Math Archive: Questions from 2024-05-30

Find the time required for an object to cool from 340∘F to 240∘F by evaluating t = 10 ln⁡2 ∫240 340 1 T−70 dT, where t is time in minutes. Round your answer to four decimal places. 7.3417 minutes 5.0489 minutes 4.2110 minutes 4.6321 minutes 6.6742 minutes

The graph of f consists of line segments, as shown in the figure. Evaluate the definite integral ∫4 10 f(x)dx using geometric formulas. A. -3 B. 1 C. 0 D. -2 E. -1

Determine whether the Mean Value Theorem can be applied to the function f(x) = 2sin⁡x + sin⁡2x on the closed interval [7π, 8π]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (7π, 8π) such that f′(c) = f(8π)−f(7π) 8π−7π. MVT applies; 11π 3 MVT applies; 15π 2 MVT does not apply MVT applies: 11π 2 MVT applies: 22π 3

Determine whether the Mean Value Theorem can be applied to the function f(x) = x2 on the closed interval [7, 13]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (7, 13) such that f′(c) = f(13)−f(7) 13−(7). MVT applies; c = 10 MVT applies; c = 11 MVT applies; c = 8 MVT applies; c = 9 MVT applies; c = 12

Determine whether the Mean Value Theorem can be applied to the function f(x) = |x−2| on the closed interval [0, 4]. If the MVT can be applied, find all point(s) c in the in the open interval such that f′(c) = f(4)−f(0) 4−0. The Mean Value Theorem does apply; c = 0. The Mean Value Theorem does apply; c = 1. The Mean Value Theorem does apply; c = 14. The Mean Value Theorem does not apply. The Mean Value Theorem does apply; c = 12.

Find the relative extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. (Round your answers to three decimal places. If an answer does not exist, enter DNE. ) g(x) = 1 2π e−(x−7)2 /2 maximum (x, y) = minimum (x, y) = inflection point (x, y) = (smaller x-value) inflection point (x, y) = (larger x-value)

The graph of the function g(y) = y3 is given below. Which of the following definite integrals yields the area of the shaded region? ∫−2 2 y3 dy ∫−2 0 y3 dy ∫0 8 y3 dy ∫0 2 y3 dx ∫0 2 y3 dy

The maker of an automobile advertises that it takes 13 seconds to accelerate from 5 kilometers per hour to 65 kilometers per hour. Assuming constant acceleration, compute the acceleration in meters per second per second. Round your answer to three decimal places.

The rate of change in sales S is inversely proportional to time t (t > 1) measured in weeks. Find S as a function of t if sales after 2 and 4 weeks are 250 units and 400 units respectively. S(t) = 50[3ln⁡t ln⁡2 + 2] S(t) = 650[ln⁡t ln⁡150 + 50] S(t) = 50[3ln⁡t ln⁡2 + 13] S(t) = 50[13ln⁡t ln⁡2 + 2] S(t) = 650[ln⁡t ln⁡150 + 1]

The maker of an automobile advertises that it takes 8 seconds to accelerate from 15 kilometers per hour to 70 kilometers per hour. Assuming constant acceleration, compute the distance, in meters, the car travels during the 8 seconds. Round your answer to two decimal places. 49.50 m 97.78 m 94.44 m 188.89 m 61.11 m

Suppose the demand equation for a product is p = 80,000 400+7x. Find the average price p on the interval 40 ≤ x ≤ 50. Round your answer to the nearest dollar. $5 $164 $114 $1,120 $103

Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function 2x2 − x − 1 and the x-axis over the interval [2, 13]. Round your answer to the nearest integer. 1,295 1,320 1,195 995 1,020

The resistance R of a certain rype of resistor is R = 0.003T4 − 2T + 100 where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical number of the function. Round numerical values in your answer to the nearest whole number. T = 18∘ T = 13∘ T = 7∘ T = 4∘ T = 6∘

Use a computer algebra system to graph the function 7 3x 3−x and determine all absolute extrema on the interval [0, 3]. absolute maximum: (2, 14/3) absolute minimum: (0, 0)(3, 0) absolute maximum: (3, 7) absolute minimum: (0, 0) absolute maximum: (2, 14/3) absolute minimum: (0, 3) absolute maximum: (1, 73) absolute minimum: (0, 0)(3, 0) absolute maximum: (3, 7) absolute minimum: (0, 3)

