An airplane takes off from an airport at sea level and climbs at the constant rate of 5m/s . The outside air temperature T varies with altitude h according to the law T=15-0.0065h where T is measured in degree Celsius and h in meters above sea level . Find the rate of change of T with respect to time .
dT/dt = -.0065 dh/dt
so plug in you number for dh/dt
To find the rate of change of T with respect to time, we need to use the chain rule of calculus. The chain rule states that if we have a function y = f(g(x)), where y depends on g(x), and g(x) depends on x, then the derivative of y with respect to x is given by dy/dx = (dy/dg) * (dg/dx).
In our case, T depends on h, and h depends on time t. So we have T = f(h(t)) and h = g(t).
Given that T = 15 - 0.0065h, we can rewrite it as T(t) = 15 - 0.0065h(t).
We are also given that h(t) = 5t, since the airplane is climbing at a constant rate of 5m/s.
Using the chain rule, we can find the derivative of T(t) with respect to time t:
dT/dt = (dT/dh) * (dh/dt)
To find dT/dh, we differentiate T(t) with respect to h:
dT/dh = -0.0065
To find dh/dt, we differentiate h(t) with respect to t:
dh/dt = 5
Now we can substitute these values into the chain rule equation:
dT/dt = (dT/dh) * (dh/dt)
= (-0.0065) * (5)
= -0.0325
Therefore, the rate of change of T with respect to time is -0.0325 degrees Celsius per second.
To find the rate of change of T with respect to time, we need to use the chain rule of calculus. The chain rule states that if we have a function T which is a composition of two functions T = f(g(h)), then the derivative of T with respect to time can be calculated using the formula:
dT/dt = (df/dg) * (dg/dh) * (dh/dt)
In this case, we have T = f(g(h)), where f(x) = 15 - 0.0065x, g(h) = h, and h(t) = 5t, where t is the time.
Let's calculate the derivatives step by step:
1. df/dg: To find the derivative of f with respect to g, we differentiate f(x) = 15 - 0.0065x with respect to x. The derivative is df/dg = -0.0065.
2. dg/dh: To find the derivative of g with respect to h, we differentiate g(h) = h with respect to h. The derivative is dg/dh = 1.
3. dh/dt: To find the derivative of h with respect to t, we differentiate h(t) = 5t with respect to t. The derivative is dh/dt = 5.
Now we can substitute these values into the chain rule formula:
dT/dt = (df/dg) * (dg/dh) * (dh/dt)
= (-0.0065) * (1) * (5)
= -0.0325
Therefore, the rate of change of T with respect to time is -0.0325 degrees Celsius per second.
An airplane takes off and climbs at a constant rate of 1400 feet per minute.
Write an equation to model the relationship between the plane’s altitude, a, and the time in minutes, t.