Question

every 3 months, the government releases median weekly earnings of full - time wage and salary workers. the line graph shows the median weekly earnings of workers in the years 2019 and 2020. the mathematical model ( d = - 0.09t^2 + 7.5t + 920.5 ) describes a worker’s median weekly earnings, ( d ), in dollars, reported ( t ) months after the beginning of 2019.
a. use the line graph to estimate a worker’s median weekly earnings 3 months into 2020 (that is, 15 months since the beginning of 2019).
b. use the formula to find a worker’s median weekly earnings 3 months into 2020. how does this compare with the estimate in part (a)?
a. according to the line graph, a worker’s median weekly earnings 3 months into 2020 was about $\square$
(round to the nearest ten as needed.)

Explanation:

Part a

Step1: Analyze the line graph

Looking at the line graph, the x - axis represents the number of months since the beginning of 2019, and the y - axis represents the median weekly earnings in dollars. We need to find the value on the y - axis when the x - value (t) is 15 (3 months into 2020, since 2019 has 12 months, 12 + 3=15). From the graph, we can see that at t = 15, the median weekly earnings seem to be around 1050 (by visually inspecting the graph and rounding to the nearest ten).

Step1: Identify the value of t

We are given that t = 15 (3 months into 2020, 12+3 = 15 months since the beginning of 2019). The formula for median weekly earnings is $d=- 0.09t^{2}+7.5t + 920.5$.

Step2: Substitute t = 15 into the formula

First, calculate $t^{2}$ when t = 15: $t^{2}=15^{2}=225$. Then, calculate $-0.09t^{2}$: $-0.09\times225=- 20.25$. Next, calculate $7.5t$: $7.5\times15 = 112.5$. Now, substitute these values into the formula: $d=-20.25 + 112.5+920.5$.

Step3: Simplify the expression

First, add - 20.25 and 112.5: $-20.25 + 112.5=92.25$. Then, add 92.25 and 920.5: $92.25+920.5 = 1012.75$. Rounding to the nearest ten, we look at the ones digit (2). Since 2 < 5, we round down. So, $d\approx1010$.

Now, comparing with the estimate from part (a) (1050), the value from the formula (1010) is less than the estimate from the graph.

Answer:

1050

Part b