Suppose the temperature in degress Fahrenheit of a metal bar of length 2 feet depends on the position x according the function

Question

Suppose the temperature in degress Fahrenheit of a metal bar of length 2 feet depends on the position x according the function

T(x) = 40 + 20x(2-x^4)

Find the average temperature in the bar _______

Is there a point where the actual temperature equals the average temperature?

Answer 1

T_Av = 1/2∫₀²[40 + 20x(2 - x⁴)]dx = 1/2(40x + 20x² - 10x⁶/3)₀² = - 160/3 = - 53.33°F

40 + 20x(2 - x⁴) = - 160/3; 2 + x(2 - x⁴) = - 8/3; 3x⁵- 6x -14 = 0; x = 1.5031137 feet.

At point x = 1.5031137 temperature is - 160/3°F