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 + 7.4t + 819.9 ) 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: Identify the time point

We need to find the median weekly earnings at \( t = 15 \) months (3 months into 2020, since 2019 has 12 months) from the line graph.

Step2: Estimate from the graph

Looking at the line graph, when \( t = 15 \) (months since beginning of 2019), the median weekly earnings (y - axis) is around 950 (by observing the graph's grid and the point at \( t = 15 \)). Rounding to the nearest ten, it is 950.

Step1: Substitute \( t = 15 \) into the formula

The formula is \( d=- 0.09t^{2}+7.4t + 819.9 \). Substitute \( t = 15 \) into the formula.
First, calculate \( t^{2}=15^{2} = 225 \).
Then, calculate each term:

• For the first term: \( - 0.09\times225=-20.25 \)
• For the second term: \( 7.4\times15 = 111 \)
• The third term is 819.9.

Step2: Calculate \( d \)

Now, sum up the three terms: \( d=-20.25 + 111+819.9 \)
First, \( - 20.25+111 = 90.75 \)
Then, \( 90.75 + 819.9=910.65\approx911 \) (wait, there might be a miscalculation. Wait, let's recalculate:

Wait, \( -0.09t^{2}+7.4t + 819.9 \) with \( t = 15 \):

\( t^{2}=225 \), so \( - 0.09\times225=-20.25 \)

\( 7.4\times15 = 111 \)

Then \( -20.25+111 = 90.75 \)

Then \( 90.75 + 819.9=910.65\approx911 \)? But wait, maybe I made a mistake in the formula. Wait, the original formula: \( d=-0.09t^{2}+7.4t + 819.9 \). Wait, maybe the coefficient of \( t^{2} \) is \( - 0.09 \) or \( - 0.09t^{2} \)? Wait, let's check again.

Wait, \( t = 15 \):

\( -0.09\times(15)^{2}+7.4\times15 + 819.9 \)

\( -0.09\times225=-20.25 \)

\( 7.4\times15 = 111 \)

So \( -20.25+111 = 90.75 \)

\( 90.75 + 819.9 = 910.65\approx911 \). But the graph estimate was 950. There is a discrepancy. Wait, maybe the formula is \( d=-0.09t^{2}+7.4t + 819.9 \) or maybe a typo? Wait, maybe the coefficient of \( t^{2} \) is \( - 0.09 \) or \( - 0.9 \)? Let's check with \( t = 15 \):

If the formula was \( d=-0.9t^{2}+7.4t + 819.9 \), then \( -0.9\times225=-202.5 \), \( 7.4\times15 = 111 \), \( -202.5 + 111=-91.5 \), \( -91.5+819.9 = 728.4 \), which is too low.

Wait, maybe the original formula is \( d = - 0.09t^{2}+74t + 819.9 \)? Let's try \( t = 15 \):

\( -0.09\times225=-20.25 \), \( 74\times15 = 1110 \), \( -20.25+1110 = 1089.75 \), \( 1089.75+819.9 = 1909.65 \), too high.

Wait, maybe the formula is \( d=-0.09t^{2}+7.4t + 819.9 \), and the graph is for 2019 - 2020, so \( t = 15 \) is 15 months since 2019, so March 2020. Looking at the graph, the y - axis is median weekly earnings, with grid lines. Let's re - examine the graph: the x - axis is months since beginning of 2019 (0, 3, 6, 9, 12, 15, 18, 21, 24). The y - axis: 800, 850, 900, 950, 1000, 1050, 1100. At \( t = 15 \), the point is around 950. Now, using the formula:

Wait, maybe I miscalculated the formula. Let's do it again:

\( d=-0.09t^{2}+7.4t + 819.9 \)

\( t = 15 \)

\( t^{2}=225 \)

\( -0.09\times225=-20.25 \)

\( 7.4\times15 = 111 \)

So \( -20.25+111 = 90.75 \)

\( 90.75+819.9 = 910.65\approx911 \)

So the formula gives approximately 911, while the graph estimate is 950. The formula's result is lower than the graph's estimate.

Answer:

950

Part b