Question

a) The point at which a company's costs equal its revenues is the break-even point. C represents the cost, in dollars, of x units of a product and R represents the revenue, in dollars, from the sale of x units. Find the number of units that must be produced and sold in order to break even. That is, find the value of x for which C=R.

C=15x+32,000 and R=17x.

How many units must be produced and sold in order to break even?

b) A bicycle travels at a speed of 55 miles per hour for x hours. Find an expression for the distance that the bicycle travels.

c) The price per unit, p, and the demand, x, for a particular material is related by the linear equation p =140 -7/8 X, while the supply is given by the linear equation

p=7/8x. At what value of p does supply equal demand?

d) Suppose that a cyclist began a 476 mi ride across a state at the western edge of the state, at the same time that a car traveling toward it leaves the eastern end of the state. If the bicycle and car met after

8.5 hr and the car traveled 32.2 mph faster than the bicycle, find the average rate of each.

The car's average rate is

The bicycle's average rate is

e) A baseball team has home games on Thursday and Saturday. The two games together earn $4640.00 for the team. Thursday 's game generates$300.00 less than Saturday 's

game. How much money was taken in at each game? How much money did Thursday 's game generate? How much money did Saturday 's game generate?

Answer 1

a). At the break-even point, C(x) = R(x) or, 15x+32000 = 17x or, 17x-15x = 32000 or, 2x = 32000 so that x = 16000. Thus, 16000 units must be produced and sold in order to break- even.

b). We know that distance = speed * time. Hence, if a bicycle travels at a speed of 55 miles per hour for x hours, then the distance that the bicycle travels is 35x.

c). The demand of an item , where p is the per unit price and x is the no. of units is given by p= 140-(7/8)x or, (7/8)x = 140-p or, x = (8/7)(140-p) = 160- 8p/7.

The supply of the item is given by p= (7/8x) or, 8x=- 7/p or, x = (7/8p). When demand is equal to supply, we have 160- 8p/7 = (7/8p) or, 8p(160- 8p/7) = 7 or, 1280p -64p²/7 = 7 or, 8960p – 64p² = 7 or, 64p² -8960p +7 =0.

A graph of 64p² -8960p +7 is attached. The graph crosses the p-axis, when p = 0 or, p = 139.999, say 140. Since p cannot be 0, hence the demand and supply will be equal when p = 140.

d).

Let the average rate of the bicycle and the car be x mph and y mph respectively. In 8.5 hours, the bicycle and the car would travel 8.5x and 8.5 miles respectively. Therefore, 8.5x+8.5 y = 476 or, x+y = 476/8.5 = 56….(1). Further, the car traveled 32.2 mph faster than the bicycle so that y = x+32.2…(2). Now, on substituting y = x+32.2 in the 1^st equation, we get x+x+32.2 = 56 or, 2x = 56-32.2 = 23.8 so that x = 23.8/2 = 11.9. Then y = 11.9+32.2 = 44.1.

Thus,

The car's average rate is 44.1 mph.

The bicycle's average rate is 11.9 mph.

e). Let the Thursday 's game and the Saturday 's game generate $ x and $ y respectively. Then x+y = 4640…(1). Also, x = y-300…(2). Now, on substituting x = y-300 in the 1^st equation, we get y-300+y = 4640 or, 2y = 4640+300 = 4940 so that y = 4940/2 = 2470. Then x = 2470-300 = 2170.Thus, Thursday 's game generated $ 2170 and Saturday 's game generated $ 2470.