A15 foot ladder is leaning against a wall of a house. the base of the ladder is pulled away from the wall at a rate of 3 ft/sec. find the rate at which the area of the triangle is changing when the base of the ladder is 9 ft from the wall.

7.875 ft/s

Explanation:

L = 15 ft

dx/dt = 3 ft/s

x = 9 ft

Let the top of ladder is coming down with the rate of dy/dt.

use Pythagorean theorem

L^2 = x^2 + y^2     (1)

Differentiate both sides with respect to t

0 = 2 x dx/dt + 2y dy/dt

x dx/dt = - y dy/dt

When, x = 9 ft then y = ? . Put this in equation (1)

15^2 = 9^2 + y^2

225 - 81 = y^2

y  = 12 ft

So

dy/dt = - x (dx/dt) / y = - 9 (3) / 12 = - 9/4 ft/s

Let A be the area of the triangle

A = 1/2 (base)(height)

A = 1/2 (x y)

Differentiate both sides with respect to t

dA/dt = 0.5 (y dx/dt + x dy/dt)

dA/dt = 0.5 [ 12 (3) - 9 (9/4)]

dA/dt = 0.5(36 - 81 /4) = 31.5 / 4 = 7.875 ft/s

yup its correct its: c) b b

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.