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3 years ago

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Circle A has a center at the origin and a point M located on the circle at (1,0). Circle B has a center at (-2,1) and point N lo

cated on a circle at (3,1).

1 answer:

kvasek [131]3 years ago

5 0

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2 years ago

Simplify the expression and combine like terms. 3(x-2)-2(x+1)3(x−2)−2(x+1)

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2 years ago

A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2

Artemon [7]

Answer:

a) The height decreases at a rate of $\frac{7}{12}$ ft/sec.

b) The area increases at a rate of $\frac{527}{24}$ ft^2/sec

c) The angle is increasing at a rate of $\frac{1}{12}$ rad/sec

Step-by-step explanation:

Attached you will find a sketch of the situation. The ladder forms a triangle of base b and height h with the house. The key to any type of problem is to identify the formula we want to differentiate, by having in mind the rules of differentiation.

a) Using pythagorean theorem, we have that $25^2 = h^2+b^2$ . From here, we have that

$h^2 = 25^2-b^2$

if we differentiate with respecto to t (t is time), by implicit differentiation we get

$2h \frac{dh}{dt} = -2b\frac{db}{dt}$

Then,

$\frac{dh}{dt} = -\frac{b}{h}\frac{db}{dt}$ .

We are told that the base is increasing at a rate of 2 ft/s (that is the value of db/dt). Using the pythagorean theorem, when b = 7, then h = 24. So,

$\frac{dh}{dt} = -\frac{2\cdot 7}{24}= \frac{-7}{12}$

b) The area of the triangle is given by

$A = \frac{1}{2}bh$

By differentiating with respect to t, using the product formula we get

$\frac{dA}{dt} = \frac{1}{2} (\frac{db}{dt}h+b\frac{dh}{dt})$

when b=7, we know that h=24 and dh/dt = -1/12. Then

$\frac{dA}{dt} = \frac{1}{2}(2\cdot 24- 7\frac{7}{12}) = \frac{527}{24}$

c) Based on the drawing, we have that

$\sin(\theta)= \frac{b}{25}$

If we differentiate with respect of t, and recalling that the derivative of sine is cosine, we get

$\cos(\theta)\frac{d\theta}{dt}=\frac{1}{25}\frac{db}{dt}$ or, by replacing the value of db/dt

$\frac{d\theta}{dt}=\frac{2}{25\cos(\theta)}$

when b = 7, we have that h = 24, then $\cos(\theta) = \frac{24}{25}$ , then

$\frac{d\theta}{dt} = \frac{2}{25\frac{24}{25}} = \frac{2}{24} = \frac{1}{12}$

7 0

3 years ago