Math Archive: Questions from 2024-05-28

A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of 4.4 feet from its low point to its high point, and that it returns to its high point every 14 seconds. Which equation best describes the motion of the buoy if at t = 0 it is at its high point? d = 115 cos⁡πt7 d = 4.4 cos⁡7πtd = 4.4 cos⁡14πtd = 4.4 cos⁡πt7 d = 115 cos⁡14πt d = 4.4 cos⁡πt14 d = 115 cos⁡πt14 d = 115 cos⁡7πt

Find the x-values (if any) at which the function f(x) = x−4 x2−14x+40 is not continuous. Which of the discontinuities are removable (limit exists) and non-removable? a) x = 4 (Removable), x = 10 (Not removable) b) No points of discontinuity. c) No points of continuity. d) x = 4 (Not removable), x = 10 (Not removable) e) x = 4 (Not removable), x = 10 (Removable)

Suppose the position function for a free-falling object on a certain planet is given by s(t) = −14t2 + v0t + s0. A silver coin is dropped from the top of a building that is 1, 366 feet tall. Find the time required for the coin to reach ground level. Round your answer to the three decimal places. 2.464 sec 9.878 sec 9.543 sec 2.640 sec 9.320 sec

Find an equation to the tangent line to the graph of the function f(x) = tan6⁡x at the point (2π/5, 849.853). The coefficients below are given to two decimal places. y = 5,361.53x − 20,953.14 y = 5,361.53x + 20,953.14 y = −17,350.27x − 20,953.14 y = 17,350.27x − 20,953.14 y = 17,350.27x + 20,953.14

Suppose the position function for a free-falling object on a certain planet is given by s(t) = −11t2 + v0t + s0. A silver coin is dropped from the top of a building that is 1, 378 feet tall. Find velocity of the coin at impact. Round your answer to the three decimal places. −235.236 ft/sec −246.236 ft/sec −257.185 ft/sec −111.364 ft/sec −123.118 ft/sec

Find an equation to the tangent line for the graph of f(x) = tan8⁡x at the point (4π/5, 0.078). Round coefficients to two decimal places. A. y = −1.31x + 3.36 B. y = 1.31x + 3.36 C. y = 1.06x + 3.36 D. y = 1.06x − 3.36

Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y(x) = 9 x−6 x = 9 and x = −6 x = 9 x = 9 and x = 6 x = 6 The graph has no horizontal tangents.

A buoy oscillates in simple harmonic motion y = Acos⁡ωt as waves move past it. The buoy moves a total of 13.5 feet (vertically) between its low point and its high point. It returns to its high point every 10 seconds. Determine the velocity of the buoy as a function of t. v = −6.75πsin⁡π 10t v = 6.75πsin⁡π 5t v = −1.35πsin⁡π5 t v = −1.35πsin⁡π10 v = 1.35πsin⁡π5 t

Find an equation of the tangent line to the graph of the function given below at the given point. 6x2 − 4xy + 4y2 − 14 = 0, (1,2) (The coefficients below are given to two decimal places.) y = 0.33x+2.33 y = 2.67x−0.67 y = −0.33x+2.33 y = −0.33x+0.67 y = 2.67x+0.67

A conical tank (with vertex down) is 24 feet across the top and 24 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 12 feet deep. 5/18π ft/min 5/72π ft/min 5/6π ft/min 5/12π ft/min

Suppose a buoy oscillates in a simple harmonic motion as the waves go past. It is noted that the buoy moves a total of 6 feet from its low point to its high point (see figure), and that it returns to its high point every 12 seconds. Let the function h(t) denote the height (in feet) of the buoy above equilibrium at any time t in seconds. (a) (2.5 points) Use the given information to draw at least one period of the graph of y = h(t). Assume that t = 0 is one of the times in which the buoy is at a high point. (b) (1.5 points) Write the function h(t).

