Jim Lambers Math 1B Fall Quarter 004-05 Final Exam Solution (Version A) 1. Suppose that a culture initially contains 500 bacteria, and that the population doubles every hours. What is the population after 4 hours? Your answer must be a decimal value. Solution We use the formula P P 0 t/d, where P 0 is the initial population, t denotes time, and d is the doubling time. Setting P 0 500, d, and t 4, we obtain P 500 4/ 500 8 500 56 18, 000.. Suppose that $0,000 is deposited in a savings account that earns 1% interest per year. If interest is compounded daily, what will the total balance be after 18 years? Your answer must be rounded to the nearest dollar. Solution The total balance A is given by the formula ( A P 1 + n) r nt, with P 0, 000, r 0.01, n 65, and t 18, we obtain ( A 0, 000 1 + 0.01 ) 65(18) 0, 000 1.0000797 6570 5, 916. 65. Suppose that an earthquake has an intensity of E 6.5 10 1 joules. What is the magnitude of the earthquake on the Richter scale? Your answer must be a decimal value. Note: the small, reference earthquake has an intensity of E 0 10 4.4 joules. Solution The magnitude M is given by the formula M log E E 0, where E 6.5 10 1 and E 0 10 4.4. We have M log 6.5 101 10 4.4 [log 6.5 + 1 4.4] (0.819 + 7.6) 5.6086. 1
4. Solve the equation for x. Your answer must be a decimal value. x +1 18 9 x Solution First, we use the property (a x ) y a xy to rewrite the equation as x +1 18 x. Then, dividing both sides by x, which is equivalent to multiplying by x, we obtain which simplifies to x x+1 18, (x 1) 18. Taking the natural logarithm of both sides and using the property ln x y y ln x yields 5. Solve the equation for x. x 1 ± (x 1) ln ln 18, ln 18 ln.6, 0.6. log (x 6) log (x 1) Solution Using the property log b x log b y log b (x/y), we obtain log x 6 x 1, which can be rewritten in exponential form to obtain Multiplying both sides by x 1 yields which has the solution x /7. x 6 x 1. 8x 8 x 6,
6. Suppose that 00 g of a radioactive substance decays to 0 g in 500 years. What is the half-life of the substance? Your answer must be a decimal value. Solution We use the formula A A 0 ( 1 ) t/h with A 0 00, A 0, and t 500 to obtain the equation 1 10 500/h which must now be solved for h. Taking the natural logarithm of both sides and using the property ln x y y ln x yields ln 1 10 ln 500 h, and therefore 500 ln 500 ln h ln 1 150.515 years. ln 10 10 7. Given that csc() 7.086, and that the angle of radians lies in Quadrant II, compute tan(). You must show your work, and your answer must be a decimal value. Solution First, we compute sin() 1 csc() 1 7.086 0.1411. Then, we note that because the angle of radians lies in Quadrant II, where cosine is negative, we have cos() 1 sin 0.98999. We conclude that tan() sin() cos() 0.1411 0.98999 0.145. 8. Suppose that an angle has a measure of 5 radians. What is its measure in degrees? Your answer must be a decimal value; fractions or π are not allowed. Solution We have θ deg 180 π θ rad 180 π 5 900 π 86.479. 9. Determine the reference angle for the angle 11π/6. Then use this angle to compute the exact value of tan(11π/6). You must show your work, and your answer must include either fractions or radicals. A decimal value is not acceptable.
