To estimate the cost of a flight for a distance of 1100 miles using the given table, we can use the method of linear interpolation. Here is the step-by-step solution:
1. Identify the known values around the target miles (1100 miles):
- The table gives the costs for 1000 miles and 1200 miles.
- From the table:
- At 1000 miles, the cost is [tex]$456.
- At 1200 miles, the cost is $[/tex]596.
2. Set up the linear interpolation formula:
Linear interpolation estimates the value at a given point within two known values. The formula for linear interpolation between [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] at a point [tex]\( x \)[/tex] is:
[tex]\[
y = y_1 + (x - x_1) \cdot \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For this problem:
- [tex]\( x_1 = 1000 \)[/tex] miles
- [tex]\( y_1 = 456 \)[/tex] dollars
- [tex]\( x_2 = 1200 \)[/tex] miles
- [tex]\( y_2 = 596 \)[/tex] dollars
- [tex]\( x = 1100 \)[/tex] miles (the target we need the cost for)
3. Plug the values into the formula:
[tex]\[
\text{Cost at 1100 miles} = 456 + (1100 - 1000) \cdot \frac{596 - 456}{1200 - 1000}
\][/tex]
4. Simplify the calculation:
- First calculate the difference in costs and miles:
- [tex]\( 596 - 456 = 140 \)[/tex] dollars
- [tex]\( 1200 - 1000 = 200 \)[/tex] miles
- Next, compute the fraction of the cost change per mile:
[tex]\[
\frac{140}{200} = 0.7 \text{ dollars per mile}
\][/tex]
- Then, calculate the difference between the target miles and the lower boundary:
[tex]\[
1100 - 1000 = 100 \text{ miles}
\][/tex]
- Finally, compute the estimated cost difference:
[tex]\[
100 \cdot 0.7 = 70 \text{ dollars}
\][/tex]
5. Calculate the total estimated cost at 1100 miles:
[tex]\[
456 + 70 = 526 \text{ dollars}
\][/tex]
Thus, the estimated cost of the flight for 1100 miles is $526.
Therefore, the correct answer is:
B. 526.