A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 4 feet per second. Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 19 feet from the wall. Round your answer to three decimal places. 4.097 rad/sec 7.604 rad/sec 0.378 rad/sec 0.338 rad/sec 0.641 rad/sec

A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 19 feet from the wall. Round your answer to three decimal places. 0.242 rad/sec 0.641 rad/sec 2.168 rad/sec 3.804 rad/sec 0.278 rad/sec none of these

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point. limx→−1 x2−14x−15 x+1 16 14 −16 −14 does not exist

Find the x-values (if any) at which the function f(x) = 11x2 − 5x − 5 is not continuous. Which of the discontinuities are removable? x = 0, removable x = 5 22, removable x = 5 22, not removable continuous everywhere x = 2, removable

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point. limx→−1 −8x2+6x+14 x+1 a. 6 b. 22 c. -22 d. -6 e. does not exist

A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when its base is 9 feet from the wall? Round your answer to two decimal places. Select the correct answer. −5.33 ft/sec −1.01 ft/sec −3.00 ft/sec 2.00 ft/sec 5.33 ft/sec

Find the x-values (if any) at which f(x) = x x2+4x is not continuous. A. f(x) is not continuous only at x = 0 and f(x) has a removable discontinuity at x = 0. B. f(x) is not continuous at x = 0, −4 and f(x) has a removable discontinuity at x = 0. C. f(x) is continuous for all real x. D. f(x) is not continuous only at x = −4 and f(x) has a removable discontinuity at x = −4. E. f(x) is not continuous at x = 0, −4 and both the discontinuities are nonremovable.

Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y(x) = 6x (x−9)2 x = 6 x = 9 and x = −6 The graph has no horizontal tangents. x = −9 x = −9 and x = 6

Find the constants a and b such that the function is continuous on the entire real line. f(x) = {7, x ≤ −3 ax+b, −3 < x < 4 −7, x ≥ 4 a = b =

Find the x-values (if any) at which the function f(x) = x+2 x2+6x+8 is not continuous. Which of the discontinuities are removable? no points of discontinuity x = −2 (not removable), x = −4 (removable) x = −2 (removable), x = −4 (not removable) no points of continuity x = −2 (not removable), x = −4 (not removable)

Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = 3. (i) lim x→3+ f(x) (ii) lim x→3− f(x) (iii) lim x→3 f(x) a. 1, 1, 1, not continuous b. 2, 2, 2, continuous c. 4, 4, 4, not continuous d. 2, 2, 2, not continuous e. 1, 1, 1, continuous

An airplane flies at an altitude of y = 5 miles toward a point directly over an observer (see figure). The speed of the plane is 500 miles per hour. Find the rates (in radians per hour) at which the angle of elevation θ is changing when the angle is θ = 45∘, θ = 60∘, and θ = 80∘. (a) θ = 45∘ rad/hr (b) θ = 60∘ rad/hr (c) θ = 80∘ (Round your answer to two decimal places.) rad/hr

A buoy oscillates in simple harmonic motion y = Acos⁡ωt as waves move past it. The buoy moves a total of 16.5 feet (vertically) between its low point and its high point. It returns to its high point every 10 seconds. Determine the velocity of the buoy as a function of t. v = −8.25πsin⁡π 10t v = 8.25πsin⁡π 5t v = −1.65πsin⁡π 5t v = −1.65πsin⁡π 10t v = 1.65πsin⁡π 5t

A conical tank (with vertex down) is 18 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep. Select the correct answer. 1 15π ft/min 2 75π ft/min 4 5π ft/min 8 25π ft/min 2 25π ft/min

Find an equation of the tangent line to the graph of the function at the given point. y = 1 + ln ⁡7xy = e 7x − y, (1/7, 1)

Find an equation to the tangent line to the graph of the function f(x) = tan8⁡x at the point (4π/5, 0.078). The coefficients below are given to two decimal places. y = 1.31x+3.36 y = −1.31x+3.36 y = 1.06x−3.36 y = 1.06x+3.36 y = −1.31x−3.36

