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Bumek [7]
3 years ago
10

Calculus Application

Mathematics
1 answer:
Fiesta28 [93]3 years ago
7 0
<h3>Answer:</h3>

-8/3 ft/s

<h3>Step-by-step explanation:</h3>

<u>We are given:</u>

distance of the top of the ladder from the ground (h) = 12 ft

height of the ladder = 20 ft

rate of change of the distance of the base of ladder from the wall (dx/dt):

2 ft/s

<u>Finding the distance of the base of the ladder from the wall:</u>

From the Pythagoras's Theorem, we know that:

hypotenuse² = height² + base²

<em>replacing the given values</em>

20² = 12² + x²

400 = 144 + x²

x² = 256                            [subtracting 144 from both sides]

x = 16 ft                            [taking the square root of both sides]

<u>The rate of change of the height of the Ladder from the ground:</u>

We know that:

h = 12 ft

(\frac{dh}{dt}) = ?

x = 16 ft

(\frac{dx}{dt}) = 2 ft/s

According to the Pythagoras's Theorem:

20² = x² + h²

<em>differentiating both sides with respect to time</em>

\frac{d(400)}{dt} = \frac{d(x^{2} + h^{2})}{dt}

0 = \frac{d(x^{2})}{dt} +  \frac{d(h^{2})}{dt}

0 = \frac{d(x^{2})}{dx}(\frac{dx}{dt}) +  \frac{d(h^{2})}{dh}(\frac{dh}{dt})

0 = 2x(\frac{dx}{dt}) +  2h(\frac{dh}{dt})

<em>replacing the variables</em>

0 = 2(16)(2) +  2(12)(\frac{dh}{dt})

0 = 64 +  32(\frac{dh}{dt})

-64 =32(\frac{dh}{dt})                                [subtracting 64 from both sides]

\frac{-64}{32} =(\frac{dh}{dt})                                    [dividing both sides by 32]

\frac{dh}{dt} = \frac{-8}{3} ft/s

Hence, the ladder will slide down at a speed of 8/3 feet per second

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