Question

the life expectancy of american women t years after 1975 is given by ( a(t)=0.5t + 78 ). complete parts (a) through (c) below. (a) find and interpret the a - intercept of the graph of this function. choose the correct interpretation of the a - intercept below. a. the a - intercept is 78 which means the life expectancy of american women was 78 years in the year 1975 b. the a - intercept is which means the life expectancy of american women was 0 years in the year (b) what is the slope of the graph of this function? the slope of the graph of this function is (simplify your answer)

Explanation:

Part (a)

Step 1: Recall the formula for A-intercept

The A-intercept occurs when \( t = 0 \) (since the A-intercept is the value of \( A(t) \) when the independent variable \( t = 0 \)). The function is \( A(t)=0.5t + 78 \).

Step 2: Substitute \( t = 0 \) into the function

Substitute \( t = 0 \) into \( A(t) \): \( A(0)=0.5(0)+78 \).
Simplify the expression: \( A(0)=0 + 78=78 \).

Step 3: Interpret the A-intercept

Since \( t = 0 \) corresponds to the year 1975 (because \( t \) is the number of years after 1975), the A-intercept of 78 means that in the year 1975, the life expectancy of American women was 78 years. So option A is correct.

Part (b)

Step 1: Recall the slope-intercept form of a linear function

The slope - intercept form of a linear function is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. In the function \( A(t)=0.5t + 78 \), we can compare it with the slope - intercept form \( y = mx + b \) (where \( y = A(t) \), \( x=t \), \( m \) is the slope, and \( b \) is the A - intercept).

Step 2: Identify the slope

By comparing \( A(t)=0.5t + 78 \) with \( y=mx + b \), we can see that the coefficient of \( t \) (which is the independent variable, similar to \( x \) in the slope - intercept form) is \( 0.5 \). So the slope \( m = 0.5 \).

Answer:

Part (a)

The correct option is A. The A - intercept is 78 which means the life expectancy of American women was 78 years in the year 1975.

Part (b)

The slope of the graph of this function is \( 0.5 \)