Solve this to mathematical equation to me

  x = 3/2

Step-by-step explanation:

I like to write the equation in the form f(x) = 0 and use a graphing calculator to solve it. Here, that can be achieved by subtracting the right side from both sides:

  9^x -2^(x +1/2) -2^(x +7/2) +3^(2x -1) = 0

See the attachment for the graphical solution.

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This sort of equation is solved by rearranging it to the form ...

  a^x = b

Then the solution is ...

  x = log(b)/log(a)

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Separating powers of 2 and powers of 3, we get ...

  9^x +3^(2x-1) = 2^(x +7/2) +2^(x +1/2)

  3^(2x) +3^(2x)·3^(-1) = 2^x(2^(7/2) +2^(1/2)) . . . . use 9=3^2; factor out 2^x

  3^(2x)·(4/3) = 2^x(9√2) . . . . . simplify

  (9/2)^x = (9√2)·(3/4) . . . . . . . multiply by (3/4)2^-x

Taking the log of both sides, and using 9=3^2, we get ...

  x·(2log(3) -log(2)) = (2log(3) +(1/2)log(2) +log(3) -2log(2) = 3log(3) -(3/2)log(2)

Dividing by the coefficient of x gives ...

  x = (3log(3) -(3/2)log(2))/(2log(3) -log(2))

  x = 3/2

21,600π

Step-by-step explanation:

The children move in a circular path.  The circumference of that circle is:

C = 2πr

C = 2π (15 ft)

C = 30π ft

The children travel 12 circumferences in 1 minute, so their velocity is:

v = d / t

v = (12 × 30π ft) / (1 min)

v = 360π ft/min

Converting to ft/hr:

v = 360π ft/min × (60 min / hr)

v = 21600π ft/hr