Multiple Choice Question: A horn is generated by a circle which moves in the following manner:

Situation
A horn is generated by a circle which moves in the following manner: The plane of the circle is always parallel to the yz-plane with center always on the xy-plane, and its diameter is a line segment cut off by the curve x = y² and x = 9y² on the 1^st quadrant of the xy-plane.

What is the diameter of the circle at x = 3?

A. 1.49 C. 1.37

B. 1.25 D. 1.15
What is the lateral area of the horn from x = 0 to x = 3?

A. 10.45 C. 7.26

B. 8.32 D. 6.51
What is the volume generated from x = 0 to x = 3?

A. 1.32 C. 1.13

B. 1.57 D. 1.63

Jhun Vert Sun, 07/28/2024 - 01:03

Solution

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$D = (y \text{ of } x = y^2) - (y \text{ of } x = 9y^2)$

$D = x^{1/2} - \frac{1}{3}x^{1/2}$

$D = \frac{2}{3}x^{1/2}$

Part 1

At x = 3
$D = \frac{2}{3}(3^{1/2})$

$D = 1.1547 ~ \text{units}$

Part 2

$dA = \pi D ~ dx$

$\displaystyle A = \dfrac{2\pi}{3}\int_0^3 x^{1/2} ~ dx$

$A = \dfrac{2\pi}{3} \left[ \dfrac{x^{3/2}}{3/2} \right]_0^3$

$A = \frac{4}{9} \pi \left( 3^{3/2} - 0^{3/2} \right)$

$A = 7.255 ~ \text{unit}^2$

Part 3

$dV = \frac{1}{4} \pi D^2 ~ dx$

$\displaystyle V = \dfrac{\pi}{4} \int_0^3 \left( \dfrac{2x^{1/2}}{3} \right)^2 ~ dx$

$\displaystyle V = \dfrac{\pi}{9} \int_0^3 x ~ dx$

$V = \dfrac{\pi}{9} \left[ \dfrac{x^2}{2} \right]_0^3$

$V = \dfrac{\pi}{9} \left[ \dfrac{3^2}{2} - \dfrac{0^2}{2} \right]_0^3$

$V = \dfrac{\pi}{2} ~ \text{units}^3 = 1.571 ~ \text{units}^3$

A. 1.49	C. 1.37
B. 1.25	D. 1.15

A. 10.45	C. 7.26
B. 8.32	D. 6.51

A. 1.32	C. 1.13
B. 1.57	D. 1.63

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