Question

15. The temperature on an unevenly heated metal plate positioned in the first quadrant of the xy-plane is given by 25xy + 25 C(x, y) = 7 (x - 1)2 + (y - 1)2 +1° Assume that temperature is measured in degrees Celsius and that x and y are each measured in inches. (Note: At no point in the following questions should you expand the denominator of C(x,y).) a. Determine doce (2.x) and |(2,3) b. If an ant is on the metal plate, standing at the point (2,3), and starts walking in the direction parallel to the positive y axis, at what rate will the temperature the ant is experiencing change? Explain, and include appropriate units. C. If an ant is walking along the line y = 3 in the positive x direction, at what instantaneous rate will the temperature the ant is experiencing change when the ant passes the point (1, 3)? Now suppose the ant is stationed at the point (6,3) and walks in a straight line towards the point (2,0). Determine the average rate of change in temperature (per unit distance traveled) the ant encounters in moving between these two points. Explain your reasoning carefully. What are the units on your answer?

PLEASE DO LETTER d.)

PLEASE DO LETTER f.)

The plane from e.) is 4(x-2)+6(y-1)+(z-1)=0 or 4x+6y+z=15

Answer 1

Q15) Ans - We have given that craiy2= 2524+25 (x-1)² + (Y-172+1 @ claim - Determine on the car and one I can Differentiate equation wirrt a we get dc - [CX-102714-12+1], 254 – [254Y+75]. 264-1) [ (x-17²+ (4-17²4 17² = [x²2x+1 +4224+1+1]asy-a [2502Y - 25ay +250-25 [(x-17²+14-172+17² [x² + y²_2x-2y+3] 254 - 2x25 [X²Y_ay+a-1] [02-172714-172+172 25024 +254 ²5opy-504²+754-50x²y + 5014 - 50450 [ca-11²+14-12 2+1] 2 - 2543_25x2y -504²_500+754 +50 [ca-122714-17²+1]2 ad Similarly differentiate equ ® w.rit. y wegeto - [(x-112+(4-17² +1.250 [25x4+25] .204-1) [ (x-172+ (4-1727172 - [ 2² +42-24-24 +3] . 250 - 2 [25x4² 253 Y+254-25] [ (21-17² + 14-17²+17² 25x² +25242 50x2 Sony +150 - 50ny?+ 50xy-soy+so [ (1-17² + 14-172+17²

Tac - 25x² - 25x4²_50d? - soy + 150+50 toy [ (2-12²+ (4-12 ²+1]2 - If an ant is on the metal plate standing at the poin (213) and start walking in the direction parallel to the positive yani's Now we have to find the rate will be temperature the ant is experiencing change. 1. l. oclo ay1 (213) ac = 25x3 25xy? - 50x12504 +752750 - [(21-123+ (4-12 +1] 2 01 T(23) = R51838 – 25x24ca72 -50X6232 50x3 475x24 50 - [ (2-1)² + (3 - 12² + 17² 200- 450-200-150 + 150+50 [ +4+)32 - - 400 36 I say I c22) * 11:14 (213) - If an ant is walking x direction, it passes along the line y=3 in positive point (113) i.e. find acl /ch3) 2543 - 25034 -50y? Sox+754 +50 [(x-1)²+(4-1) ²+1] 2 x **13)3 25x4112x3-50X(37-50X1+75x3 +50 [ (8-12²+ 8-11² 1] ² 2011 (13) 675-75-450-50+225150 [o+2²+ 132

эс 1 ox1c113) 375 25 o 15 ) Instantaneous rate will the temperature is is Q16] Ans. We have given that Fly) = 8-x* - 343 - (x) @ claim - Determine facany) ffy (69). Differentiate egn oft Wiritia we get, I fala142= 8-22 7. P ra similarly differentiate eam & worty we get Tfy caly)=-641 6 Parametric equation & for tangent line to trace fraud at 2=2. fray) = 8.22-3y² = 0 our 342=8-22 dy 2 - 8-12 At a=2 4 =/8- (272 - o Parametric eq R for tangent line to frale +624 fcally) - 8-21²-3Y²=0 22 8-342 n = √ 8-342 At y = 1 x V 8-3 =15