Question

1. (3 points) the percent of the total work force that is female is given by the function ( p(t) ) below, where ( t ) is the number of years past 1970: ( p(t)=-0.0034 t^{2}+0.45 t + 34 ).

a. (1 point) from the equation, identify the maximum point on the graph. you may use desmos or the vertex formula.
b. (2 point) what does this maximum point represent in the context of females in the work force in the years past 1970? please write a complete sentence including both the ( x ) and ( y ) components.

Explanation:

Step1: Find the \(t\)-coordinate of the vertex

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\)-coordinate (in this case \(t\)-coordinate) of the vertex is given by \(t=-\frac{b}{2a}\).
Here, \(a=- 0.0034\) and \(b = 0.45\).

$$t=-\frac{0.45}{2\times(-0.0034)}=\frac{0.45}{0.0068}\approx66.18$$

Step2: Find the \(p(t)\)-coordinate of the vertex

Substitute \(t = 66.18\) into the function \(p(t)=-0.0034t^{2}+0.45t + 34\).

$$p(66.18)=-0.0034\times(66.18)^{2}+0.45\times66.18+34$$

$$=-0.0034\times4380.79+29.78+34$$

$$=-14.89+29.78 + 34$$

$$=48.89$$

Answer:

a. The maximum point is \((66.18,48.89)\)
b. The maximum point \((66.18,48.89)\) represents that approximately \(66\) years past \(1970\) (i.e., in the year \(1970 + 66=2036\)), the percentage of the female workforce reaches its maximum value of approximately \(48.89\%\)