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 38 feet. x = 76 2+π feet, y = 38 2+π feet x = 114 2+π feet y = 38 2+π feet x = 38 4+π feet y = 76 4+π feet x = 76 4+π feet y = 38 4+π feet x = 38 4+π feet y = 114 4+π feet

Use a graphing utility to graph the function f(x) = 16 8−x and locate the absolute extrema of the function on the interval [0, 8). absolute minimum: (4, 4) no absolute maximum absolute maximum: (0, 2) absolute minimum: (8, 0) absolute minimum: (0, 2) absolute maximum: (8, 16) absolute minimum: (0, 2) no absolute maximum absolute minimum: (0, 0) no absolute maximum

Use differentials to approximate the quantity. (Round your answer to four decimal places. ) 38.4 4 38.4 4 ≈ Please explain how you obtained the above approximation by answering the following questions. a) In order to use the differential, we need a function. What is the most appropriate function for this problem? b) What is the differential of this function? dy = c) What is the most appropriate value for x? d) What is the corresponding value of dx? e) What is the corresponding value of dy? (Round your answer to four decimal places.)

The resistance R of a certain type of resistor is R = 0.003T4 − 5T + 100 where R is measured in ohms and the temperature T is measured in degrees Celsius. Use a computer algebra system to find dR dT dR dT = 0.003T3−5 2 0.003T4−5T+100 dR dT = 0.012T4−5T 2 0.003T4−5T+100 dR dT = 0.012T3−5 2 0.003T4−5T+100 dR dT = 3T3 500 0.003T4−5T+100 dR dT = 0.003T3−5 2 0.003T4−5

A long distance phone service charges $0.35 for the first 8 minutes and $0.05 for each additional minute or fraction thereol. Use the greatest integer function to write the cost C of a call in terms of time t (in minutes). C = {0.350 < t ≤ 80.35+0.05[|t−8|] t > 8 C = {0.350 < t ≤ 80.35+0.05[|t−7|] t > 8 C = {0.350 < t ≤ 8 0.35+0.05(t−8) t > 8 C = {0.350 < t ≤ 8 0.35+0.05[|t−8|] t > 8, is not an integer 0.35+0.05(t−7) t > 8, t is an integer C = {0.350 < t ≤ 80.35+0.05[|t−7|] t > 8, t is not an integer 0.35+0.05(t−8) t > 8, t is an integer

Find the length and width of a rectangle that has perimeter 64 meters and a maximum area. 8 m; 24 m. 16 m; 16 m 20 m; 13 m. 17 m; 15 m. 1 m; 31 m.

Use Newton's Method to approximate the zero(s) of the function f(x) = x3+x+1 accurate to three decimal places. 0.682 0.649 0.703 −0.694 −0.682

An object is thrown vertically upward from the surface of a celestial body at a velocity of 24 meters per second. Its distance from the surface at t seconds is given by s(t) = −0.4t2 + 24t. a. How fast is the object moving 2 seconds after being thrown? b. How long after the object is thrown does it reach its maximum height? c. How high will the object go? a. The object's velocity after 2 seconds is m/sec. (Round to the nearest tenth as needed.) b. It takes the object sec to reach its maximum height. (Round to the nearest tenth as needed.) c. The object reaches a maximum height of m. (Round to the nearest tenth as needed.)

A ball is thrown vertically upwards from a height of 8 ft with an initial velocity of 30 ft per second How high will the ball go? Note that the acceleration of the ball is given by a(t) = −32 feet per second per second. 50.1875 ft 18.5469 ft 53.1875 ft 22.0625 ft 34.1875 ft

The graph of f consists of line segments, as shown in the 10 figure. Evaluate the definite integral ∫ 2 10 f(x)dx using 2 geometric formulas. a. 3.5 b. 1.5 C. 4.5 d. 0.5 e. 2.5

Rate of Change A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, then the top will move down the wall at a rate of r = 2x 625 − x2 ft/sec where x is the distance between the base of the ladder and the house, and r is the rate in feet per second. (a) Find the rate r when x is 7 feet. (b) Find the rate r when x is 15 feet. (c) Find the limit of r as x approaches 25 from the left.

