78. A couple seeking to buy a home decides that a monthly payment of $800 fits their budget. Their bank’s interest rate is 7.5%. The amount they can borrow A is a function of the time t in years that it will take to repay the debt. If we denote this function by , then the following table defines the function. Source: Comprehensive Mortgage Payment Tables, Publication No. 492, Financial Publishing Co., Boston Choose the expression in the form describes how long they will have to make payments if they need to borrow $103,000.
​ 85. If a test that has reliability r is lengthened by a factor , the reliability R of the new test is given by the reliability is r = 0.5, the equation becomes . When the test length is unchanged, the reliability r = R(1) = 0.5. If the length of the test is doubled, the reliability r = R(2). What percent improvement is there in the reliability when the test length is doubled? 86. If 140 feet of fence is to be used to fence in a rectangular yard, then the resulting area of the fenced yard is given by A = x(70-x), where x is the width of the rectangle. Is A a function of x? 87. If a box with a square cross section is to be sent by a delivery service, there are restrictions on its size such that its volume is given by the function , where x is the length of each side of the cross section (in inches). What restrictions must be placed on x (the domain) so that the volume makes physical sense?
174. Suppose the cost C of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by . Because p is the percent particulate pollution, we know . Use the restriction on p and experiment with a C-range to obtain an accurate graph of the equation. Determine the p-intercept and explain its meaning.
209. A glass of skim milk supplies 0.1 mg of iron, 8.5 g of protein, and 1 g of carbohydrates. A quarter pound of lean red meat provides 3.4 mg of iron, 22 g of protein, and 20 g of carbohydrates. Two slices of whole grain bread supply 2.2 mg of iron, 10 g of protein, and 12 g of carbohydrates. If a person on a special diet must have 9.1 mg of iron, 62.5 g of protein, and 53 g of carbohydrates, how many glasses of skim milk, how many servings of meat, and how many two-slice servings of whole grain bread will supply this? 210. A social agency is charged with providing services to three types of clients, A, B, and C. A total of 450 clients are to be served, with $140,000 available for counseling and $100,000 available for emergency food and shelter. Type A clients require an average of $200 for counseling and $300 for emergencies, type B clients require an average of $500 for counseling and $200 for emergencies, and type C clients require an average of $300 for counseling and $100 for emergencies. How many of each type of client can be served?
230. A company distributes college logo sweatshirts and sells them for $50 each. The total cost function is linear, and the total cost for 100 sweatshirts is $4,480, whereas the total cost for 250 sweatshirts is $7,480. Find the equation for the total cost function where x is the number of sweatshirts produced and sold. 231. Some of the graphs of total revenue R, total cost C, variable cost VC, fixed cost FC, and profit P are shown as functions of the number of units x. Choose the function represented by the line labeled b. 232. As the price of a commodity increases, what happens to supply?
247. Suppose that in a certain market, the demand function for a product is given by and the supply function is given by , where p is the price per unit in dollars and q is the quantity produced. If the government levies a tax of $7 per item on the supplier, who passes the tax on to the consumer as a price increase, find the equilibrium price and quantity after the tax is levied. Round q to the nearest whole number and round p to one decimal place.