Calculus - check answers

I'd appreciate corrections for the ones I get wrong. :) Thanks!

6. A point on a damped spring has motion given by s(t)=2e^(-1.5t) s in2(pi)t, where s is measured in centimeters and t is measured in seconds. Choose which of these is the graph for the velocity function for 0 < or equal to t < or equal to 4
(not sure about how to solve this one)

7. The edge of a cube was found to have a length of 50 cm with a possible error in measurement of 0.1 cm. Based on the measurement, you determine that the volume is 125,000 cm^3. Use tangent line approximation to estimate the percentage error in volume.


8. An inverted conical tank (with vertex down) is 14 feet across the top and 24 feet deep. If water is flowing in at a rate of 12 ft^3/min, find the rate of change of the depth of the water when the water is 10 feet deep.

0.007 ft/min
*0.449 ft/min
0.018 ft/min
0.051 ft/min
0.065 ft/min

9. For the function f(x)=Inx/x^2, find the approximate location of the critical point in the interval (0, 5).

(0.5, −2.773)
(1, 0)
(1.649, 0.184)
*(2, 0.173)
(0.778, −1.813)

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  1. #6
    s(t)=2e^(-1.5t) sin2πt
    so the period is 2π/2π = 1
    You want the graph that starts out at (0,0) and increases, and you want 4 periods.
    The first max will be at t=1/4, so s(1/4) = 2e^(-3/8) ≈ 1.37
    Looks like B to me

    dv = 3s^2 ds = 3*2500*0.1 = 750
    750/125000 = 0.006 = 0.6%
    How did you get your answer?

    using similar triangles, when the water has a depth of y, the radius of the surface is 7/24 y. So, the volume of the water is
    V = 1/3 π r^2y = π/3 (7/24)^2 * y^3
    dV/dt = π/3 (7/24)^2 * 3y^2 dy/dt
    Thus, when y=10,
    12 = (7/24)^2 * π * 100 dy/dt
    dy/dt = 1728/(1225π) ≈ 0.449 ft/min
    good work

    f(x) = lnx/x^2
    f'(x) = (1-2lnx)/x^3
    f'(x)=0 when 1-2lnx=0, or x=√e ≈ 1.649
    How did you arrive at x=2?

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  2. Thank you so much for your time and help!
    #9 I resolved and got the answer: (1.649, 0.184) but it wouldnt let me edit the post.
    #7 was a guess because I was quite confused on how to solve it

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