An electrical heating element produces heat depending on the resistance of the element and the current passed through it. the heat produced can be given by the formula h = i2r where h is the heat generated, i is the current, and r is the resistance. if the element has a fixed current of 2 amps passing through it and a variable current of x amps, it is able to produce a heat of 10x3 + 80, depending on the variable resistance for different additional values of current x. determine the formula for the variable resistance.

The heat produced by current I is
H = I²R
where
R =  resistance.
According to the formula, heat produced is proportional to the square of the current.

When a current of I = 2 amps is applied, the heat produced is 
H = 10x³ + 80.
This heat includes heat due to a fixed current of 2 amps, and heat due to a variable current of x amps.

Because the heat produced is proportional to the square of the current, write the expressions as
H = (10x)*(x²) + 20*(2²)

The second term on the right is heat due to the fixed current of 2 amps, written as
20*(2²).
Therefore the fixed resistance is R = 20 ohms, and the square of the fixed current is 2².

The first term represents heat due to variable resistance, written as
(10x)*(x²).
Therefore the variable resistance is 10x, and the square of the variable current is x².

The variable current is  10x.