David is in high school and is working hard to maintain his GPA. He has found that his grades are directly affected by the number of weekly hours outside of school that he spends studying. This relationship can be modeled by the equation $$G=-\frac16t^2+5t+-67/2$$ where $$G$$ represents his GPA and $$t$$ represents the number of hours per week that he spends studying. How many hours a week should he study to maximize his GPA? What will his GPA be if he studies for this amount of time? Does this answer make sense?
If we were to graph this equation using a graphing calculator, we would see that the equation is a parabola that opens downward. Recall that to find the x-coordinate of the vertex of such a graph we can use $$x=-b/2a$$ where the equation is in the form $$y=ax^2+bx+c$$. (In this example we are using $$G$$ and $$t$$ instead of $$y$$ and $$x$$ but the form of the equation is the same.) In our example $$t=-b/2a$$ where $$a=-\frac16$$ and $$b=5$$. This tells us that the time David needs to spends studying to maximize his GPA is $$t=-5/(2*-\frac16)$$ or $$t=15 hrs$$. To find out what his maximum GPA is, we insert our solution for $$t$$ into the original equation and solve. $$G=-\frac16(15^2)+5(15)-67/2$$ or $$G=4$$. This answer makes sense because the highest GPA a person can receive is a $$4.0$$.
Mindy is a surveyor and is trying to determine the height of a nearby office building. She is standing 22 feet from the base of the building and determines that the angle between the ground and the top of the building from her position is 76 degrees. Using this information, find the height of the building.
Begin by visualizing the scenario as a right triangle where the distance between Mindy and the top of the building is the $$hypotenuse$$, the distance from Mindy to the base of the building is the side $$adjacent$$ to the known angle, and the height of the building is the side $$opposite$$ the known angle. We know that the length of the side $$adjacent$$ to the angle is 22 feet. Additionally, we know that the tangent of an angle is equal the length of the $$opposite$$ side divided by the length of the $$adjacent$$ side (tangent equals opposite over adjacent). Thus here $$tan(76)=height/22 ft$$. To solve for the height, we multiply both sides of the equation by 22 ft and find that $$height=22ft*tan(76)$$ or $$height=88.24 ft$$.
Richard is going on a business trip and will need to rent a car for the duration of the trip. The car rental company charges a flat rate of $$$85$$ with an additional $$$.75$$ for each mile the car travels. Write an equation that shows how much the rental will cost based on the number of miles driven.
The total cost of the vehicle can be broken into two pieces. One piece remains the same no matter what. This is the $$$85$$ flat rate charge. The other piece will vary based on how many miles the car is driven. We know that every mile driven results in a $$$.75$$ increase. So if the car drives one mile, the charge will be $$$.75*1$$, if the car drives two miles, the charge will be $$$.75*2$$ and so on. Let's call the number of miles driven variable $$m$$. So the charge based on the number of miles driven will be $$$.75*m$$. The total cost of the vehicle (Let's call this variable $$C$$ ) will be the sum of these two charges. In other words, $$C=85 + .75m$$.