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

Answer:

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

Step-by-step 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,

$\frac{da}{dT}$ = $\frac{d(LW)}{dT}$

$\frac{da}{dT}$ = W $\frac{dL}{dT}$ + L $\frac{dW}{dT}$

The values ​​they give us are

$\frac{dL}{dT}$ = 1.1 × 10⁻⁵ cm/ºC

$\frac{dW}{dT}$ = 8.9 × 10⁻⁶ cm/ºC

W = 1.6  cm

L= 2.6 cm

Substituting the values ​​and calculating

$\frac{da}{dT}$ = (1.6 × 1.1 × 10⁻⁵) + (2.6 × 8.9 × 10⁻⁶)

$\frac{da}{dT}$ = (1.76 × 10⁻⁵) + (2.31 × 10⁻⁵)

$\frac{da}{dT}$ =  4.07 × 10⁻⁵ cm²/ºC

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