Final answer:
To find the numbers b such that the average value of 3(10x-6x²) is b, integrate the expression and divide by the range of x values. The average value is b = 15.
Step-by-step explanation:
To find all numbers b such that the average value of 3(10x-6x²) is b, we need to first find the average value of the expression 3(10x-6x²). The average value is calculated by integrating the expression and dividing it by the range of x values.
Integrating the expression 3(10x-6x²) gives us 150x² - 30x³. To find the range of x values, we need to solve for when the expression equals zero. Setting 150x² - 30x³ = 0, we can factor out an x² to get x²(150 - 30x) = 0. This equation has two solutions: x = 0 and x = 5.
The range of x values is from 0 to 5. Now, we can calculate the average value by integrating 3(10x-6x²) from x = 0 to x = 5 and dividing it by the range of x values. Evaluating the integral and dividing by 5, we get the average value as b = 15.