Arc Length and Surface Area

Arc Length and Surface Area: Level 3 Challenges


Lise jogs along the curve defined by f(x)=2(x1)323f(x) =\dfrac{2(x - 1)^{\frac{3}{2}}}{3} from (1,f(1))(1, f(1)) to (4,f(4)).(4, f(4)). Steve jogs along the straight line connecting those two points. Steve and Lise both start from x=1x = 1 at the same time and Lise jogs at a speed of 73 units/s\frac{7}{\sqrt{3}} \mbox{ units}\mbox{/s}. What is the speed at which Steve must run (in  units/s\mbox{ units}\mbox{/s}) so that he arrives at (4,f(4))(4, f(4)) at the same time as Lise?

The coordinates of a dot PP moving in the xyxy plane at time tt are given by: x=5etcostx = 5e^{-t}\cos t and y=5etsinty = 5e^{-t} \sin t. Let LaL_a be the distance traveled by PP in the interval 0ta0 \leq t \leq a. If limaLa=z\displaystyle \lim_{a \to \infty} L_a = z, what is the value of z2z^2?

A delivery drone flying at constant speed 15 m/s15 \text{ m/s} and constant height 2700 m2700 \text{ m} toward a destination drops its goods. If the trajectory of the falling goods until it hits the ground can be described by the equation y=2700x275,y=2700-\frac{x^2}{75}, where xx is the horizontal distance it travels and yy is its height above the ground, what is the distance (not horizontal displacement) traveled by the goods until it hits the ground?

Note: You can use a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C.\displaystyle \int \sqrt{a^2+u^2}\,du=\frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln(u+\sqrt{a^2+u^2})+C.

There exists a unique, positive-valued, non-constant, continuous and differentiable function y=f(x)y = f(x) such that

(i) over any specified interval, the area between f(x)f(x) and the xx-axis is equal to the arclength of the curve, and

(ii) f(0)=1f(0) = 1.

If S=12f(x)dxS = \displaystyle\int_{-1}^{2} f(x) dx, then find 1000S\lfloor 1000S \rfloor.

Consider the curve y=exy={ e }^{ -x } in the first quadrant. Now it's rotated about the x-axisx\text{-axis} to obtain a solid of revolution. What is its surface area to 4 decimal places?


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