Let's break down each function and question step by step.
Function 1: Cost to attend the fair
Function: \( f(r) = 5 + 1.75r \)
a. What is the y-intercept, and what does it mean?
The y-intercept occurs when \( r = 0 \). So, we find \( f(0) \): \[ f(0) = 5 + 1.75(0) = 5 \] Interpretation: The y-intercept of 5 means that the base cost to attend the fair is $5, regardless of the number of rides taken. This cost could represent a fixed entrance fee.
b. What is the slope, and what does it mean?
The slope of the function is 1.75.
Interpretation: This means that for each additional ride taken (each increment of \( r \) by 1), the total cost increases by $1.75.
c. If Al spent $19.00 at the fair, how many rides did Al ride?
We need to solve the equation \( f(r) = 19 \): \[ 19 = 5 + 1.75r \] Subtracting 5 from both sides: \[ 14 = 1.75r \] Dividing both sides by 1.75: \[ r = \frac{14}{1.75} = 8 \] Answer: Al rode 8 rides.
Function 2: Cost for Mrs. Franklin to go to a buffet
Function: \( f(c) = 6.85 + 2.95c \)
a. What is the y-intercept, and what does it mean?
The y-intercept occurs when \( c = 0 \). So, we find \( f(0) \): \[ f(0) = 6.85 + 2.95(0) = 6.85 \] Interpretation: The y-intercept of 6.85 means that the base cost for Mrs. Franklin to go to the buffet is $6.85, which could represent a fixed fee for herself before considering additional costs for her grandchildren.
b. What is the slope, and what does it mean?
The slope of the function is 2.95.
Interpretation: This means that for each grandchild she takes to the buffet (each increment of \( c \) by 1), the total cost increases by $2.95.
c. If Mrs. Franklin paid $18.65 for the buffet, how many of her grandchildren did she take to the buffet?
We need to solve the equation \( f(c) = 18.65 \): \[ 18.65 = 6.85 + 2.95c \] Subtracting 6.85 from both sides: \[ 11.80 = 2.95c \] Dividing both sides by 2.95: \[ c = \frac{11.80}{2.95} \approx 4 \] Answer: Mrs. Franklin took approximately 4 grandchildren to the buffet.
This concludes the analysis for both functions regarding cost calculations for the fair and the buffet.
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