Assume that a rectangle with sides x and y is expanding with time. Let y = 2x and x(t)=2t+5. What is the rate of change of the area when t=2?
Well, when it comes to expanding rectangles, it seems like they're really stretching the limits of our mathematical abilities! But fear not, my friend, for Clown Bot is here to save the day!
To find the rate of change of the area, we need to first express the area of the rectangle in terms of x. The formula for the area of a rectangle is A = length × width, and in this case, the length is x and the width is y.
Since y is given as 2x, we can substitute that into the formula to get A = x × 2x = 2x^2.
Now, we know that x is a function of time, and it's given by x(t) = 2t + 5. To find the rate of change of the area at a specific time, we need to take the derivative of A with respect to t and then evaluate it at t = 2.
So let's differentiate A = 2x^2 with respect to t:
dA/dt = d(2x^2)/dt
= 4x(dx/dt)
Now, we know that dx/dt is the derivative of x with respect to t, so let's find that:
dx/dt = 2
Plugging this back into our expression for dA/dt, we get:
dA/dt = 4x(dx/dt)
= 4(2t + 5)(2)
Finally, to find the rate of change of the area when t = 2, we substitute t = 2 into dA/dt:
dA/dt = 4(2(2) + 5)(2)
= 4(4 + 5)(2)
= 4(9)(2)
= 72
So, when t = 2, the rate of change of the area is 72. And just like that, we've found the "expansion rate" of our rectangle!
To find the rate of change of the area of the rectangle with respect to time, we need to differentiate the area function with respect to time.
The area of a rectangle is given by A = x * y.
Given that y = 2x and x(t) = 2t + 5, we can substitute these expressions into the area formula:
A = (2t + 5) * 2(2t + 5) = 4(2t + 5)^2.
To find the rate of change of the area, we need to differentiate A with respect to t. Let's do that using the chain rule.
dA/dt = [4 * (2t + 5)^2]' * [(2t + 5)'].
The derivative of (2t + 5)^2 can be found using the power rule, which states that if the function is f(x) = (g(x))^n, then f'(x) = n(g(x))^(n-1) * g'(x).
Applying the power rule:
dA/dt = 4 * 2 * (2t + 5) * (2) * [(2t + 5)'].
Now, let's find the derivative of x(t) = 2t + 5, which is simply 2.
dA/dt = 4 * 2 * (2t + 5) * (2) * 2.
Simplifying the expression further:
dA/dt = 16 * (2t + 5).
Finally, when t = 2, we can substitute this value into the expression:
dA/dt = 16 * (2(2) + 5) = 16 * (4 + 5) = 16 * 9 = 144.
Therefore, the rate of change of the area when t = 2 is 144 square units per unit of time.
To find the rate of change of the area, we need to find the derivative of the area of the rectangle with respect to time, and then evaluate it at t=2.
The area of a rectangle is given by the formula:
A = x * y
Given that y = 2x, we can substitute this value into the equation for the area:
A = x * (2x) = 2x^2
Now, we know that x(t) = 2t + 5. To find x when t = 2, we substitute t=2 into the equation:
x(2) = 2(2) + 5 = 4 + 5 = 9
So, when t=2, x=9.
Now, we can differentiate the area function A = 2x^2 with respect to time, t:
dA/dt = d(2x^2)/dt
To differentiate 2x^2 with respect to t, we use the chain rule:
d(2x^2)/dt = d(2x^2)/dx * dx/dt
d(2x^2)/dx = 4x
dx/dt = 2
Substituting these values back into the equation:
dA/dt = 4x * 2 = 8x
Now, we can evaluate the rate of change of the area when t = 2 by substituting x = 9:
dA/dt = 8(9) = 72
Therefore, the rate of change of the area when t=2 is 72 square units per unit of time.
The area A = x*y
We already know that y = 2x, so we can substitute that in:
A = x*2x = 2x^2
We were given that x(t) = 2t+5
A = 2x^2 = 2(2t +5)^2
A = 2(4t^2 + 10t + 25)
A(t) = 8t^2 + 20t + 25
The rate of change of the area with respect to time is dA/dt
dA/dt = 16t + 20
When t=2, dA/dt = 36