A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to temperature T when the width is 1.6 cm and the length is 2.8 cm if d l divided by dt equals 1.9 times 10 Superscript negative 5 Baseline cm divided by degrees Upper C and dw divided by dt equals 8.5 times 10 Superscript negative 6 Baseline cm divided by degrees C. Round to one decimal place.

When we have a thermal expansion problem we must have the relationship of the change in length as a function of the temperature, which are given in this problem, so we can write the expression for the area of a rectangle

a = L W

They ask us to find the rate of variation of this area depending on the temperature, so we can derive this expression with respect to the temperature

da / dT = d(LW) / dt

We use the derivative of a product since the two magnitudes change

da / dT = W dL/dT + L dW/dT

The values they give us are

[tex]\frac{dL}{dT}[/tex] = 1.9 10⁻⁵ cm/ºC

[tex]\frac{dW}{dT}[/tex] = 8.5 10⁻⁶ cm/ºC

W = 1.6 cm

L= 2.8 cm

Substituting the values and calculating

[tex]\frac{da}{dT}[/tex] = 1.6 1.9 10⁻⁵ + 2.8 8.5 10⁻⁶

[tex]\frac{da}{dT}[/tex] = 3.04 10⁻⁵ + 2.38 10⁻⁵

[tex]\frac{da}{dTy}[/tex]= 5.42 10⁻⁵ cm²/ºC

The variation rate is 5.42 10⁻⁵ cm²/ºC