Exam 8: Special Topics in Algebra

A certain area of forest is populated by two species of animals, which scientists refer to as A and B for simplicity. The forest supplies two kinds of food, referred to as $\mathrm { F } _ { 1 }$ and $\mathrm { F } _ { 2 }$ . For one year, species A requires $1.45$ units of $\mathrm { F } _ { 1 }$ and 1 units of $\mathrm { F } _ { 2 }$ . Species $B$ requires $2.0$ units of $\mathrm { F } _ { 1 }$ and $1.9$ units of $\mathrm { F } _ { 2 }$ . The forest can normally supply at most 833 units of $F _ { 1 }$ and 453 units of $F _ { 2 }$ per year. What is the maximum total number of these animals that the forest can support?

Draw the graph of the equation. - $x=y^{2}+4 y-3$

The vertex of a parabola occurs where the derivative of its equation is 0, so an equation associated with the equation of a parabola can be solved to find the x-coordinate of the vertex. Find the vertex of the parabola, given the equation of the parabola and its associated equation. -Between $x = - 8$ and $x = - 1$ , the largest value of the expression $y = x ^ { 3 } + 12 x ^ { 2 } + 36 x + 9$ occurs where $3 x ^ { 2 } + 24 x + 36 = 0$ . Of the values of $x$ that satisfy $3 x ^ { 2 } + 24 x + 36 = 0$ , which gives the maximum value of $y = x ^ { 3 } + 12 x ^ { 2 } + 36 x + 9 ?$

Draw the graph of the equation. - $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 81 } = 1$

A railroad tunnel is shaped like a semi-ellipse. The height of the tunnel at the center is 20 ft and the vertical clearance must be 10 ft at a point 15 ft from the center. Find an equation for the ellipse.

Use the given feasible region determined by the constraint inequalities to find the minimum possible value of the objective function. - $f = 4 x + 3 y$ subject to the constraints $\left\{ \begin{array} { l } 2 x + y \leq 12 \\x + 2 y \leq 14 \\x + y \leq 8 \\x \geq 0 , y \geq 0\end{array} \right.$

John does 12 pushups on the first day of a 30-day month, and then increases the number of pushups by 2 pushups a day. How many pushups has he done by the end of the month?

Find the next three terms of the arithmetic sequence. -3, 9, 15, 21, . . .

Graph the inequality. - $y < - 4 x + 1$

Find the sum of all the integers from -32 to 50.

Evaluate the sum. - $\sum _ { \mathrm { i } = 1 } ^ { 6 } 5 ^ { \mathrm { i } }$

A certain ball when dropped from a height rebounds 107 of the original height. How high will the ball rebound after the fifth bounce if it was dropped from a height of 32 feet? Round your answer to the nearest tenth of a Foot.

Find the first five terms of the sequence defined by the given general term. - $a _ { n } = n - 5$

Graph the region that satisfies the constraints. -Bubba's Computer World sells personal computers. To advertise its computers to target groups, they advertise with x one-minute spots on local television at a cost of $264 per minute, and y one-minute spots on radio, at a Cost of $220 per minute. Research shows that they sell one computer for every 3 minutes of television Advertising, and one computer for every 6 minutes of radio advertising. Suppose the company has at most $55,000 to spend on advertising and that they sell at least 50 computers per month from this advertising.

Graph the circle. - $(x+3)^{2}+y^{2}=1$

The vertex of a parabola occurs where the derivative of its equation is 0, so an equation associated with the equation of a parabola can be solved to find the x-coordinate of the vertex. Find the vertex of the parabola, given the equation of the parabola and its associated equation. -Between $x = - 5$ and $x = 10$ , the largest value of the expression $y = x ^ { 3 } - 6 x ^ { 2 } - 63 x - 13$ occurs where $3 x ^ { 2 } - 12 x - 63 = 0$ . Of the values of $x$ that satisfy $3 x ^ { 2 } - 12 x - 63 = 0$ , which gives the maximum value of $y = x ^ { 3 } - 6 x ^ { 2 } - 63 x - 13 ?$

Graph the region that satisfies the constraints. -A company produce two types of printers, the Inkjet and the Laserjet. It takes 5 hours to make the Inkjet and 10 hours to make the Laserjet. The company can make at most 150 printers per day and has 1000 labor-hours Available per day. Let x represent the number of Inkjet printers produced in a day and y represent the number Of Laserjet printers produced in a day.

Rewrite the expression as a sum, difference, or multiple of logarithms. -Find where $f ^ { \prime } ( x ) = 0$ if $f ^ { \prime } ( x ) = \left( x ^ { 2 } - 9 \right) \left[ 2 ( x - 8 ) ^ { 2 } \right] + ( x - 8 ) ^ { 3 } ( 5 x )$

Evaluate the sum. - $\sum _ { i = 1 } ^ { 4 } \frac { 3 + i } { i }$