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 2.6 cm if d l divided by dt equals 1.1 times 10 Superscript negative 5 Baseline cm divided by degrees Upper C and dw divided by dt equals 8.9 times 10 Superscript negative 6 Baseline cm divided by degrees C. Round to one decimal place.

Answer 1

The variation rate is 4.07 × 10⁻⁵ cm²/ºC

Explanation:

The relationship of the change in length as a function of the temperature, which are given in this problem can be written by the expression for the area of ​​a rectangle

a = L × W

Differentiating both sides,

`(da)/(dT)` =
`(d(LW))/(dT)`

`(da)/(dT)` = W
`(dL)/(dT)` + L
`(dW)/(dT)`

The values ​​they give us are

`(dL)/(dT)` = 1.1 × 10⁻⁵ cm/ºC

`(dW)/(dT)` = 8.9 × 10⁻⁶ cm/ºC

W = 1.6 cm

L= 2.6 cm

Substituting the values ​​and calculating

`(da)/(dT)` = (1.6 × 1.1 × 10⁻⁵) + (2.6 × 8.9 × 10⁻⁶)

`(da)/(dT)` = (1.76 × 10⁻⁵) + (2.31 × 10⁻⁵)

`(da)/(dT)` = 4.07 × 10⁻⁵ cm²/ºC

The variation rate is 4.07 × 10⁻⁵ cm²/ºC