Find an equation to the tangent line to the graph of the function f(x) = tan6⁡x at the point (2π/5, 849.853). The coefficients below are given to two decimal places. y = 17,350.27x−20,953.14 y = −17,350.27 x−20,953.14 y = 5,361.53x−20,953.14 y = 17,350.27x+20,953.14 y = 5,361.53x+20,953.14

An airplane flies at an altitude of 6 miles toward a point directly over an observer. Consider θ and x as shown in the figure. (a) Write θ as a function of x. θ = (b) Find θ when x = 15 miles and x = 3 miles. (Enter your answers in radians rounded to two decimal places.) x = 15 miles θ ≈x = 3 miles θ ≈

A man 6 feet tall walks at a rate of 8 feet per second away from a light that is 15 feet above the ground (see figure). (a) When he is 10 feet from the base of the light, at what rate is the tip of his shadow moving? ft/sec (b) When he is 10 feet from the base of the light, at what rate is the length of his shadow changing? ft/sec

A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. If water is flowing into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep. 81/20π ft/min 9/100π ft/min 9/40π ft/min 81/50π ft/min 81/200π ft/min

A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure). When he is 13 feet from the base of the light, at what rate is the tip of his shadow moving? 50 ft/sec 3/50 ft/sec 1/2 ft/sec 9/2 ft/sec 50/3 ft/sec

A water tank in the shape of an inverted circular cone has a base radius of 2 m and height of 6 m. If water is being pumped into the tank at a rate of 3.9 m3/min, find the rate at which the water level is rising when the water is 4.8 m deep. (Round your answer to three decimal places if required) m/min

An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the figure. The speed of the plane is 350 mph. Find dθ dt in radians per hour when x = 5 miles.

Find the x-values (if any) at which f is not continuous. If there are any discontinuities, determine whether they are removable. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) f(x) = {1 5x + 1, x ≤ 5 6 − x, x > 5 removable discontinuities x = nonremovable discontinuities x =

Find an equation of the line that is tangent to the graph of f and parallel to the given line. Function Line f(x) = 2x2 4x − y + 8 = 0 y =

A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. if water is poured into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep. A. 9/40π ft/min B. 9/100π ft/min C. 81/20π ft/min D. 81/50π ft/min

A population of 350 bacteria is introduced into a culture and grows in number according to the equation P(t) = 350(1 + 6t 60 + t2) where t is measured in hours. Find the rate at which the population is growing when t = 4. Round your answer to two decimal places. 16.00 bacteria per hour 125.07 bacteria per hour 28.71 bacteria per hour 27.63 bacteria per hour 32.81 bacteria per hour

A population of 350 bacteria is introduced into a culture and grows in number according to the equation P(t) = 350(1 + 6t 60 + t2) where t is measured in hours. Find the rate at which the population is growing when t = 4. Round your answer to two decimal places. 16.00 bacteria per hour 125.07 bacteria per hour 28.71 bacteria per hour 27.63 bacteria per hour 32.81 bacteria per hour

Suppose the position function for a free-falling object on a certain planet is given by s(t) = −16t2 + v0t + s0. A silver coin is dropped from the top of a building that is 1, 362 feet tall. Determine the average velocity of the coin over the time interval [1, 2]. −48 ft/sec −53 ft/sec 16 ft/sec 48 ft/sec −16 ft/sec

A population of 530 bacteria is introduced into a culture and grows in number according to the equation below, where t is measured in hours. Find the rate at which the population is growing (in bacteria/hour) when t = 5. (Round your answer to two decimal places. ) P(t) = 530(1 + 5t 40 + t2) P′(5) = bacteria/hour bacteria/hour

Find the x-values (if any) at which the function f(x) = x+6 x2+13x+42 is not continuous. Which of the discontinuities are removable? no points of continuity no points of discontinuity x = −6 (removable), x = −7 (not removable) x = −6 (not removable), x = −7 (not removable) x = −6 (not removable), x = −7 (removable)

