1.

A rectangular field is to be enclosed by840feet of fence. One side of the field is a building, so fencing is not required on that side. Determine the dimensions of the rectangle that maximize its area. Length of side perpendicular to the building,x= feet Length of side parallel to the building = feet

2.

A billboard designer has decided that a sign should have2-ft margins at the top and bottom and1-ft margins on the left and right sides. Furthermore, the billboard should have a total area of5000ft²(including the margins). Ifxdenotes the left-right width (in feet) of the billboard, determine the value ofxthat maximizes the area of the printed region of the billboard. x= feet Use this value ofxto compute the maximum area of the printed region. Maximum area of printed region = square feet

3.

By cutting away anx-by-xsquare from each corner of a rectangular piece of cardboard and folding up the resulting flaps, a box with no top can be constructed.If the cardboard is30inches long by30inches wide, determine the value ofxthat yields the maximum volume of the resulting box. x= inches Use this value ofxto compute the maximum volume of the box. Maximum volume of box = cubic inches

4.

A rectangular box is to have a square base and a volume of252ft³. If the material for the base costs10cents per square foot, material for the top costs18cents per square foot, and the material for the sides costs12cents per square foot, determine the dimensions of the box (in feet) thatminimizethe total cost of materials used in constructing the rectangular box. Width of the base,x= feet Height of the box,y= feet

5.

The HiWay Robery Financial Group has determined to advisexclients at their upcoming series of get-rich-quick seminars, the price per clientp(x) in dollars that must be charged is p(x) =14004x. The financial group also determines that the total cost of advisingxclients is given by C(x) =448x+700. How many clients does the financial group need to have in order to maximize their profit? Number of clients =

What pricepshould they charge each client in order to maximize their profit? Price per client,p= dollars

Compute their maximum profit. Maximum profit = dollars

6.

Mt. Nittany Property Management Company and Happy Valley Rentals are two different businesses that manage numerous apartments around town. Both companies use the same monthly profit model given by P(x) =15x²+2340x31200 wherexrepresents the number of apartments rented in a month andP(x) is in dollars. (a) Assuming Mt. Nittany Property Management Company manages 100 apartments, find the number of apartments they should rent out each month to maximize profits and calculate their maximum monthly profits. Number of apartments =

Maximum monthly profit = dollars

(b) Assuming Happy Valley Rentals manages 50 apartments, find the number of apartments they should rent out each month to maximize profits and calculate their maximum monthly profits. Number of apartments =

Maximum monthly profits = dollars

7.

The demand equation for video games is given by x=30015p wherexis the number of video games andpis in dollars. Determine the value ofpthat maximizes total revenue. p= dollars

Use this value ofpto compute the maximum total revenue. Maximum total revenue = dollars