Question

An unevenly heated metal plate has temperature T(x,y) in degrees Celsius at a point (x,y). If T(2,1) = 107, Tx(2,1) = 9, and Ty(2,1) = −8, estimate the temperature at the point (2.03,0.95).

Answer 1

Using Taylor's approximation, the estimated temperature at the point (2.03,0.95) given the temperature and gradients at (2,1) is approximately 107.67°C.

Step-by-step explanation:

The student has asked about calculating the temperature at a point (2.03,0.95) on a metal plate with a known temperature and temperature gradients at the point (2,1). Using Taylor's approximation, we can estimate the temperature at (2.03,0.95) knowing that T(2,1) = 107, Tx(2,1) = 9, and Ty(2,1) = −8.

The change in x is 0.03 and the change in y is −0.05. Applying the linear approximation, the estimated temperature T(x,y) at point (2.03,0.95) can be approximated by:

T(2+0.03,1–0.05) ≈ T(2,1) + Tx(2,1)*Δx + Ty(2,1)*Δy

T(2.03,0.95) ≈ 107 + 9*(0.03) + (−8)*(−0.05)

T(2.03,0.95) ≈ 107 + 0.27 + 0.4

T(2.03,0.95) ≈ 107.67

Therefore, the estimated temperature at the point (2.03,0.95) is approximately 107.67°C.

Answer 2

Answer: 107.67°C

Step-by-step explanation:

I guess that we could do a Taylor expansion around the point (2, 1)

Remember that a Taylor expansion around the point (a, b) is:

`T(x,y) = T(a, b) + (dT(a,b))/(dx)(x - a) + (dT(a,b))/(dy)(y - b) + .....`

Where the latter terms need higher orders of the derivates of T, so we can not find them, regardless of that, this expansion will be accurate near the point (a, b),

Then, using this, we can write our expanssion as:

T(x,y) = 107 + 9*(x - 2) - 8*(y - 1)

Now we evaluate this in x = 2.03 and y = 0.95

T(2.03, 0.95) = 107 + 9(2.03 - 2) - 8*(0.95 - 1) = 107.67

Then a good estimation of the temperature at the point (2.03,0.95) is 107.67°C