The altitude of a triangle is increasing at a rate of 2.500 centimeters/minute while the area of the triangle is increasing at a rate of 1.500 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 10.500 centimeters and the area is 93.000 square centimeters?
Given
A=bh/2, b=2*93/10.5=17.714
differentiate with respect to t (by the product rule on the right hand side).
The only unknown left is the rate of change of the base.
Hint: it is negative.
let the area be A
let the base be x
let the height be y
given: at a time of t minutes,
dA/dt = 1.5 cm^2/min
dy/dt = 2.5 cm^2/min
find:
dx/dt when A = 93 and y = 10.5
A = xy/2 or
2A = xy (equ#1)
differentiate implicitly with respect to t, using the product rule
2dA/dt =x(dy/dt) + y(dx/dt) (equ#2)
we know A=93 when y = 10.5, so x = 17.714
sub into equ#2
2(1.5) = 17.714(2.5) + 10.5dx/dt
solve for dx/dt
My answer is -3.9319, but it's wrong.
I have -3.932 too.
Can you check the numbers in the question?
I copied & pasted this from my online homework. It said the answer is wrong.
I also had -3.932
What answer did the book have?
I use an online homework program called WebWork. It just tells you if your answer is correct or not.
Try -3.93, -3.932.
The answer is -3.9300. Thanks!
To find the rate at which the base of the triangle is changing, we can use the formula for the area of a triangle:
Area = (1/2) * base * altitude.
Differentiate both sides of the formula with respect to time:
d(Area)/dt = (1/2) * (d(base)/dt) * altitude + (1/2) * base * (d(altitude)/dt).
Given that the altitude is increasing at a rate of 2.500 centimeters/minute and the area is increasing at a rate of 1.500 square centimeters/minute, we can substitute these values into the equation:
1.500 = (1/2) * (d(base)/dt) * 10.500 + (1/2) * base * 2.500.
Now we can solve for the rate at which the base is changing, which is d(base)/dt.
Rearranging the equation, we have:
1.500 = 5.250 + 1.250 * base.
Subtracting 5.250 from both sides gives:
1.500 - 5.250 = 1.250 * base.
-3.750 = 1.250 * base.
Dividing both sides by 1.250, we find:
base = -3.750 / 1.250.
Therefore, the base of the triangle is changing at a rate of -3 centimeters/minute when the altitude is 10.500 centimeters and the area is 93.000 square centimeters.