Use the rectangles in the following graph to approximate the area of the region bounded by y = sin⁡x, y = 0, x = 0, and x = π. 1.0519 0.7850 1.1775 1.5700 3.1400

A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of r = 2x 625−x2 ft/sec, where x is the distance between the base of the ladder and the house. Find the rate r when x is 24 feet. r = 48/7 ft/sec r = 7/24 ft/sec r = 48/5 ft/sec r = 7/48 ft/sec r = 24/7 ft/sec

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. Not drawn to scale (a) Write θ as a function of x. θ(x) = (b) The speed of the plane is 320 miles per hour. Find dθ/dt when x = 10 miles and x = 5 miles. (Round your answers to three decimal places. ) when x = 10 dθ dt = rad/h when x = 5 dθ dt = rad/h

Find the length and width of a rectangle that has perimeter 56 meters and a maximum area. 7, 21 1, 27 14, 14 15, 13 18, 11

The graph of the function g(y) = y2 is given below. Which of the following definite integrals yields the area of the shaded region? a. ∫−2 2 y2 dy b. ∫0 2 y2 dy c. ∫−2 0 y2 dy d. ∫0 4 y2 dy e. ∫0 2 y2 dx

A sphere has a volume of 4.0 cubic inches. If the sphere's volume can vary between 3.2 cubic inches and 5.5 cubic inches, how can the radius vary? Round your answer to five decimal places. Radius can vary between 0.18475 inch and 2.48475 inches. Radius can vary between 0.87404 inch and 1.14587 inches. Radius can vary between 1.58533 inches and 2.07839 inches. Radius can vary between 1.45113 inches and 1.73825 inches. Radius can vary between 0.91416 inch and 1.09503 inches.

Use differentials to approximate the value of 124 3. Round your answer to four decimal places. 5.0067 4.9767 4.9867 4.9967 4.9667

Find the length and width of a rectangle that has perimeter 48 meters and a maximum area. 12 m; 12 m. 16 m; 9 m. 1 m; 23 m. 13 m; 11 m. 6 m; 18 m.

Find the time required for an object to cool from 320∘F to 250∘F by evaluating t = 10 ln⁡2∫250 320 1 T−50 dT where t is time in minutes. Round your answer to four decimal places. 3.0048 minutes 4.3296 minutes 2.7317 minutes 3.2752 minutes 4.7626 minutes

Approximate the positive zero(s) of the function f(x) = x4 − 2 cos⁡x to three decimal places. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001 . Select one: a. -1.116 b. 1.014 c. 0.991 d. -1.014 e. 1.065

Use left endpoints and 12 rectangles to find the approximation of the area of the region between the graph of the function cos⁡(10x) and the x-axis over the interval [0, π/10]. Round your answer to four decimal places. −0.0038 0.0462 0.0762 0.0262 −0.0138

Approximate the positive zero(s) of the function f(x) = x3 − 3cos⁡x to three decimal places. Use Newton's Method and continue the process until two successive approximations differ by less than 0.001 . −1.105 1.082 1.156 −1.207 1.105

Find the length and width of a rectangle that has an area of 968 square feet and whose perimeter is a minimum. l = 242 feet; w = 4 feet l = 484 feet; w = 2 feet l = w = 22√2 feet; l = 121 feet; w = 8 feet l = 484√2 feet; w = √2 feet

The rate of change in sales S is inversely proportional to time t(t > 1) measured in weeks. Find S as a function of t if sales after 2 and 4 weeks are 200 units and 350 units respectively. A. S(t) = 550[ln⁡t ln⁡3 + 150] B. S(t) = 50[3 ln⁡t ln⁡2 + 1] C. S(t) = 50[11 ln⁡t ln⁡2 + 1] D. S(t) = 550[ln⁡t ln⁡3 + 1] E. S(t) = 50[3 ln⁡t ln⁡2 + 11]

Find the cubic function of the form f(x) = ax3 + bx2 + cx + d where a ≠ 0 and the coefficients a, b, c, d are real numbers, which satisfies the conditions given below. Relative maximum: (3, 3) Relative minimum: (5, 1) Inflection point: (4, 2) f(x) = 12 x3 − 6x2 + 45 2 x + 24 f(x) = 12 x3 − 6x2 + 41 2 x +23 f(x) = 12 x3 − 6x2 + 45 2 x − 24 f(x) = 12 x3 − 6x2 + 41 2 x − 23 f(x) = −12 x3 + 6x2 − 45 2 x + 24