Question

an athlete whose event is the shot put releases the shot. when the shot is released at an angle of 35°, its path can be modeled by the formula

( y = - 0.01 x ^ { 2 } + 0.7 x + 6.2 )

in which ( x ) is the shots horizontal distance, in feet, and ( y ) is its height, in feet. this formula is shown by one of the graphs, (a) or (b), in the figure. use the formula to answer the questions below.

use the formula to determine the shots maximum distance.

the maximum distance is approximately 78.0 feet.

(round to the nearest tenth as needed.)

which graph, (a) or (b), shows the shots path?

graph (b)

graph (a)

Explanation:

Step1: Analyze the quadratic function

The given function is \(y = - 0.01x^{2}+0.7x + 6.2\), which is a quadratic function of the form \(y = ax^{2}+bx + c\) (\(a=-0.01\), \(b = 0.7\), \(c = 6.2\)). The maximum distance occurs when \(y = 0\). So we set up the equation \(-0.01x^{2}+0.7x + 6.2=0\). Multiply through by \(- 100\) to get \(x^{2}-70x - 620=0\).

Step2: Use the quadratic formula

The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For \(x^{2}-70x - 620=0\) (\(a = 1\), \(b=-70\), \(c=-620\)), we have \(x=\frac{70\pm\sqrt{(-70)^{2}-4\times1\times(-620)}}{2\times1}=\frac{70\pm\sqrt{4900 + 2480}}{2}=\frac{70\pm\sqrt{7380}}{2}=\frac{70\pm85.9}{2}\).
We get two solutions: \(x_{1}=\frac{70 + 85.9}{2}=\frac{155.9}{2}=78.0\) and \(x_{2}=\frac{70-85.9}{2}=\frac{-15.9}{2}=-7.95\). Since distance \(x\geq0\), we take \(x = 78.0\).

Now, for the graph:
The function \(y=-0.01x^{2}+0.7x + 6.2\) is a parabola opening downwards (\(a=-0.01<0\)). Graph (a) is a higher - arching parabola (assuming larger maximum height in the context of the axes shown, as the coefficient of \(x^{2}\) is smaller in magnitude compared to a steeper - dropping parabola). But more precisely, when \(x = 0\), \(y=6.2\). Graph (a) has a non - zero \(y\) - intercept (the value of \(y\) when \(x = 0\)) and the general shape of a parabola opening downwards. Graph (b) seems to have a lower \(y\) - intercept (closer to the \(x\) - axis at \(x = 0\)).

Answer:

graph (a)