SOLUTION: Please help me solve this equation: Consider the function {{{ f(x)= root(3,x) }}}, {{{ 1<=x<=8 }}}. a) Find the average value of f on the interval [1, 8]. b) Find c such tha

Algebra ->  Test -> SOLUTION: Please help me solve this equation: Consider the function {{{ f(x)= root(3,x) }}}, {{{ 1<=x<=8 }}}. a) Find the average value of f on the interval [1, 8]. b) Find c such tha      Log On


   

Question 350476: Please help me solve this equation: Consider the function +f%28x%29=+root%283%2Cx%29+, +1%3C=x%3C=8+.
a) Find the average value of f on the interval [1, 8].
b) Find c such that fave. = f (c).
c) Sketch the graph of f and a rectangle whose area is the same as the area under
the graph of f .
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me solve this equation: Consider the function +f%28x%29=+root%283%2Cx%29+, +1%3C=x%3C=8+.
a) Find the average value of f on the interval [1, 8].
For the average value on this interval you will use:
%28int%28root%283%2C+x%29%2C+dx%2C+1%2C+8%29%29%2F%288-1%29
Writing the radcial in fractional exponent form will probably help with finding the integral:
%28int%28x%5E%281%2F3%29%2C+dx%2C+1%2C+8%29%29%2F%288-1%29
The indefinite integral of x%5E%281%2F3%29dx is x%5E%284%2F3%29%2F%284%2F3%29+%2B+C+=+%283x%5E%284%2F3%29%29%2F4+%2B+C. The definite integral, from 1 to 8, will be:
%283%288%29%5E%284%2F3%29%29%2F4+-+%283%281%29%5E%284%2F3%29%29%2F4
8%5E%284%2F3%29+=+8%5E%28%281%2F3%29%2A4%29+=+%288%5E%281%2F3%29%29%5E4+=+2%5E4+=+16. So now we have
%283%2816%29%29%2F4+-+3%2F4
48%2F4+-+3%2F4
45%2F4
Now the average value is:
%2845%2F4%29%2F7+=+45%2F28

b) Find c such that fave. = f (c).
So we solve:
f%28c%29+=+45%2F28
root%283%2C+c%29+=+45%2F28
Cube both sides:
c+=+91125%2F21952

c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
The graph of f (without the limited domain) looks like:
graph%28400%2C+400%2C+-1%2C+11%2C+-1%2C+5%2C+x%5E%281%2F3%29%29
Note:
  • Truncate this graph so that only the part between x = 1 and x = 8 shows.
  • I think the intention is for the graph of the rectangle to be on the same graph. Specifically, the bottom of the rectangle should be the on the x axis between 1 and 8 and the sides will be 45/28 in height extending up from the x axis (at x = 1 and x = 8). If done well, the top of the rectangle should intersect the graph of f at x = c = 91125/21952 (which is approximately 4.15)