Linear and Quadratic Functions Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 15
Section 1.1 :: Slopes and Equations of Lines 2 / 15
Exercise Heart Rate To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minute) should be in the target heart rate zone. The lower limit of this zone is found by taking 70% of the difference between 220 and your age. The upper limit is found by using 85%. Source: Physical Fitness 3 / 15
Exercise Heart Rate To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minute) should be in the target heart rate zone. The lower limit of this zone is found by taking 70% of the difference between 220 and your age. The upper limit is found by using 85%. Source: Physical Fitness (a) Find formulas for the upper and lower limits ( u and ℓ ) as linear equations involving the age x . (b) What is the target heart rate zone for a 20-year-old? 4 / 15
Exercise Heart Rate (c) What is the target heart rate zone for a 40-year-old? (d) Two women in an aerobics class stop to take their pulse and are surprised to find that they have the same pulse. One woman is 36 years older than the other and is working at the upper limit of her target heart rate zone. The younger woman is working at the lower limit of her target heart rate zone. What are the ages of the two women, and what is their pulse? 5 / 15
Global Warming In 1990, the Intergovernmental Panel on Climate Change predicted that the average temperature on Earth would rise 0 . 3 ◦ C per decade in the absence of international controls on greenhouse emissions. Let t measure the time in years since 1970, when the average global temperature was 15 ◦ C. Source: Science News (a) Find a linear equation giving the average global temperature in degrees Celsius in terms of t , the number of years since 1970. 6 / 15
Global Warming In 1990, the Intergovernmental Panel on Climate Change predicted that the average temperature on Earth would rise 0 . 3 ◦ C per decade in the absence of international controls on greenhouse emissions. Let t measure the time in years since 1970, when the average global temperature was 15 ◦ C. Source: Science News (b) Scientists have estimated that the sea level will rise by 65 cm if the average global temperature rises to 19 ◦ C. According to your answer to part (a), when would this occur? 7 / 15
Section 1.2 :: Linear Functions and Applications 8 / 15
Supply and Demand Suppose that the demand and price for a certain model of a youth wristwatch are related by p = D ( q ) = 16 − 1 . 25 q , where p is the price (in dollars) and q is the quantity demanded (in hundreds). Find the price at each level of demand. (a) 0 watches (b) 400 watches (c) 800 watches 9 / 15
Supply and Demand Suppose that the demand and price for a certain model of a youth wristwatch are related by p = D ( q ) = 16 − 1 . 25 q , where p is the price (in dollars) and q is the quantity demanded (in hundreds). Find the quantity demanded for the watch at each price. (d) $ 8 (e) $ 10 (f) $ 12 10 / 15
Supply and Demand Suppose that the demand and price for a certain model of a youth wristwatch are related by p = D ( q ) = 16 − 1 . 25 q , where p is the price (in dollars) and q is the quantity demanded (in hundreds). Suppose the price and supply of the watch are related by p = S ( q ) = 0 . 75 q , where p is the price (in dollars) and q is the quantity supplied (in hundreds) of watches. Find the equilibrium quantity and the equilibrium price. 11 / 15
Marginal Cost of Coffee The manager of a restaurant found that the cost to produce 100 cups of coffee is $ 11.02, while the cost to produce 400 cups is $ 40.12. Assume the cost C ( x ) is a linear function of x , the number of cups produced. (a) Find a formula for C ( x ). (b) What is the fixed cost? (c) Find the total cost of producing 1000 cups. (d) Find the total cost of producing 1001 cups. (e) Find the marginal cost of the 1001st cup. (f) What is the marginal cost of any cup and what does this mean to the manager? 12 / 15
Section 2.2 :: Quadratic Functions 13 / 15
Cancer From 1975 to 2007, the age-adjusted incidence rate of invasive lung and bronchial cancer among women can be closely approximated by f ( t ) = − 0 . 040194 t 2 + 2 . 1495 t + 23 . 921 , where t is the number of years since 1975. Source: National Cancer Institute Based on this model, in what year did the incidence rate reach a maximum? In what years was the rate increasing? Decreasing? 14 / 15
Maximizing the Height of an Object If an object is thrown upward with an initial velocity of 32 ft / second, then its height after t seconds is given by h = 32 t − 16 t 2 . (a) Find the maximum height attained by the object. (b) Find the number of seconds it takes the object to hit the ground. 15 / 15
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