Answer:
The temperature is increasing at the rate of 6.5 degree celcius per second.
Step-by-step explanation:
We are given the following information:
Temperature of a point is given by T(x,y)
After t seconds a bug reaches a point [tex]x = \sqrt{2 + t}, y = 3 + \frac{1}{2}t[/tex]
At t = 2, x = 2, y = 4
The temperature function satisfies:
[tex]T_x(2,4) = 8\\T_y(2,4) = 9[/tex]
We have to find:
[tex]\displaystyle\frac{dT}{dt} = T_t(x(t), y(t))\\\\= T_x(x(t),y(t)).x'(t) + T_y(x(t),y(t)).y'(t) \\\\= T_x(x(t),y(t)).\displaystyle\frac{1}{2\sqrt{(t+2)}} + T_y(x(t),y(t)).\displaystyle\frac{1}{2} \\\\\displaystyle\frac{dT(x,y)}{dt}_{at~t=2} = \displaystyle\frac{dT(2,4)}{dt}_{at~t=2} = 8.\displaystyle\frac{1}{4} + 9.\displaystyle\frac{1}{2} = 2 + 4.5 = 6.5[/tex]
Hence, the temperature is increasing at the rate of 6.5 degree celcius per second.
