The length of a rectangle is given by 2t + 2 and its height is √t, where t is time in seconds and the dimensions are in centimeters. Find the rate of change of the area with respect to time.
A'(t)=
A = 2 (t+1)(t^.5)
dA/dt = 2 [ (t+1)(.5 t^-.5) + t^.5 (1) ]
= 2 [ .5 t^.5 +.5 t^-.5 + t^.5 ]
= 2 [ 1.5 t^.5 + .5 t^-.5 ]
= 3 sqrt t + 1/sqrt t
= (3 t + 1 )/sqrt t
Well, if the length is given by 2t + 2 and the height is √t, then the formula for the area of the rectangle would be (2t + 2)(√t).
To find the rate of change of the area with respect to time, we can differentiate this formula with respect to t.
Using the product rule, the derivative of the area formula would be:
A'(t) = (2)(√t) + (2t + 2)(1/2√t)
But since we're talking about a rate of change, we can simplify this a bit more:
A'(t) = 2√t + (2t + 2)/2√t
And if we simplify further, we can get:
A'(t) = 2√t + (t + 1)/√t
So, the rate of change of the area with respect to time is 2√t + (t + 1)/√t. Now, I hope that answer squared you away!
To find the rate of change of the area with respect to time, you need to find the derivative of the area function with respect to time.
Given that the length of the rectangle is 2t + 2 and its height is √t, the area (A) of the rectangle can be calculated as:
A(t) = length * height
= (2t + 2)(√t)
To find A'(t), the derivative of A(t) with respect to t, you can use the product rule of differentiation:
A'(t) = (2t + 2)(d/dt(√t)) + (√t)(d/dt(2t + 2))
First, differentiate √t with respect to t. The derivative of √t is 1/(2√t).
Second, differentiate 2t + 2 with respect to t. The derivative of 2t is 2, and the derivative of 2 is 0.
A'(t) = (2t + 2)(1/(2√t)) + (√t)(2)
= (t + 1)/(√t) + 2√t
Simplifying further:
A'(t) = (t + 1)/(√t) + 2√t
Therefore, the rate of change of the area with respect to time is A'(t) = (t + 1)/(√t) + 2√t.
To find the rate of change of the area with respect to time, we first need to find the equation for the area of the rectangle.
The area of a rectangle is calculated by multiplying its length by its height. In this case, the length is given by 2t + 2 and the height is √t.
So, the equation for the area of the rectangle is:
A = (2t + 2) * √t
To find the rate of change of the area with respect to time, we need to differentiate this equation with respect to t.
Using the product rule of differentiation, we differentiate each term separately and then combine them.
First, let's differentiate the term (2t + 2):
d/dt (2t + 2) = 2
Now, let's differentiate the term √t:
d/dt (√t) = (1/2) * t^(-1/2)
Now, let's apply the product rule to find the derivative of the area A:
dA/dt = (2t + 2) * (1/2) * t^(-1/2) + √t * 2
Simplifying this expression, we get:
A'(t) = (t + 1) / √t + 2√t
So, the rate of change of the area with respect to time is A'(t) = (t + 1) / √t + 2√t.