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- The volume V of a sphere of radius r changes over time t. a. Find an equation relating dV>dt to dr>dt. b. At what rate is the volume changing if the radius increases at 2 in>min when the radius is 4 inches? c. At what rate is the radius changing if the volume increases at 10 in3>min when the radius is 5 inches?arrow_forwardThe flow rate of a pipe, Q in (m/s) is given by Q = dV where V is dt volume (in m) and t is time in (s). The volume in a pipe at time t is given in Table 1. Table 1 4 6 t, s 2 8 10 V(t), m3 7.56 3.87 5.09 6.34 4.59 Estimate the flow rate Q at time t = 4s by using: (a) 2-Point Forward Difference (b) 2-Point Backward Difference (c) 3-Point Central Differencearrow_forwardFor many species of fish, the allometric relationship between the weight W and the length L is approximately W = kL3, where k is a constant. Find the rate of change of the weight as a corresponding rate of change of the length. (dL/dt)arrow_forward
- The instantaneous flow rate of kerosene flowing through a pipe in 1000 kg/h is reported to be as follows Time, min 0 15 30 45 60 75 90 105 120 Flow rate, kg/s x 10-3 2.7 2.1 1.8 1.8 2.0 2.3 2.6 3.0 3.6 Calculate the total quantity of oil flowing in the pipe during the two-hour period using Simpson’s one-third rularrow_forwardA mass of 1 kgkg is attached to the end of a spring whose restoring force is 180 NmNm. The mass is in a medium that exerts a viscous resistance of 156 NN when the mass has a velocity of 6 msms. The viscous resistance is proportional to the speed of the object. Suppose the spring is stretched 0.07 mm beyond the its natural position and released. Let positive displacements indicate a stretched spring, and suppose that external vibrations act on the mass with a force of 7sin(3t)7sin(3t) NN at time tt seconds. Find an function to express the steady-state component of the object's displacement from the spring's natural position, in mm after tt seconds. (Note: This spring-mass system is not "hanging", so there is no gravitational force included in the model.)arrow_forwardFlank wear data were collected in a series of turning tests using a coated carbide tool on hardened alloy steel at a feed of 0.30 mm/rev and a depth of cut 4.0 mm. At a speed of 125 m/min, flank wear = 0.12 mm at 1 min, 0.27 mm at 5 min, 0.45 mm at 11 min, 0.58 mm at 15 min, 0.73 at 20 min, and 0.97 mm at of 25 min. At a speed of 175 m/min, flank wear = 0.22 mm at 1 min, 0.47 mm at 5 min, 0.70 mm at 9 min, 0.80 mm at 11 min, and 0.99 mm at 13 min. The last value in each case is when final tool failure occurred. (a) On a single piece of linear graph paper, plot flank wear as a function of time for both speeds. Using 0.50 mm of flank wear as the criterion of tool failure, determine the tool lives for the two cutting speeds. (b) On a piece of natural log-log paper, plot your results determined in the previous part. From the plot, determine the values of n and C in the Taylor Tool Life Equation. (c) As a comparison, calculate the values of n and C in the Taylor equation solving…arrow_forward
- Suppose a raindrop evaporates as it falls but maintains its spherical shape. Assume that the rate at which the raindrop evaporates (that is, the rate at which it loses mass) is proportional to its surface area, where the constant of proportionality is –0.01. The density (mass per volume) of water at 3.98°C is 1 g/cm3. The surface area of a sphere is 4πr2, and its volume is 4πr3/3, where r is the radius. Assume no air resistance. (Project 8 models the motion of this raindrop under the influence of air resistance.) Assume that the initial radius is 0.3 cm. Determine the raindrop’s initial mass. Write a differential equation for the rate of change of mass as a function of r. Write an equation for r as a function of massarrow_forwardTo investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Velocity (m/s) Swimmer Water Guar Syrup 1 0.90 0.93 0.92 0.98 3 1.00 0.95 4 1.10 1.14 1.20 1.23 6. 1.25 1.23 7 1.25 1.28 1.30 1.30 9 1.35 1.31 10 1.40 1.42 11 1.40 1.41 12 1.50 1.53 13 1.65 1.59 14 1.70 1.70 15 1.75 1.80 16 1.80 1.78arrow_forwardTo investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Swimmer Velocity (m/s) Water Guar Syrup 1 0.90 0.94 2 0.92 0.98 3 1.00 0.95 4 1.10 1.15 5 1.20 1.24 6 1.25 1.24 7 1.25 1.28 8 1.30 1.30 9 1.35 1.31 10 1.40 1.42 11 1.40 1.41 12 1.50 1.54 13 1.65 1.57 14 1.70 1.70 15 1.75 1.80 16 1.80 1.78 17 1.80 1.81 18 1.85 1.86 19 1.90 1.87 20 1.95 1.95 The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use ?d = ?water − ?guar syrup.) State the appropriate null and alternative hypotheses. H0: ?d < 0 Ha: ?d = 0 H0: ?d = 0 Ha: ?d < 0…arrow_forward
- To investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Swimmer Velocity (m/s) Water Guar Syrup 1 0.90 0.93 2 0.92 0.98 3 1.00 0.95 4 1.10 1.14 5 1.20 1.23 6 1.25 1.23 7 1.25 1.28 8 1.30 1.30 9 1.35 1.34 10 1.40 1.42 11 1.40 1.44 12 1.50 1.53 13 1.65 1.59 14 1.70 1.70 15 1.75 1.80 16 1.80 1.78 17 1.80 1.84 18 1.85 1.86 19 1.90 1.89 20 1.95 1.95 The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use μd = μwater − μguar syrup.) Find the test statistic and P-value. (Round your test statistic to one decimal place and your…arrow_forwardTo investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Swimmer Velocity (m/s) Water Guar Syrup 1 0.90 0.94 2 0.92 0.97 3 1.00 0.95 4 1.10 1.15 5 1.20 1.24 6 1.25 1.24 7 1.25 1.27 8 1.30 1.30 9 1.35 1.32 10 1.40 1.43 11 1.40 1.42 12 1.50 1.54 13 1.65 1.58 14 1.70 1.70 15 1.75 1.80 16 1.80 1.77 17 1.80 1.82 18 1.85 1.86 19 1.90 1.88 20 1.95 1.95 The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use ?d = ?water − ?guar syrup.) Find the test statistic and P-value. (Use a table or SALT. Round your test statistic to one…arrow_forwardTo investigate the fluid mechanics of swimming, twenty swimmers each swam a specified distance in a water-filled pool and in a pool where the water was thickened with food grade guar gum to create a syrup-like consistency. Velocity, in meters per second, was recorded and the results are given in the table below. Swimmer Velocity (m/s) Water Guar Syrup 1 0.90 0.94 2 0.92 0.97 3 1.00 0.95 4 1.10 1.15 5 1.20 1.24 6 1.25 1.24 7 1.25 1.27 8 1.30 1.30 9 1.35 1.31 10 1.40 1.42 11 1.40 1.41 12 1.50 1.54 13 1.65 1.59 14 1.70 1.70 15 1.75 1.80 16 1.80 1.77 17 1.80 1.81 18 1.85 1.86 19 1.90 1.89 20 1.95 1.95 The researchers concluded that swimming in guar syrup does not change mean swimming speed. Are the given data consistent with this conclusion? Carry out a hypothesis test using a 0.01 significance level. (Use μd = μwater − μguar syrup.) State the appropriate null and alternative hypotheses. H0: μd ≠ 0 Ha: μd = 0 H0: μd = 0 Ha: μd ≠ 0…arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning