Concept explainers
CONSUMPTION FUNCTIONS A certain economy’s consumption function is given by the equation
where
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- Size of High Schools The farm population has declined dramatically in the years since World War II, and with that decline, rural school districts have been faced with consolidating in order to be economically efficient. One researcher studied data from the early 1960s on expenditures for high schools ranging from 150 to 2400 in enrollment. He considered the cost per pupil as a function of the number of pupils enrolled in the high school, and he found the approximate formula C=7430.402n+0.00012n2 where n is the number of pupils enrolled and C is the cost, in dollars, per pupil. a. Make a graph of C versus n. b. What enrollment size gives a minimum per-pupil cost? c. If a high school had an enrollment of 1200, how much in per-pupil cost would be saved by increasing enrollment to the optimal size found in part b?arrow_forwardThe maximum weight M that can be supported by a beam is jointly proportional to its width w and the square of its height h and inversely proportional to its length L. a Write an equation that expresses this proportionality. b Determine the constant of proportionality if a beam 4 in. wide, 6 in. high, and 12 ft long can support a weight of 4800 lb. c If a 10-ft beam made of the same material is 3 in. wide and 10 in. high, what is the maximum weight it can support?arrow_forwardFlood Control A river is 8 feet above its flood stage. The water is receding at a rate of 3 inches per hour. Write a mathematical model that shows the number of feet above flood stage after t hours. Assuming the water continually recedes at this rate, when will the river be 1 foot above its flood stage?arrow_forward
- Applications Waste management The corrosive waste is industrial sewage limits the useful life of the piping in a waste processing plant to 12 years. The piping system was originally worth 137,000, and it will cost the company 33, 000 to remove it at the end of its 12-year useful life. Find a depreciation equation.arrow_forwardDemand A company has found that the daily demand x for its boxes of chocolates is inversely proportional to the price p. When the price is $5, the demand is 800 boxes. Approximate the demand when the price is increased to $6.arrow_forwardProfit The demand equation for a microwave oven is given by p=1400.0001x, where p is the unit price (in dollars) of the microwave oven and x is the number of units sold. The cost equation for the microwave oven is C=80x+150,000, where C is the total cost (in dollars) and x is the number of units produced. The total profit P obtained by producing and selling x units is modeled by P=xpC. (a) Find the profit function P in terms of x. (b) Find the profit when 250,000 units are sold. (c) Find the unit price when 250,000 units are sold. (d) Find (if possible) the unit price that will yield a profit of 10 million dollars. If not possible, explain why.arrow_forward
- 70x-x? Given the revenue function of a company R(x): then the maximum revenue isarrow_forwardBig fish eat little fish, and little fish eat algae. So, the population b of big fish is a function of the population I of little fish, and the population of little fish depends on the amount a of algae available. We can model the big fish population as b(1) = 0.5/+3 hundred big fish, where I is measured in thousands of little fish. The little fish population can be modelled as (a) 0.3a-1 thousand little fish, where a is the number of tons of algae. Match following functions and statements. [Choose ] Compute b(10) 53 When there are ten thousands little fish, there are 5300 big fish. Compute I(10) When there are 10 tons of algae, there are two thousands little fish 50 Compute b(I(10)) 5. 530 b(10) means 20 When there are 10 tons of algae, there are 5 hundred big fish |(10) means [Choose ]arrow_forwardSolve for f(g(x)) and g(f(x))arrow_forward
- Evaluate the function f(x)=4x²-2x for the given values of x. (a) f(-1) (b) ƒ (0) (c) / (1) (d)/(2) (e) / (3) Part 1 of 5 (a)/(-1)- Checkarrow_forwardFind the x-intercepts and y-intercept of the following function. f(x)=(x−1)(x+1)(x+2) a) x-intercepts: (2,0), (1,0), and (−1,0). y-intercept: (0,2) b) x-intercepts: (−2,0), (−1,0), and (1,0). y-intercept: (0,2) c) x-intercepts: (−2,0), (−1,0), and (1,0). y-intercept: (0,−2) d) x-intercepts: (2,0), (1,0), and (−1,0). y-intercept: (0,−2)arrow_forward
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