7-2 Solving Exponential Equations and Inequalities
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1 Write an exponential function for the graph that passes through the given points. 16. (0, 6.4) and (3, 100) Substitute 100 for y and 6.4 for a and 3 for x into an exponential function and determine the value of 22. A savings account offers 0.7% interest compounded bimonthly. If $110 is deposited in this account, what will the balance be after 15 years? Substitute $110 for P, for r, 6 for n and 15 for t and simplify. An exponential function that passes through the given points is. 20. Find the balance of an account after 7 years if $700 is deposited into an account paying 4.3% interest compounded monthly. Substitute $700 for P, for r, 12 for n and 7 for t and simplify. 23. A college savings account pays 13.2% annual interest compounded semiannually. What is the balance of an account after 12 years if $21,000 was initially deposited? Substitute $21,000 for P, for r, 2 for n and 12 for t and simplify. 21. Determine how much is in a retirement account after 20 years if $5000 was invested at 6.05% interest compounded weekly. Substitute $5000 for P, for r, 52 for n and 20 for t and simplify. esolutions Manual - Powered by Cognero Page 1
2 26. Solve each inequality esolutions Manual - Powered by Cognero Page 2
3 30. SCIENCE A mug of hot chocolate is 90 C at time t = 0. It is surrounded by air at a constant temperature of 20 C. If stirred steadily, its temperature in Celsius after t minutes will be y(t) = (1.071) t. Find the temperature of the hot chocolate after 15 minutes. Find the temperature of the hot chocolate after 30 minutes. c. The optimum drinking temperature is 60 C. Will the mug of hot chocolate be at or below this temperature after 10 minutes? Substitute 15 for t in the equation and simplify. 31. ANIMALS Studies show that an animal will defend a territory, with area in square yards, that is directly proportional to the 1.31 power of the animal s weight in pounds. If a 45-pound beaver will defend 170 square yards, write an equation for the area a defended by a beaver weighing w pounds. Scientists believe that thousands of years ago, the beaver s ancestors were 11 feet long and weighed 430 pounds. Use your equation to determine the area defended by these animals. Substitute 170 for y, 45 for b, and 1.31 for x in the exponential function. Substitute 30 for t in the equation and simplify. c. Substitute 10 for t in the equation and simplify. The equation for the area a defended by a beaver weighting w pounds is Substitute 430 for w in the equation and solve for y. So, temperature of the hot chocolate will be below 60 C after 10 minutes. esolutions Manual - Powered by Cognero Page 3
4 32. Solve each equation esolutions Manual - Powered by Cognero Page 4
5 CCSS MODELING In 1950, the world population was about billion. By 1980, it had increased to about billion. Write an exponential function of the form y = ab x that could be used to model the world population y in billions for 1950 to Write the equation in terms of x, the number of years since (Round the value of b to the nearest ten-thousandth.) Suppose the population continued to grow at that rate. Estimate the population in c. In 2000, the population of the world was about 6.08 billion. Compare your estimate to the actual population. d. Use the equation you wrote in part a to estimate the world population in the year How accurate do you think the estimate is? Explain your reasoning. Substitute for y, for a, and 30 for x in the exponential function and solve for 37. The exponential function that model the situation is. Substitute 50 for x in the equation and simplify. c. The prediction was about 375 million greater than the actual population. d. Substitute 70 for x in the equation and simplify. Because the prediction for 2000 was greater than the actual population, this prediction for 2020 is probably even higher than the actual population will be at the time. esolutions Manual - Powered by Cognero Page 5
6 39. TREES The diameter of the base of a tree trunk in centimeters varies directly with the power of its height in meters. A young sequoia tree is 6 meters tall, and the diameter of its base is 19.1 centimeters. Use this information to write an equation for the diameter d of the base of a sequoia tree if its height is h meters high The General Sherman Tree in Sequoia National Park, California, is approximately 84 meters tall. Find the diameter of the General Sherman Tree at its base. The equation that represent the situation is. Substitute 84 for h in the equation and solve for d. 44. CHALLENGE Solve for x: = 4 x. Simplify the exponential equation. Use a graphing calculator to solve the exponential equation for x. Enter 4x as Y1 and E22 as Y2. Adjust the viewing window to see both graphs and the intersection. Use the CALC function to find the intersection of the two graphs. The diameter of the General Sherman Tree at its base is about 1001 cm. So, the solution to the equation is about x = OPEN ENDED What would be a more beneficial change to a 5-year loan at 8% interest compounded monthly: reducing the term to 4 years or reducing the interest rate to 6.5%? Reducing the term will be more beneficial. The multiplier is for the 4-year and for the 6.5%. esolutions Manual - Powered by Cognero Page 6
7 46. CCSS ARGUMENTS Determine whether the following statements are sometimes, always, or never true. Explain your reasoning. 2 x > 8 20x for all values of x. The graph of an exponential growth equation is increasing. c. The graph of an exponential decay equation is increasing. Always; 2 x will always be positive, and 8 20x will always be negative. Always; by definition the graph will always be increasing even if it is a small increase. c. Never; by definition the graph will always be decreasing even if it is a small decrease. 48. PROOF Show that 27 2x 81 x + 1 = 3 2x x + 1. esolutions Manual - Powered by Cognero Page 7
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Simplify each expression. 1. 2. 3. MULTIPLE CHOICE Identify all values of x for which is undefined. A 7, 4 B 7, 4 C 4, 7, 7 D 4, 7 D Simplify each expression. 4. esolutions Manual - Powered by Cognero
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