91Ó°ÊÓ

An athlete whose event is the shot put releases the shot with the same initial velocity but at different angles. The figure shows the parabolic paths for shots released at angles of \(35^{\circ}\) and \(65^{\circ} .\) Exercises \(57-58\) are based on the functions that model the parabolic paths. When the shot whose path is shown by the red graph is released at an angle of \(65^{\circ},\) its height, \(g(x),\) in feet, can be modeled by $$ g(x)=-0.04 x^{2}+2.1 x+6.1 $$ where \(x\) is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the red graph. a. What is the maximum height, to the nearest tenth of a foot, of the shot and how far from its point of release does this occur? b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw? c. From what height was the shot released?

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x),\) in feet, can be modeled by $$ f(x)=-0.8 x^{2}+2.4 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, \(f(x),\) in feet, can be modeled by $$ f(x)=-0.8 x^{2}+3.2 x+6 $$ where \(x\) is the ball's horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of a foot. c. Graph the function that models the ball's parabolic path.

Among all pairs of numbers whose sum is \(20,\) find a pair whose product is as large as possible. What is the maximum product?

Among all pairs of numbers whose difference is \(16,\) find a pair whose product is as small as possible. What is the minimum product?

Among all pairs of numbers whose difference is \(24,\) find a pair whose product is as small as possible. What is the minimum product?

Describe how to find the possible rational zeros of a polynomial function.

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

How does the linear factorization of \(f(x)\), that is, $$f(x)=a_{n}\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right)$$ show that a polynomial equation of degree \(n\) has \(n\) roots?