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Calculus / Thomas' Calculus: Early Transcendentals 13 / Chapter 3 / Problem 8AAE

Thomas' Calculus: Early Transcendentals | 13th Edition | ISBN: 9780321884077 | Authors: George B. Thomas Jr., Maurice D. Weir, Joel R. Hass

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Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to

Chapter 3, Problem 8AAE

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QUESTION:

Problem 8AAE

Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (–12,9 ) and (12, –9) How wide should the gondola be?

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QUESTION:

Problem 8AAE

Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (–12,9 ) and (12, –9) How wide should the gondola be?

ANSWER:

Solution

Step 1 of 1

Here It is given that

${x}^{2}+{y}^{2}=225$

Differentiating with respect to x , we get

$2x+2y\frac{dy}{dx}=0$

$\frac{dy}{dx}=-\frac{x}{y}$

Now the slope of tangent at $(12,-9)$

${\frac{dy}{dx}}_{(12,-9)}=\frac{12}{9}$

${\frac{dy}{dx}}_{(12,-9)}=\frac{4}{3}$

Hence the equation of the tangent is

$\frac{y+9}{x-12}=\frac{4}{3}$

By cross multiplication , we get

$3(y+9)=4(x-12)$

As the height is 15 +8 = 23 ft

Substituting $y=23$ , we get

$3(23+9)=4(x-12)$

$3(32)=4x-48$

=> $x=36$

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