:
[tex]\[ 7 - 8t = 343 \][/tex]
Next, solve for [tex]\( t \)[/tex]:
1. Subtract 7 from both sides:
[tex]\[ -8t = 343 - 7 \][/tex]
[tex]\[ -8t = 336 \][/tex]
2. Divide by -8:
[tex]\[ t = \frac{336}{-8} \][/tex]
[tex]\[ t = -42 \][/tex]
So, the value of [tex]\( t \)[/tex] that makes [tex]\( f(t) = 49 \)[/tex] is [tex]\( t = -42 \)[/tex].
### Summary
a. [tex]\( f(t) = \frac{7 - 8t}{7} \)[/tex]
b. [tex]\( f(-7) = 9 \)[/tex]
c. [tex]\( t = -42 \)[/tex]
I hope this helps! Let me know if you have any more questions.
