The altitude (i.e., height) of a triangle is increasing at a rate of 2 cm/minute while the area of the triangle is increasing at a rate of 4 square cm/minute. At what rate is the base of the triangle changing when the altitude is 10.5 centimeters and the area is 97 square centimeters?
Area of triangle:
A = (1/2)bh
When A = 97 and h = 10.5,
A = (1/2)bh
97 = (1/2)(b)(10.5)
b = 18.48
Differentiate area with respect to time:
A = (1/2)bh
dA/dt = (1/2)( h*db/dt + b*dh/dt )
Substitute the values:
4 = (1/2)( 10.5*db/dt + 18.48*2)
Solve for db/dt.
hope this helps~ `u`
To find the rate at which the base of the triangle is changing, we can use the relationship between the area, altitude, and base of a triangle.
Let's denote the altitude of the triangle as h (in centimeters) and the base as b (in centimeters). The formula for the area of a triangle is:
Area = (1/2) * base * altitude
Now we have the following information:
- The altitude is increasing at a rate of 2 cm/minute, so dh/dt = 2 cm/minute.
- The area is increasing at a rate of 4 square cm/minute, so dA/dt = 4 square cm/minute.
- The altitude is 10.5 centimeters and the area is 97 square centimeters.
We want to find db/dt, the rate at which the base of the triangle is changing.
To find db/dt, we can differentiate the formula for the area of a triangle with respect to t:
dA/dt = (1/2) * (db/dt * h + b * dh/dt)
Plugging in the given values, we have:
4 = (1/2) * (db/dt * 10.5 + b * 2)
Simplifying the equation, we get:
8 = db/dt * 10.5 + 2b
Since we want to find db/dt at the point when h = 10.5 and A = 97, we can substitute h = 10.5 and A = 97 into the equation:
8 = db/dt * 10.5 + 2b
At this point, we have two unknowns, db/dt and b, and one equation. To solve for db/dt, we need another equation relating b and h. However, from the given information, we don't know the relationship between the base and altitude of the triangle. Without this information, we cannot determine the rate at which the base is changing. Additional information or a specific relationship between the base and altitude is required in order to solve the problem.