Jim Lambers Math 1B Fall Quarter Final Exam Solution (Version A)

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1 Jim Lambers Math 1B Fall Quarter 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/ , 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, ) 65(18) 0, , Suppose that an earthquake has an intensity of E 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 joules. Solution The magnitude M is given by the formula M log E E 0, where E and E We have M log [log ] ( )

2 4. Solve the equation for x. Your answer must be a decimal value. x x Solution First, we use the property (a x ) y a xy to rewrite the equation as x 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,

3 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 /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 years. ln 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() Then, we note that because the angle of radians lies in Quadrant II, where cosine is negative, we have cos() 1 sin We conclude that tan() sin() cos() 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 π 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.

4 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/ / 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 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 Verify the identity Solution We have 1 + sec θ sin θ + tan θ 1 + sec θ sin θ + tan θ csc θ sin θ + sin θ 4

5 1. Solve the equation sin θ ( 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 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

6 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 sin sin 101 b sin 69, c 15 sin sin 69 sin , c sin 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 cos , c 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 , ( ) 10 sin 7 α sin β 180 α γ 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

7 Solution First, we compute r x + y ( 5) Then, we use the relation tan θ y x 6 5, and the fact that x < 0, to obtain ( θ tan 1 6 ) 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