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egoroff_w [7]
2 years ago
7

The temperature in degrees Celsius on the surface of a metal plate is T(x, y) = 19 − 4x2 − y2 where x and y are measured in cent

imeters. Estimate the average temperature when x varies between 0 and 2 centimeters and y varies between 0 and 4 centimeters. °C
Mathematics
1 answer:
Inessa [10]2 years ago
6 0

Answer:

Average value of temperature will be 8.335^{\circ}C

Step-by-step explanation:

We have given the temperature in degree Celsius T(x,y)=19-4x^2-y^2

It is given that x varies between 0 to 2

So 0\leq x\leq 2

And y varies between 0 and 4

So 0\leq y\leq 4

So area between the region A = 4×2 = 8cm^2

Now average temperature is given by

Average\ value=\frac{1}{A}\int \int T(x,y)dA

Average\ value=\frac{1}{8}\int \int (19-4x^2-y^2)dxdy

=\frac{1}{8}\int \int (19x-4\frac{x^3}{3}-y^2x)dy

As limit of x is 0 to 2

=\frac{1}{8}\int  (19\times 2-4\times \frac{2^3}{3}-y^2\times 2)-(19\times 0-4\times \frac{0^3}{3}-y^2\times 0))dy=\frac{1}{8}\int(38-\frac{32}{3}-2y^2)dy

==\frac{1}{8}(38y-\frac{32}{3}y-\frac{2y^3}{3})

As limit of y is 0 to 4

So Average\ value=\frac{1}{8}(38\times 4-\frac{16}{3}\times 4-\frac{2\times 4^3}{3})-0=8.335^{\circ}C

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