To analyze the function \( C(f) = 15 + 1.50f \), where \( C \) represents the total cost in dollars of a bouquet with \( f \) flowers, we can answer the questions as follows:
Part A: Maximum Spending
The function shows that the base cost of any bouquet is \( 15 \) dollars, and there is an additional charge of \( 1.50 \) dollars for each flower. Given that the bouquet cannot hold more than 20 flowers, we can calculate the maximum cost by substituting \( f = 20 \) into the cost function.
\[ C(20) = 15 + 1.50 \times 20 \]
Calculating this:
\[ C(20) = 15 + 30 = 45 \]
Thus, the most a customer can spend on a bouquet is $45.
Part B: Graphing the Function
To graph the cost function \( C(f) = 15 + 1.50f \), we will consider the following:
-
Domain: The function applies for \( f \) in the range \( 0 \leq f \leq 20 \).
-
Intercepts:
- Y-intercept: This occurs when \( f = 0 \): \[ C(0) = 15 + 1.50 \times 0 = 15 \]
- X-intercept: This occurs when \( C = 0 \): \[ 0 = 15 + 1.50f \rightarrow 1.50f = -15 \rightarrow f = -10 \] (This means \( C \) never reaches 0 in the context of this function.)
-
Endpoint Calculation: We already calculated the cost when \( f = 20 \), which is \( 45 \).
Now let's plot the points:
- At \( f = 0 \), \( C = 15 \).
- At \( f = 20 \), \( C = 45 \).
Now we can make a straight line on the graph connecting these two points:
- Start at the point \( (0, 15) \).
- End at the point \( (20, 45) \).
Graphing the Function
To graph this, you can visualize or create a graph with the following points:
- \( (0, 15) \)
- \( (20, 45) \)
The line will be straight with a slope, as the function is linear, and you will see it rising as flowers are added to the bouquet.
Final Notes
- The graph will only be plotted in the first quadrant since both \( C \) and \( f \) are positive in this context (the number of flowers and cost).
- The linear function indicates that for each additional flower, the cost increases uniformly by $1.50.
1 answer