An unevenly heated metal plate has temperature T(x,y) in degrees Celsius at a point (x, y). If...

Question:

An unevenly heated metal plate has temperature {eq}T(x,y) {/eq} in degrees Celsius at a point {eq}(x, y) {/eq}. If {eq}T(2, 1) =138 {/eq}, {eq}T_x(2, 1) =8 {/eq} and {eq}T_y(2, 1) =-11 {/eq}, estimate the temperature at the point (2.05, 0.95). Please include units in your answer.

Estimating the Temperature Value:

By using the given temperature function value and its partial derivatives values at the point, we have to estimate the given temperature function value, {eq}T(2.05, 0.95) {/eq}.

For estimation, we have to use this formula {eq}\ \displaystyle T(x, y) =T(a, b) + T_{x}(a, b)(x-a)+T_{y}(a, b)(y-b) {/eq}.

Answer and Explanation:

The given values are {eq}\displaystyle T(2, 1) =138 {/eq}, {eq}\displaystyle T_x(2, 1) =8 {/eq} and {eq}\displaystyle T_y(2, 1) =-11 {/eq}.

Estimating the temperature at the point {eq}\displaystyle T (2.05, 0.95) {/eq}:

{eq}\begin{align*} \\ \displaystyle T(x, y) &=T(a, b) + T_{x}(a, b)(x-a)+T_{y}(a, b)(y-b) \\ \displaystyle &=138 + (8)(x-2)+(-11)(y-1) \\ \displaystyle &=138+8x-16-11y+11 \\ \displaystyle &=8x-11y+138-16+11 \\ \displaystyle T(x, y) &=8x-11y+133 \\ \displaystyle T (2.05, 0.95) & \approx T(x, y)=8x-11y+133 \\ \displaystyle T (2.05, 0.95) & \approx 8(2.05)-11(0.95)+133 \\ \displaystyle T (2.05, 0.95) & \approx 138.95 \end{align*} {/eq}

Therefore, the estimated value of the temperature at the point is {eq}\ \displaystyle \mathbf{\color{blue}{ 138.95 }} {/eq} units.


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