Question

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.

Answer 1

Answer:

The variation rate is 5.42 10⁻⁵ cm²/ºC

Explanation:

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

$\frac{dL}{dT}$ = 1.9 10⁻⁵ cm/ºC

$\frac{dW}{dT}$ = 8.5 10⁻⁶ cm/ºC

W = 1.6  cm

L= 2.8 cm

 Substituting the values ​​and calculating

$\frac{da}{dT}$ = 1.6 1.9 10⁻⁵ + 2.8 8.5 10⁻⁶

$\frac{da}{dT}$ = 3.04 10⁻⁵ + 2.38 10⁻⁵

$\frac{da}{dTy}$=  5.42 10⁻⁵ cm²/ºC

The variation rate is 5.42 10⁻⁵ cm²/ºC