Question

the functions f(x) = -2.9x + 286 and g(x) = 0.01x² - 5.1x + 370 model the chance, as a percent, that a 60-year-old will survive to age x. the bar graph to the right shows the chances of 60-year-olds surviving to various ages.
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?

a. select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answers.)
a. f(70) = 83, which means that the chance of a 60-year-old surviving to the age of 70 is 83 %.
b. f(70) = , which means that the chance of a 60-year-old surviving to the age of is 70%.

b. select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answers.)
a. g(70) = , which means that the chance of a 60-year-old surviving to the age of is 70%.
b. g(70) = 62, which means that the chance of a 60-year-old surviving to the age of 70 is 62 %.

c. determine the better model for the chance of surviving to age 70. choose the correct answer below.
a. the functions f(x) and g(x) are equally good models.
b. the function g(x) is a better model.
c. neither f(x) nor g(x) are good models.
d. the function f(x) is a better model.

Explanation:

Part a

Step1: Substitute \( x = 70 \) into \( f(x) \)

The function is \( f(x)=-2.9x + 286 \). Substitute \( x = 70 \):
\( f(70)=-2.9\times70 + 286 \)

Step2: Calculate the value

First, calculate \( -2.9\times70=-203 \). Then, \( -203 + 286 = 83 \). So \( f(70) = 83 \), which means the chance of a 60 - year - old surviving to age 70 is 83%.

Step1: Substitute \( x = 70 \) into \( g(x) \)

The function is \( g(x)=0.01x^{2}-5.1x + 370 \). Substitute \( x = 70 \):
\( g(70)=0.01\times70^{2}-5.1\times70 + 370 \)

Step2: Calculate each term

First, \( 0.01\times70^{2}=0.01\times4900 = 49 \). Second, \( -5.1\times70=-357 \). Then, \( 49-357 + 370=49 + 13 = 62 \). So \( g(70)=62 \), which means the chance of a 60 - year - old surviving to age 70 is 62%.

From the bar graph, the chance of a 60 - year - old surviving to age 70 is 88% (from the blue bar at age 70). The value of \( f(70) = 83 \) is closer to 88 than \( g(70)=62 \). So the function \( f(x) \) is a better model.

Answer:

\( f(70)=\boldsymbol{83} \), interpretation: The chance of a 60 - year - old surviving to age 70 is 83%.

Part b