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Difficulty: MediumLinear Functions and Graphs

A state department of transportation models the relationship between the age of a highway, in years, and its road quality index on a scale from 00 to 100100. The index decreases at a constant rate with respect to the age of the highway. At age 44 years, the highway has a road quality index of 8282. At age 1212 years, the highway has a road quality index of 6666. According to the model, after how many years from its construction will the highway's road quality index reach 5050?

Answer: 20 years

Answer

According to the model, the highway's road quality index will reach 5050 after 2020 years.
Since the relationship is linear, the rate of change is constant. Using the points (4,82)(4, 82) and (12,66)(12, 66), the slope is 66−8212−4=−2\frac{66 - 82}{12 - 4} = -2. The linear function can be written in point-slope form as I−82=−2(t−4)I - 82 = -2(t - 4), which simplifies to I=−2t+90I = -2t + 90. Setting I=50I = 50 yields the equation 50=−2t+9050 = -2t + 90. Subtracting 9090 from both sides gives −40=−2t-40 = -2t, and dividing by −2-2 results in t=20t = 20. Therefore, the index reaches 5050 after 2020 years.

Step-by-Step Solution

1
Calculate the slope (rate of change) of the linear function.
The slope is −2-2.
Since the index decreases at a constant rate, the relationship is linear. The slope is calculated as the change in the road quality index divided by the change in the highway's age: 66−8212−4=−168=−2\frac{66 - 82}{12 - 4} = \frac{-16}{8} = -2.
2
Determine the linear equation relating the road quality index, II, and the age, tt.
I=−2t+90I = -2t + 90
Using the point-slope formula with the point (4,82)(4, 82) and slope −2-2: I−82=−2(t−4)⇒I−82=−2t+8⇒I=−2t+90I - 82 = -2(t - 4) \Rightarrow I - 82 = -2t + 8 \Rightarrow I = -2t + 90.
3
Solve for the age, tt, when the index II is 5050.
t=20t = 20
Substitute 5050 for II in the equation: 50=−2t+9050 = -2t + 90. Subtract 9090 from both sides to get −40=−2t-40 = -2t. Divide by −2-2 to find t=20t = 20.

Key Concept

Writing and solving linear equations from two points
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Topics

Linear Functions and Graphs

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