Calculus Problem Set.docx

PROBLEM SET #3 CALCULUS

1. As a bucket is raised a distance of 30 feet from the bottom of a well, water leaks out at a uniform rate. Find the work done if the bucket originally contains 24 pounds of water and one-third leaks out. Assume that the weight of the empty bucket is 4 pounds, and disregard the weight of the rope. a. 600 lbs-ft b. 540 lbs-ft c. 720 lbs-ft d. 2880 lbs ft 2. Two lines through the point (1, –3) are tangent to the parabola = 𝑥 2 . Determine one point of tangency. A. (-1, -1) B. (3, -9) C. (-1, 1) D. (-3, 9) 3. A building with a rectangular base is to be constructed on a lot in the form of a right triangle with legs 18 m and 24 m. If the building has one side along the hypotenuse of the triangle find the dimensions of the base of the building for maximum floor area a. 150 sq.m b. 210 sq.m c. 100 sq.m d. 108 sq.m

10. Find the arc length of the graph of (𝑥 + 3)2 = 8(𝑦 – 1)3 from A( - 2,1) to B(5, 3). A. 8.44 B. 2.084 C. 7.36 D. 6.17 11. Suppose h(x) is an even, continuous function given 2 0 that ∫0 ℎ(𝑥)𝑑𝑥 = 7 and∫−5 ℎ(𝑥)𝑑𝑥 = −4. Find the value of −2

∫−5 ℎ(𝑥)𝑑𝑥 A. -11 B. 3

C. 11

D. -3

12. Find the 100th derivative of 𝑓(𝑡) = 7𝑡 A. 7𝑡−1 Ln100 7 C. 7𝑡+1 Ln100 7 B. 7𝑡−1 Ln99 7 D.7𝑡 Ln100 7 13. Determine the maximum value of f(x) in the equation 𝑓(𝑥) = 3 cos 𝑥 − 6 sin 𝑥 A. 3√2 B. 3√5 C. 6√2 D. 6√5 1

4. A building with a rectangular base is to be constructed on a lot in the form of a right triangle with legs 18 m and 24 m. If the building has one side along the hypotenuse of the triangle find the dimensions of the base of the building for maximum floor area a. 150 sq.m b. 210 sq.m c. 100 sq.m d. 108 sq.m 5. Two men A and B, start from the same point S of a circular track of 100m radius. A runs at 10m/sec, while B runs 8m/sec. in opposite directions. Find the rate at which the distance between them (chord) is changing, when the central angle between them is 120 degrees. a. 9m/sec b. 3m/sec c. 6 m/sec d. 7m/sec 6. Between 0°C and 30°C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula 𝑉 = 999.87 − 0.06426𝑇 + 0.0085043𝑇 2 − 0.0000679𝑇 3 Find the temperature at which water has its maximum density. A. 5° B. 3.5° C. 4.2° D. 4° 7. Determine the volume generated when the area bounded by the curves y=sinx and y=cosx from x=0 to x=𝜋/4 is revolved about the y-axis. A. 1.794 B. 0.348 C. 0.897 D. 0.696 8. Flour sifted onto waxed paper forms a conical pile whose radius and height are always equal, although both increases with time. The volume of flour on the waxed paper is increasing at the rate of 7.26 cm3/s. How fast is the height of the flour increasing when the volume is 29 cm3? A. 0.294 cm/s C. 0.252 cm/s B. 0.221 cm/s D. 0.336 cm/s 9. For the polar curve 1-2cosx , calculate the length of the inner loop A. 2.682 B. 13.364 C. 21.23 D. 2pi

14. Evaluate: Lim sin( ) 𝑥→0

A. e2x B. Does Not Exist

𝑥

C. 0.9848 D. -0.9848

15. Water is pouring into a swimming pool. After t hours, there are t + √𝑡 gallons in the pool. At what rate is the water pouring into the pool when t = 9hrs. a. 7/6 gph c. 8/7 gph b. 6/5 gph d. 5/4 gph 16. The base of an isosceles triangle is 8 feet long. If the altitude is 6 feet long and is increasing 3 inches per minute, at what rate are the base angles changing? A.

1 52

° Per Min

C. 0.0183° Per Sec

B. 1.102° Per Sec

D.

15 13

Rad Per Hour

17. Determine the area of the region outside r=3+2sinθ, and inside r=2. A. 1.486 B. 2.196 C. 2.476 D. 1.978 18. Find the arc length of the cardioid r=2+2cosθ A. 4π B. 8π C. 16π D. 16 A particle moves along the curve 𝑟 =

4 1+cos 𝜃

with a

constant speed of 4 ft per second. 19. Find vr A. 3√2 B. 2√2 C. 2√3

D. 3√3

20. Find vθ A. 3√2

D. 3√3

B. 2√2

C. 2√3