In 1995 an investor put $5000 into a fund. The graph above shows the value of this investment, in US dollars, as a function of the “date”. As indicated in the graph, we think of the “value” as v, and we think of the date as given by d .
(a)Make a mathematical statement that says:
“ The initial investment was $5000”.
(b)Is the graph concave up or concave down?
What does this fact say about the growth of the investment?
(c)When did the investment reach the value of $55,000?
(d)What is the average yearly increase from 2035 to 2045?
(e)Which is larger, the average yearly increase between 2035 and 2045, or the average yearly increase between 2015 and 2025? Explain how you decided.
You have decided to purchase a car for $18,750. You have 20% of the purchase price to use as a down payment. The remaining payment will be financed at a 10% yearly interest. The graph above shows the monthly payment you will have to make, depending on how long you choose to take to pay off the loan.
(a)What is the actual amount you borrowed?
(b)Estimate (from the graph) your monthly payment if you want to pay off the loan in 3 years.
(c)If you can afford to pay only $300 each month, how many years will you have to keep paying that?
(d)How much less will you have to pay each month if you choose to pay off the loan in 4 years, compared to 3 years?
(e)What is the difference, in total dollars you have to pay, between a 3-year and a 4-year loan?
The graph above shows the temperature T = T(v) adjusted for “wind chill” as a function of velocity v of the wind when the thermometer reads 30oF. The adjusted temperature T is the temperature that has the same cooling power when there is no wind.
(a)At what wind speed is the wind-chill-adjusted temperature equal to 0oF?
(b)Your answer to (a) is the solution to an equation involving T(v).
Which equation?
(c)At what value of v would a small increase in wind speed have the greatest effect on T(v)? In other words, at what wind speed could you expect a small increase in wind speed to be most significant on how cold it feels?
(d)Suppose the wind speed is 45 miles per hour, and the thermometer reads 30oF. Judging from the shape of the graph, how significant would you expect a small increase in wind speed on how cold it feels? Do you believe that this is true? Why?
Fertilizer is used to increase crop production (“yield”). But the cost of fertilizer must be taken into account if your goal is to maximize profit. The graph above shows the added bushels per acre of corn, that results from using different amounts of fertilizer (nitrogen). The straight line in the graph shows the cost of the fertilizer needed converted to “bushels of corn” (not dollars), per acre.
(a)How many pounds of nitrogen should be used to produce maximum crop yield?
(b)The profit is the crop yield minus the fertilizer cost. For each level of fertilizer used (per acre), how can the profit be read off the graph?
(c)How many pounds per acre of nitrogen should be applied to maximize profit?
(d)Can you figure out how many pounds of nitrogen you can buy for the price of one bushel of corn, based on the graph shown? Explain.
The graph above shows the percentage by which Social Security Checks increased each year (compared to the previous year) as a result of cost-of-living adjustments for the years 1996-2002. (The data points were connected by straight line segments to improve readability of the graph.)
(a)What was the cost-of-living increase of Social Security Checks in 1997?
If someone received a Social Security Check for $765 each month in 1996, how much were the monthly checks in 1997?
(b)When was the cost-of-living increase the greatest? What was the cost of living increase in that year?
(c)When was the cost of living increase adjustment 2.5%?
(d)Did Social Security Checks increase or decrease from2001 to 2002?
Explain.
The graph above depicts the average monthly profit for Slim’s Used Car Sales based on data Slim has gathered from his five franchises over the last few years.
(a)Slim pays rent for the lots, salaries for his sales people, utilities, health insurance, (and a few other such things) each month, regardless of how many cars are sold. Estimate Slim’s monthly “fixed cost”.
(b)Slim also has “variable costs”, such as prices he pays for the cars, labor for washing the cars, some minor repairs, and so on. Slim receives money (revenue) from his customers for each car he sells. What has to happen so that Slim’s revenue outweighs his total cost?
(This is often called the “break-even point”.)
(c)Describe under what circumstances Slim’s profit is increasing.
(d)Give plausible reasons why Slim’s profits sometimes decrease.
(e)On the basis of the information presented in the graph, how many cars should Slim try to sell each month to maximize his profit?