Solve the following equations for all solutions of x

2sin^2x+3cos-3=0

Remember that sin^2x + cos^2x = 1 so sin^2x = 1 - cos^2x
2sin^2 x + 3cosx - 3 = 0
2(1 - cos^2x) + 3cosx - 3 = 0
2cos^2x - 3cosx + 1 = 0
This is just a quadratic equation in cosx
So let u = cosx
2u^2 - 3u + 1 = 0
This factors as
(2u-1)(u-1) = 0
So 2u = 1 and u = 1 or 2cosx = 1 and cosx = 1
cosx = 1 when x = 0cosx = 1/2 when x = pi/3 and 5pi/3
So x = {0, pi/3, 5pi/3} is the solution set for your equation.

(a) 11.75%

(b) Profit decreases by $5.88 per calculator.

Step-by-step explanation:

(a) The percentage of failures with time is given by the following expression:

Integrating this function from x = 0 to x = 1 year, gives us the percentage of failures in the first year:

11.75% of the calculators will fail within the warranty period.

(b) If the cost of a calculator is $50, and the profit per sale is $25, the average revenue per calculator is $75. Considering no income in failed calculators, the new cost per calculator is:

The effect of warranty replacement on profit is given by the difference in cost per calculator:

Profit decreases by $5.88 per calculator.