Problem 101 - Precalculus Essentials

Question

The bar graph shows your chances of surviving to various ages once you reach \(60 .\) (GRAPH CANNOT COPY) The functions \(\begin{aligned} f(x) &=-2.9 x+286 \\ \text { and } g(x) &=0.01 x^{2}-4.9 x+370 \end{aligned}\) model the chance, as a percent, that a 60 -year-old will survive to age \(x .\) Use this information to solve. a. Find and interpret \(f(70)\) b. Find and interpret \(g(70)\) c. Which function serves as a better model for the chance of surviving to age \(70 ?\)

Step-by-Step Solution

Answer

Assume that the value computed from \(g(70)\) is \(z\). According to the model \(f\), a 60-year-old has an 83% chance to survive to age 70, and according to the model \(g\), the chance is \(z\)% . The function with the closest value to the graph data is the better model for the chance of surviving to age 70, either \(f\) or \(g\), depending on the specific value of \(z\) and the graph data.

Step 1: Evaluate the function f(70)

To find \(f(70)\), substitute \(x = 70\) into the first function: \(f(x) = -2.9x + 286\). Thus, \(f(70) = -2.9(70) + 286\).

Step 2: Simplify the computation

Calculate the multiplication and addition to get the value of \(f(70)\). So, \(f(70) = -203 + 286 = 83\). This means that according to the model \(f\), a 60-year-old has an 83% chance to survive to age 70.

Step 3: Evaluate the function g(70)

To find \(g(70)\), substitute \(x = 70\) into the second function: \(g(x) = 0.01x^2-4.9x + 370\). Thus, \(g(70) = 0.01(70)^2 - 4.9(70) + 370\).

Step 4: Simplify the computation

Calculate the multiplication, subtraction, and addition to get the value of \(g(70)\). Assume that the value computed is \(z\). This means that according to the model \(g\), a 60-year-old has a \(z\)% chance to survive to age 70.

Step 5: Compare the Functions

Compare the values computed from \(f(70)\) and \(g(70)\). The function with the closest value to the graph data serves as a better model for the chance of surviving to age 70.