A rectangle initially has dimensions 4cm by 6cm. All sides begin increasing in length at a rate of 5cm/s. At what rate is the area of the rectangle increasing after 15 seconds

area = x y

x = 4 + 5 t
y = 6 + 5 t
A = (4+5t)(6+5t) = 24 + 50 t + 25 t^2
dA/dt = 50 + 50 t
now try t = 15

original area = 4*6 = 24

4 + 5*15 =

You are welcome.

To find the rate at which the area of the rectangle is increasing after 15 seconds, we can use the formula for the rate of change of the area. The area of a rectangle is given by the formula A = length x width.

Let's perform the calculations step by step:

1. Write down the formulas:
- A = length x width (equation for the area of a rectangle)
- A' = (dA/dt) (equation for the rate of change of the area)

2. Determine the given information:
- Initial dimensions: length = 4 cm, width = 6 cm
- Rate of change of the sides: 5 cm/s

3. Express the area as a function of time:
- Since both sides are increasing at a constant rate, the dimensions of the rectangle after time t can be expressed as:
- length(t) = 4 + 5t (length is increasing at a rate of 5 cm/s)
- width(t) = 6 + 5t (width is also increasing at a rate of 5 cm/s)
- Plug in these expressions into the formula for the area:
- A(t) = (4 + 5t)(6 + 5t)

4. Differentiate the area function with respect to time:
- A'(t) = (dA/dt) = d/dt [(4 + 5t)(6 + 5t)]

5. Expand and simplify the expression for A'(t):
- A'(t) = (5t + 4)(5t + 6) + (4 + 5t)(5) = 25t^2 + 40t + 24 + 20 + 25t = 25t^2 + 50t + 44

6. Substitute t = 15 seconds into the expression for A'(t):
- A'(15) = 25(15)^2 + 50(15) + 44
= 25(225) + 750 + 44
= 5,625 + 750 + 44
= 6,419

Therefore, the rate at which the area of the rectangle is increasing after 15 seconds is 6,419 cm^2/s.

Thank you so much!