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7. [-/5 Points] DETAILS SCALCET9 4.7.065. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = -. Consider the cost function C(x) given below. (Round your answers to the nearest X cent.) C(x) = 24,000 + 270x + 6x3/2 (a) Find the total cost (in dollars) at a production level of 1,000 units. $ (b) Find the average cost (in dollars per unit) at a production level of 1,000 units. $ per unit (c) Find the marginal cost (in dollars per unit) at a production level of 1,000 units. $ per unit (d) Find the production level (in units) that will minimize the average cost. units (e) What is the minimum average cost (in dollars per unit)? $ per unitOrnithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. island 4 km: D B nest 8 km A bird with these tendencies is released from an island that is 4 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Points B and D are 8 km apart. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. (Round your answers to two decimal places.) (a) In general, if it takes 1.3 times as much energy to fly over water as land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area? km from B (b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. Assuming the bird's energy expenditure is minimized, determine a function for the ratio WV - in terms of x, the distance from B to C. W = (c) What should the value of - be in order for the bird to fly directly to its nesting area D?Example 6 Video Exampled) A store has been selling 300 TV monitors a week at $450 each. A market survey indicates that for each $40 rebate offered to buyers, the number of monitors sold will increase by IO a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue? Solution If x is the number of monitors sold per week, then the weekly increase in sales is . For each increase of 80 units sold, the price is decreased by $40. So for each addltlonal unlt sold, the decrease In pnce WI" be E x 40 and the demand function Is 990 : 450-( ){xBOO) :600( ) The revenue function is RV) = XPEX) = Since R'(x) = , we see that R'[X) = 0 when x = |:| . This value of x gives an absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward}. 'I11e corresponding price is mm: = |:l and the rebate is 450 300 : \\:| . Therefore, to maximize revenue, the store should offer a rebate of $ E