Find the rate of change of area in respect to temperature of a rectangle.

Its a steel plate that expands when heated. I have to find that rate of change when width is 1.7cm and length is 2.9. dl/dt=1.0 x 10^-5 (this is an exponent...) and dw/dt=8.0 x 10^-6

A = l* w

dA/dt = l*dw/dt + w*dl/dt

Plug in the values and calculate

L+W

To find the rate of change of the area with respect to temperature of a rectangle, we can use the formula:

Rate of change of area with respect to temperature (dA/dt) = (dA/dl) * (dl/dt) + (dA/dw) * (dw/dt)

Given that dl/dt = 1.0 x 10^-5 and dw/dt = 8.0 x 10^-6, we need to find (dA/dl) and (dA/dw) first.

The formula for the area of a rectangle is A = l * w, where A represents the area, l represents the length, and w represents the width.

Differentiating the area equation with respect to the length, we get:

(dA/dl) = (d/dl) (l * w)
= (d/dl) (l) * w (constant w)
= w

Similarly, differentiating the area equation with respect to the width, we get:

(dA/dw) = (d/dw) (l * w)
= l

Substituting the values of (dA/dl) = w = 1.7 cm and (dA/dw) = l = 2.9 cm into the rate of change formula, we have:

Rate of change of area with respect to temperature (dA/dt) = (1.7 cm * 1.0 x 10^-5) + (2.9 cm * 8.0 x 10^-6)
= 1.7 x 10^-5 cm (10^-5 + (2.32 x 10^-5))

Calculating the result, we have:

Rate of change of area with respect to temperature (dA/dt) = 1.7 x 10^-5 cm (3.32 x 10^-5)
= 5.644 x 10^-10 cm²/s

To find the rate of change of the area of a rectangle with respect to temperature, we need to use the concept of partial derivatives. The area of a rectangle is given by the formula A = length * width.

In this case, we are dealing with the temperature dependence of the rectangle's dimensions. Let's denote the length as L, width as W, and temperature as T.

Given the values:
Length (L) = 2.9 cm,
Width (W) = 1.7 cm,
dl/dt = 1.0 x 10^-5 cm/°C,
dw/dt = 8.0 x 10^-6 cm/°C.

We want to find dA/dT, the rate of change of area (A) with respect to temperature (T).

Using the product rule of differentiation, we have:

dA/dT = (dA/dL) * (dL/dT) + (dA/dW) * (dW/dT)

First, let's find the partial derivatives (dA/dL) and (dA/dW):

(dA/dL) = W (since W is constant with respect to L)
(dA/dW) = L (since L is constant with respect to W)

Now, let's find the partial derivatives (dL/dT) and (dW/dT):

(dL/dT) = dl/dt
(dW/dT) = dw/dt

Substituting the given values into the formula, we get:

dA/dT = (W) * (dl/dt) + (L) * (dw/dt)
= (1.7 cm) * (1.0 x 10^-5 cm/°C) + (2.9 cm) * (8.0 x 10^-6 cm/°C)

Calculating this expression gives you the rate of change of the area with respect to temperature.