The ordering and transportation cost C for the components used in manufacturing a product is C = 120(240 x2 + x x+50), x ≥ 1 where C is measured in thousands of dollars and x is the order size in hundreds. Find the rate of change of C with respect to x for x = 21. Round your answer to two decimal places. 3.13 thousand dollars per hundred 7.41 thousand dollars per hundred -8.02 thousand dollars per hundred -5.03 thousand dollars per hundred -6.20 thousand dollars per hundred

Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = −4. (i) lim x→−4+ f(x) (ii) lim x→−4− f(x) (iii) lim x→−4 f(x) 1, 1, 1, continuous −4, −4, −4, continuous 2, 2, 2, continuous 1, 1, 1, not continuous 2, 2, 2, not continuous

Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = −3 (i) lim x→−3+ f(x) (ii) lim x→−3− f(x) (iii) lim x→−3 f(x) A. 2, 2, 2, continuous B. 2, 2, 2, not continuous C. 3, 3, 3, not continuous D. −3, −3, −3, continuous E. 3, 3, 3, continuous

Determine the following limit. (Hint: Use the graph to calculate the limit.) lim x→2 (5 − x) A. 2 B. 3 C. 5 D. 7 E. does not exist

The ordering and transportation cost C for the components used in manufacturing a product is given below, where C is measured in thousands of dollars and x is the order size in hundreds. (Round your answers to two decimal places.) C = 145(239 x2 + x x+22), x ≥ 1 (a) Find the rate of change of C with respect to x when x = 10. C′(10) = thousand dollars /100 components (b) Find the rate of change of C with respect to x when x = 15. C′(15) = thousand dollars 100 components (c) Find the rate of change of C with respect to x when x = 20. C′(20) = thousand dollars 100 components What do these rates of change imply about increasing order size? The cost per item with increasing order size.

A buoy oscillates in simple harmonic motion y = Acos⁡(ωt) as waves move past it. The buoy moves a total of 4.8 feet (vertically) from its low point to its high point. It returns to its high point every 12 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t = 0. y = (b) Determine the velocity of the buoy as a function of t. v =

A buoy oscillates in simple harmonic motion y = Acos⁡(ωt) as waves move past it. The buoy moves a total of 3 feet (vertically) from its low point to its high point. It returns to its high point every 4 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t = 0. y(t) = (b) Determine the velocity of the buoy as a function of t. v(t) =

A water tank, shown to the right, is shaped like an inverted cone with height 4 m and base radius 0.5 m. a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank? Use 1000 kg/m3 for the density of water and 9.8 m/s2 for the acceleration due to gravity. b. Is it true that it takes half as much work to pump all the water out of the tank when it is filled to half its depth as when it is full? Explain. a. Draw a y-axis in the vertical direction (parallel to gravity) and choose the bottom of the tank as the origin. For 0 ≤ y ≤ 4, find the cross-sectional area A(y) in terms of y. A(y) = (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression.)

Use a triple integral to find the volume of the wedge bounded by the parabolic cylinder y = x2 and the planes z = 9 − y and z = 0. The volume of the wedge is (Type an exact answer.)

Use a triple integral to find the volume of the solid bounded by the surfaces z = 2ey and z = 2 over the rectangle {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ ln⁡2}. The volume of the solid is (Type an exact answer.)

Use a triple integral to compute the volume of the wedge of the square column |x| + |y| = 19 created by the planes z = 0 and x + y + z = 19. The volume of the wedge is

The coordinate axes in the figure run through the centroid of a solid wedge parallel to the labeled edges. The wedge has a = 4, b = 6, and c = 9. The solid has constant density δ = 1. The square of the distance from a typical point (x, y, z) of the wedge to the line L: z = 0, y = 6 is r2 = (y−6)2 + z2. Calculate the moment of inertia of the wedge about L. IL = (Simplify your answer. Type an integer or fraction.)