2 The revenue equation (in hundreds of millions of dollars) for barley production in a certain country is approximated by R(x) = 0.0695x + 1.4116x + 2.1031 where x is in hundreds of millions of bushels. Find the marginal-revenue equation and use it to find the marginal revenue for the production of the given number of bushels. (a) The marginal-revenue equation is R'(x) = (Round to four decimal places as needed.)
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- The demand function for the Viking Boat’s 34-ft Sundancer yacht isp = 200 − 0.01x ln x,where x denotes the number of yachts and p is the price per yacht in hundreds of dollars.(a) Find the revenue function and the marginal revenue function for this model of yacht.(b) Use the result of part (a) to estimate the revenue realized from the sale of the 500-th34-ft Sundance yacht. (Note: This is an estimate, so be careful to use the correct functionfrom part (a) and to evaluate it at the correct value of x.)(c) Now compute the exact revenue realized from the sale of the 500-th 34-ft Sundance yacht. (d) Comment on the approximation provided in part (b).Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.a. When will the cities have the same population?b. Suppose City C has a current population of y0 < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y0 and p such that Cities A and C have the same population in 10 years?The marginal cost of the xth box of light bulbs is 10 + [ln(x + 1)] (x + 1)2 , and the fixed cost is $1,000. Find the total cost C(x) to make x boxes of bulbs. C(x) =
- 1. The cost (in pesos) of producing x corkboards is given by x2 + 20x + 500, while the price (in pesos) of each corkboard if there is a demand for x units is given by p(x) = 100 − x.(a) Find the marginal profit of producing and selling 15 corkboards. (b) At what production level is the profit at a maximum?2. Due north, Seokjin is walking towards the most famous spot in a park at a rate of 1.3 meters per second. Due east, Yoongi is leaving the same spot at a rate of 1 meter per second. If they started walking at the same time, and Seokjin is initially 79 meters away from the spot, how fast is the distance between them changing after 30 seconds?The cost of producing x teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula C(x) = 100 + 41x − 0.04x2. (a) Find the marginal cost function C'(x). C'(x) = ___________ Use it to determine how fast the cost is going up (in $) at a production level of 100 teddy bears. $ ________ per teddy bear Compare this with the exact cost of producing the 101st teddy bear (in $). The cost is increasing at a rate of $ _____ per teddy bear. The exact cost of producing the 101st teddy bear is $ ________ . Thus, there is a difference of $ ___________ . (b)Find the average cost function C, and evaluate C(100) (in $). C(x)= _______ C(100) = $ ______ per teddy bear What does the answer tell you? The average cost of producing the first hundred teddy bears is $ _________ per teddy bear.The population of Normal, Illinois grows proportionally to its population at time t. At an initial population of 2100, it grows at a rate of 8% over 10 years. What is the value of the proportionality constant?
- One method of slowing the growth of an insect populationwithout using pesticides is to introduce into the populationa number of sterile males that mate with fertile femalesbut produce no offspring. If P represents the number offemale insects in a population, S the number of sterile malesintroduced each generation, and r the population’s naturalgrowth rate, then the female population is related to time tby equation given at the end. Suppose an insect population with 10,000 females grows ata rate of r = 0.10 and 900 sterile males are added. Evaluatethe integral to give an equation relating the female populationto time. (Note that the resulting equation can’t besolved explicitly for P.)The cost function of a company as a function of units produced x is C(x)=181+43x−0.02x2. What is the cost to produce unit 25? What is the marginal cost at x=24?A company has determined the demand and supply functions for a product are given by p = D( x) = 14 - x 2 (Demand Function) p = S( x) = 4 x + 2 (Supply Function) where x is the daily quantity and p is in dollars per unit. Find the consumer surplus at the equilibrium point.
- A company is considering a new manufacturing process in one of its plants. The new process provides substantial initial savings, with the savings declining with time x according to the rate-of-savings function S'(x)=200-x^2 where x is measured in years and S′(x) in thousands of dollars per year. At the same time, the cost of operating the new process increases with time x according to the rate-of-cost function (in thousands of dollars per year) C'(x)=0.3x^2+20/3x a. to maximize savings, for how many years should the company use the new process? b. Use Desmos to check your answer from part (a): Build up a graph and shade the appropriate area that corresponds to the maximum savings. c. What is the net savings 8 years after the process has been implemented? 12 years after? d. Use Desmos to check your answer from part (c): Build up a graph and shade the appropriate area that corresponds to the maximum savings.A company's marginal revenue function is MR(x) = 700x−1 and its marginal cost function is MC(x) = 200x−1 (both in thousands of dollars), where x is the number of units (x > 1). Find the total profit from x = 100 to x = 200. (Round your answer to the nearest thousand dollars.)