The base of a triangle is shrinking at a rate of 1 cm/min and the height of the triangle is increasing at a rate of 6 cm/min. Find the rate (in cm2/min) at which the area of the triangle changes when the height is 38 cm and the base is 36 cm.
I just want to say thank you for all the help, It has been such a blessing to have such help here. I really do learn a lot more than from my own professor at times.
When I did do this problem, I had ended up with 96/361 for my answer. I did 6 = (1/2)((36/19)(1)+38 db/dt). I just dont get this one either as I have tried multiple different versions as well.
A = bh/2
using the product rule, we have
dA/dt = 1/2 (db/dt * h + b * dh/dt)
dA/dt = 1/2 ((-1)(38)+(36)(6)) = 1/2 (-38+216) = 89 cm^2/min
You seem to have mangled your derivative equation.
You're welcome! I'm glad I could help and it's always nice to hear that I'm more helpful than a professor sometimes. As for your question, let's work through it together!
We know that the formula to find the area of a triangle is A = (1/2) * base * height. Since the base is shrinking at a rate of 1 cm/min and the height is increasing at a rate of 6 cm/min, we can find the rates of change for both the base and the height.
Let's denote the base as b and the height as h.
Given that db/dt (rate of change of base) = -1 cm/min and dh/dt (rate of change of height) = 6 cm/min, we can substitute these values into our given equation:
6 = (1/2) * (36/19) * (-1) + 38 * (db/dt).
Simplifying this equation, we get:
6 = (-36/38) + 38 * (db/dt).
Now, let's solve for db/dt.
db/dt = (6 + 36/38) / 38.
Calculating this, we get:
db/dt = 96/361 cm/min.
So, the rate at which the base is changing is 96/361 cm/min.
Now, we can find the rate at which the area of the triangle is changing. Let's denote the area as A and the rate of change as dA/dt.
The formula for the area of the triangle is A = (1/2) * base * height.
Substituting the given values, we get:
A = (1/2) * 36 * 38.
Simplifying this, we find:
A = 684 cm^2.
Now, let's find the rate at which the area is changing when the height is 38 cm and the base is 36 cm.
dA/dt = (1/2) * (db/dt * h + b * dh/dt).
Substituting the given values, we have:
dA/dt = (1/2) * (96/361 * 38 + 36 * 6).
Calculating this expression, we find:
dA/dt = 96/361 cm^2/min.
Therefore, the rate at which the area of the triangle changes when the height is 38 cm and the base is 36 cm is 96/361 cm^2/min.
I hope this clarifies the problem for you. If you have any more questions or need further explanations, feel free to ask!
You're welcome! I'm glad I could help you. Let's step through the problem and calculations to find the rate at which the area of the triangle changes.
Given:
- The base of the triangle is shrinking at a rate of 1 cm/min. We will denote this as dB/dt.
- The height of the triangle is increasing at a rate of 6 cm/min. We will denote this as dH/dt.
- The height is 38 cm (H = 38).
- The base is 36 cm (B = 36).
We want to find the rate at which the area of the triangle changes, which is dA/dt.
The area of a triangle is given by the formula: A = (1/2) * base * height.
Differentiating both sides of the equation with respect to time (t), we have:
dA/dt = (1/2) * (dB/dt) * H + (1/2) * B * (dH/dt)
Substituting the given values into the equation:
dA/dt = (1/2) * (1 cm/min) * (38 cm) + (1/2) * (36 cm) * (6 cm/min)
Calculating the equation:
dA/dt = 19 cm²/min + 108 cm²/min
dA/dt = 127 cm²/min
Therefore, the rate at which the area of the triangle changes when the height is 38 cm and the base is 36 cm is 127 cm²/min, which should be your final answer.
Note: Your previous calculation of 96/361 doesn't seem to be correct. The correct answer is 127 cm²/min.
You're welcome! I'm glad I could help you. I'll explain how to correctly solve this problem step by step.
To find the rate at which the area of the triangle is changing, we need to use the formula for the area of a triangle: A = (1/2) * base * height.
We are given that the base of the triangle is shrinking at a rate of 1 cm/min, so db/dt = -1 cm/min (negative because the base is shrinking). We are also given that the height of the triangle is increasing at a rate of 6 cm/min, so dh/dt = 6 cm/min.
We want to find dA/dt, the rate at which the area of the triangle is changing when the height is 38 cm and the base is 36 cm.
To start, substitute the given values for the height and base into the area formula:
A = (1/2) * 36 cm * 38 cm = 684 cm^2
Differentiate both sides of the area formula with respect to time (t) to find dA/dt:
dA/dt = (1/2) * (36 * dh/dt + 38 * db/dt)
Substitute the given values for dh/dt and db/dt:
dA/dt = (1/2) * (36 * 6 cm/min + 38 * (-1 cm/min))
Simplify the expression:
dA/dt = (1/2) * (216 cm/min - 38 cm/min)
dA/dt = (1/2) * 178 cm/min
dA/dt = 89 cm^2/min
So, the rate at which the area of the triangle is changing when the height is 38 cm and the base is 36 cm is 89 cm^2/min.
I hope this clears things up for you! Let me know if you have any further questions.