Solution Because the angle 11π/6 is in Quadrant IV, its reference angle is obtained using the positive x-axis. It follows that the reference angle is π 11π/6 π/6. In this quadrant, tangent is negative, so we have tan 11π 6 tan π 6 sin pi 6 cos π 6 1/ / 1. 10. Suppose that a right triangle has an angle of 1, and that the side opposite this angle has a length of 8. Compute the lengths of the other two sides. Your answers must be decimal values. Solution Let h be the length of the hypotenuse, and let a be the length of the side adjacent to the given angle. Then we have sin 1 8 h, h 8 sin 1.. It follows from the Pythagorean Theorem that a h 8 0.84. 11. Compute the exact value of cos 1 (sin(5π/4)). A decimal value is not acceptable. Solution First, we note that the angle 5π/4 lies in Quadrant III, so the reference angle is 5π/4 π π/4. Since sine is negative in this quadrant, it follows that We conclude that cos 1 ( sin 5π 4 sin 5π 4 sin π 4. ) ( ) cos 1 π cos 1 π π 4 π 4. 1. Verify the identity Solution We have 1 + sec θ sin θ + tan θ 1 + sec θ sin θ + tan θ csc θ. 1 + 1 sin θ + sin θ 4
1. Solve the equation 1 + 1 sin θ ( 1 + 1 1 sin θ csc θ. sin x sin x for x, where 0 x < π if x is in radians, and 0 x < 60 if x is in degrees. Solutions must be exact values; decimal values are not acceptable. Solution First, we use the double-angle formula for sine to obtain sin x cos x sin x, which can be rearranged and factored to obtain sin x( cos x ) 0. It follows that the equation is satisfied if sin x 0 or cos x /. The first equation is satisified of x 0 or x 180. The second equation is satisified if x 0, and using the identity cos(60 x) cos x, it follows that x 0 is the other solution. We conclude that the only solutions in the interval 0 x < 60 are x 0, 0, 180, and 0. 14. Solve the equation sin θ + sin θ 0 for θ. You must describe all real values of θ that satisfy this equation. Solutions may be in degrees or radians. Solutions must be exact values; decimal values are not acceptable. Solution Factoring, we obtain the equation sin θ( sin θ + 1) 0, and therefore the equation is satisfied if sin θ 0 or sin θ 1/. The first equation has the solutions θ kπ, where k is an integer. The second equation is satisfied at θ sin 1 ( 1/) sin 1 (1/) π/6. Because sin(π θ) sin θ, it follows that another solution is θ π ( π/6) 7π/6. We conclude that the solutions are where k is an integer. θ kπ, θ π 6 + kπ, θ 7π 6 + kπ, ) 5
15. Suppose that a triangle has angles α 10 and β 101, and that the side opposite the angle α has length a 15. Compute the third angle γ, in degrees, and the lengths of the other two sides b and c. Your answers must be decimal values. Solution The angle γ is given by From the law of sines, we have b γ 180 α β 180 10 101 69. sin 10 15 sin 101 b sin 69, c 15 sin 101 15 sin 69 sin 10 84.79, c sin 10 80.64. 16. Suppose that a triangle has sides of length a 10 and b 0, and that the angle γ formed by these two sides has a measure of 7. Compute the measures of the other two angles α and β, in degrees, and the length of the third side c. Your answers must be decimal values. Solution First, we use the law of cosines to obtain c a + b ab cos γ 100 + 400 400 cos 7 76.9, c 19.4. Next, we compute the angle α that is opposite a, because it is guaranteed to be acute. From the law of sines, We conclude that sin α 10 sin 7 19.4, ( ) 10 sin 7 α sin 1 9.5. 19.4 β 180 α γ 180 9.5 7 78.64. 17. Compute the polar coordinates (r, θ) of the point whose rectangular coordinates are (x, y) ( 5, 6). Your answers must be decimal values. The coordinate θ can be in either degrees or radians, but you must specify which measure you are using. 6
Solution First, we compute r x + y ( 5) + 6 61 7.81. Then, we use the relation tan θ y x 6 5, and the fact that x < 0, to obtain ( θ tan 1 6 ) + 180 19.8. 5 18. Compute the exact rectangular coordinates (x, y) of the point whose polar coordinates are (r, θ) (5, 5π/6). Decimal values are not acceptable. Solution First, we note that the angle 5π/6 is in Quadrant II, where sine is positive and cosine is negative. Since the reference angle is π 5π/6 5π/6, we have and x r 5 cos 5π 6 5 cos π 6 5, y r sin θ 5 sin 5π 6 5 sin π 6 5. 7