Homework: HW1 Applications of Functions (3.3 & 3.4)
The total cost (in dollars) of producing x college textbooks is C(x)=43x+11,000. (a) What are the fixed costs? (b) What is the marginal cost per book? (c) What is the total cost of producing 700 books? 37,000 books? (d) What is the average cost when 700 books are produced? when 37,000 books are produced?
ANSWER: A) 11,000 B) 43 C) 41100 AND 1602000 D) 58.71 AND 43.3 WORK: A1) Fixed cost is or b. In this case, b = 11,000 B1) Marginal cost is m. In this case, m=43 C1) x= quantity, Therefore: C(700)=43*700+11,000 AND C(37000)=43*37000+11,000 C2) C(700)=30100+11,000 C(37000)=1591000+11,000 C3) C(700)=41100 C(37000)=1602000 D1) C(700)=41100; divide both sides by 700 C(37000)=1602000; divide both sides by 37000 D2) C=58.71428571428571 (or, 58.71; 58.7; 59) C=43.297 repeating (or, 43.30, 43.3, 43)
Suppose you are the manager of a firm. The accounting department has provided cost estimates, and the sales department sales estimates, on a new product. Analyze the data they give you, determine what it will take to break even, and decide whether to go ahead with production of the new product. The product has a production cost function C(x)=135x+5,400 and a revenue function R(x)=180x. If the company can produce and sell no more than 98 units, should it do so?
ANSWER: A) 120 B) No; It will still lose money if they produce less than 120 WORK: A1) P(x) = 0 A2) 45x-5400=0 A3) 45x=5400 A4) x=120
A bullet is fired upward from ground level. Its height above the ground (in feet) at time t seconds is given by the equation below. H=−16t²+968t Find the maximum height of the bullet and the time at which it hits the ground.
ANSWER: A) 14641 B) 60.5 WORK: A1) t = -b/2a = -968/2(-16) = -968/-32 A2) t = 30.25 A3) H=−16t²+936t; plug in t A4) H=−16(30.25²)+968(30.25) A5) H=-14641 + 29282 A6) H= 14641 B1) 2 x 30.25 B2) 60.5
A researcher in physiology has decided that a good mathematical model for the number of impulses fired after a nerve has been stimulated is given by y equals negative y=−x²+40x−50, where y is the number of responses per millisecond and x is the number of milliseconds since the nerve was stimulated. (a) When will the maximum firing rate be reached? (b) What is the maximum firing rate?
ANSWER: A) 20 B) 350 WORK: Use the Quadratic Function f(x) = ax²+bx+c to find x, and then solve for y by plugging in x. To find X using this function, x= -b/2a A1) a = -1 < 0 A2) x= -b/2a A3) x = -40/2(-1) A4) x = -40/-2 A5) x = 20 B1) y=−x²+40x−50; plug in x B2) y=−1(20²)+40(20)-50 B2) y=-400+800-50 B2) y=350
In the cost function below, C(x) is the cost of producing x items. Find the average cost per item when the required number of items is produced. C(x)=6.5x+9,100 a. 200 items b. 2000 items c. 5000 items
ANSWER: A) 52 B) 11.05 C) 8.32 WORK: A1) C(200)=6.5*200+9100 A2) C(200)=1300+9100 A3) C(200)=10400; divide both sides by 200 A4) C = 52 B1) C(2000)=6.5*2000+9100 B2) C(2000)=13000+9100 B3) C(2000)=22100; divide both sides by 2000 B4) C = 11.05 C1) C(5000)=6.5*5000+9100 C2) C(2000)=32500+9100 C3) C(2000)=41600; divide both sides by 5000 C4) C = 8.32
The revenue (in millions of dollars) from the sale of x units at a home supply outlet is given by R(x)=0.13x. The profit (in millions of dollars) from the sale of x units is given by P(x)=0.083x−1.6. Find the cost equation. What is the break-even point?
ANSWER: A) C(x) = .047x -1.6 B) 19.277 WORK: A1) P(x) = R(x) - C(x); add P(X) to both sides, and subtract C(x) A2) C(x) = R(x) - P(x) A3) C(x) = .13x - .083x - 1.6 A4) C(x) = .047x+1.6 B1) P(x) = 0 B2) .083x -1.6 = 0 B3) .083x = 1.6; divide both sides by .083 B4) 19.277
Assume that the following has a linear cost function. Find the cost function, C(x), revenue function, R(x), and profit function, P(x), where x is the number of items produced (or sold). Fixed Cost: 250 Marginal Cost: 19 Item Sells For: 30
ANSWER: A) C(x)= B) R(x)= C) P(x)= WORK: A1.) C(x)=mx+b A2.) C(x)=19x+250 B1) R(x)=px B2) R(x)=30x C1) P(x)= R(x)-C(x) C2) P(x)= 30x-19x-250 C3) P(x)=11x-250
Finding the equilibrium
The equilibrium is where the supply and demand meet.
Assume that the following has a linear cost function for producing x items. Find the cost function, revenue function, and profit function. Fixed Cost = 1000 Marginal Cost = 15 Item Sells For = 45
ANSWER: A) C(x)=15x+1000 B) R(x)=45x C) P(x)=30x-1000 WORK: A1.) C(x)=mx+b A2.) C(x)=15x+1000 B1) R(x)=px B2) R(x)=45x C1) P(x)= R(x)-C(x) C2) P(x)= 45x-15x-1000 C3) P(x)=30x-1000
The Dispatch Tool Works spends $40000 to produce 120 parts, achieving a marginal cost of $320. Find the linear cost function.
ANSWER: 1600 WORK: 1.) C(x)=mx+b, We know that the marginal cost is 320 therefore C(x)=320x+b 2.) We also know it costs 40000 to produce 120 parts. Therefore, 40000 = 320*120+b 3.) 40000 = 320*120+b; 320x120 = 38400 4.) 40000 = 38400+b; Subtract 38400 from both sides 5.) 1600 = b
A cost function, C(x), is given below. Find the average cost per item when the required number of items are produced. C(x)=30x+1300, 1000 items
ANSWER: 31.3 WORK: 1.) C(x)=30x+1300; x = 1000 2.) C(1000) = 30*1000+1300 3.) C(1000) = 30000+1300 4.) C(1000) = 31300; divide both sides by 1000 5.) C = 31.3
Assume that the situation can be expressed as a linear cost function. Find the cost function. Fixed cost is $200; 30 items cost $2,000 to produce.
ANSWER: 60x+200 WORK: 1.) C(x)=mx+b 2.) B is the Fixed Cost, which is $200 there for, C(x)=mx+200 3.) 30 items cost 2,000. 30 is the quantity, which is X, therefore: 2,000=30x+200 4.) Subtract 200 from each side like so: 2,000 = 30m + 200 -200...................-200 1,800 = 30m 5.) Now divide each side by 30, like so: 1800/30 = 30m/30 60=m 6.) 60x+200
A parking garage charges a one-time fee of $10.00 plus $1.50 per half hour. Choose the function which represents the total cost, C(x), for parking for x hours.
ANSWER: C(x)=10+3x WORK: 1.) If it costs 1.50 every half hour, then the cost for one hour is 3.00 2.) C(x)=10+3x
Use the supply and demand curves in the accompanying graph. At what price are 5 items demanded?
Depending on the chart determines the answer. Look at the quantity axis and fine 5, then go up the price axis to find the price of that line. For this one it was between 70 and 80, so $75. Reverse this if you're looking for when something is supplied.
