Question

the graphs, on the right, show the percentage of jobs in a countrys labor force held by men and women from 1970 through 2015.
the formula ( p = 1.3sqrt{t} + 39 ) models the percentage of jobs in the labor force, ( p ), held by women ( t ) years after 1970. answer parts a through c.

a. use the appropriate graph to estimate the percentage of jobs in the labor force held by women in 2010.
( p = 47% )
(round to the nearest whole number as needed.)

b. use the mathematical model to determine the percentage of jobs in the labor force held by women in 2010.
( p = 47.2% )
(round to the nearest tenth as needed.)

c. according to the formula, when will 55% of jobs in the labor force be held by women?
55% of jobs in the labor force will be held by women by ( square ).
(round to the nearest whole number as needed.)

Explanation:

Step1: Set up the equation

We know the formula \( p = 1.3\sqrt{t}+39 \), and we want to find \( t \) when \( p = 55 \). So we set up the equation:
\( 55=1.3\sqrt{t}+39 \)

Step2: Isolate the square root term

Subtract 39 from both sides of the equation:
\( 55 - 39=1.3\sqrt{t} \)
\( 16 = 1.3\sqrt{t} \)

Step3: Solve for \(\sqrt{t}\)

Divide both sides by 1.3:
\( \sqrt{t}=\frac{16}{1.3}\approx12.3077 \)

Step4: Solve for \( t \)

Square both sides of the equation:
\( t = (\frac{16}{1.3})^2\approx(12.3077)^2\approx151.48 \)
Since \( t \) is the number of years after 1970, we round to the nearest whole number, so \( t\approx151 \). Then the year is \( 1970 + 151=2121 \)? Wait, that can't be right. Wait, maybe I made a mistake. Wait, let's check the calculation again.

Wait, \( 55 - 39 = 16 \), then \( \sqrt{t}=\frac{16}{1.3}\approx12.3077 \), then \( t=(12.3077)^2\approx151 \). But that seems too far. Wait, maybe the formula is \( p = 1.3\sqrt{t}+39 \). Wait, maybe I misread the formula. Wait, let's check again.

Wait, the formula is \( p = 1.3\sqrt{t}+39 \). So when \( p = 55 \):

\( 55-39 = 1.3\sqrt{t} \)

\( 16 = 1.3\sqrt{t} \)

\( \sqrt{t}=\frac{16}{1.3}\approx12.3077 \)

\( t=(12.3077)^2\approx151 \). So 1970 + 151 = 2121. But that seems unrealistic. Wait, maybe the formula is \( p = 1.3\sqrt{t}+39 \), but maybe the coefficient is 1.3 or maybe 1.3 is a typo? Wait, no, the problem says \( p = 1.3\sqrt{t}+39 \). Wait, maybe I made a mistake in the calculation. Let's recalculate \( \frac{16}{1.3} \): 16 divided by 1.3 is approximately 12.3077. Then squaring that: 12.3077 12.3077. Let's calculate 12^2 = 144, 0.3077^2≈0.0947, and 2120.3077≈7.3848. So (12 + 0.3077)^2 = 12^2 + 212*0.3077 + 0.3077^2≈144 + 7.3848 + 0.0947≈151.4795. So t≈151. So the year is 1970 + 151 = 2121. But that seems too far in the future. Wait, maybe the formula is \( p = 1.3\sqrt{t}+39 \), but maybe the initial year is different? Wait, no, the problem says t years after 1970. So maybe the answer is 2121? But that seems odd. Wait, maybe I made a mistake in the equation setup. Let's check again.

Wait, the formula is \( p = 1.3\sqrt{t}+39 \). We need to find t when p = 55. So:

\( 55 = 1.3\sqrt{t}+39 \)

Subtract 39: \( 16 = 1.3\sqrt{t} \)

Divide by 1.3: \( \sqrt{t} = 16 / 1.3 ≈ 12.3077 \)

Square: \( t ≈ (12.3077)^2 ≈ 151 \)

So 1970 + 151 = 2121. So the answer is 2121.

Answer:

2121