The altitude of a triangle is increasing at a rate of 1000 centimeters/minute while the area of the triangle is increasing at a rate of 1500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 7000 centimeters and the area is 89000 square centimeters?
I have tried to answer this in so many ways but couldn't get right answer ... Help please !
Let the base be x cm and the height be y cm
Area = (1/2)xy cm^2
dArea/dt = (1/2)(x dy/dt + y dx/dt) ------- #1
given: dy/dt = 1000 cm/min
dArea/dt = 1500 cm^2/min
find dx/dt when y = 7000 cm and Area = 89000 cm^2
...........
when y = 7000 and area = 89000
(1/2)(x)(7000) = 89000
x = 178/7
which leaves dx/dt as the only missing part of #1
1500 = (1/2)( (178/7)(1000) + 7000(dx/dt) )
3000 = 178000/7 + 7000 dx/dt
dx/dt = -157000/49000 cm/min
= - 157/49 cm/min
= appr -3.2 cm/min
At that instant, the base is decreasing at a rate of appr 3.2 cm/min
To find the rate at which the base of the triangle is changing, we can use the relationship between the area and the base of a triangle.
Let's denote the altitude of the triangle as h, the base as b, and the area as A.
We are given that the altitude is increasing at a rate of 1000 centimeters/minute (dh/dt = 1000 cm/min) and the area is increasing at a rate of 1500 square centimeters/minute (dA/dt = 1500 cm²/min).
We want to find db/dt, the rate at which the base is changing when the altitude is 7000 cm and the area is 89000 cm².
We can start by using the formula for the area of a triangle:
A = 0.5 * base * altitude
Taking the derivative with respect to time (t) on both sides, we get:
dA/dt = 0.5 * (db/dt * h + b * dh/dt)
Now, let's plug in the given values into this equation:
89000 = 0.5 * (db/dt * 7000 + b * 1000)
Now, we have an equation with two variables (db/dt and b). To solve for db/dt at a specific point (when the altitude is 7000 cm and the area is 89000 cm²), we need another equation.
To find this additional equation, we can use the fact that the area (A) of the triangle is also related to the altitude (h) and the base (b) using the formula:
A = 0.5 * base * altitude
Plugging in the given values at the specific point, we get:
89000 = 0.5 * b * 7000
Now we have a second equation, and we can solve the system of equations to find the value of b.
89000 = 0.5 * b * 7000
89000 = 3500b
Dividing both sides by 3500, we get:
b = 89000 / 3500
b = 25.43 cm
Now, we can substitute this value for b back into the first equation:
89000 = 0.5 * (db/dt * 7000 + 25.43 * 1000)
Simplifying, we get:
89000 = 3500 * db/dt + 25430
Rearranging, we get:
3500 * db/dt = 89000 - 25430
db/dt = (89000 - 25430) / 3500
Evaluating this expression, we get:
db/dt = 60 cm/min
Therefore, the rate at which the base of the triangle is changing when the altitude is 7000 cm and the area is 89000 cm² is 60 cm/min.