Final answer:
To find the two numbers, we'll use the formulas for geometric mean and harmonic mean. The geometric mean is found by taking the square root of the product of the two numbers, and the harmonic mean is found by taking the reciprocal of the average of the reciprocals of the two numbers. We are given that the geometric mean is 4 and the harmonic mean is 3.2. Solving the equations, we find that the two numbers are 20 and 0.8.
Step-by-step explanation:
To find the two numbers, we'll use the formulas for geometric mean and harmonic mean. The formula for the geometric mean is: GM = sqrt(a * b) where GM is the geometric mean and a and b are the two numbers. Similarly, the formula for the harmonic mean is: HM = 2 / (1/a + 1/b) where HM is the harmonic mean. We are given that GM = 4 and HM = 3.2. Plugging these values into the formulas, we get:
4 = sqrt(a * b)
3.2 = 2 / (1/a + 1/b)
Squaring both sides of the first equation, we get:
16 = a * b
Multiplying both sides of the second equation by a * b, we get:
3.2 * (a * b) = 2 * (b + a)
Substituting the value of (a * b) from the first equation into the second equation, we get:
3.2 * (16) = 2 * (b + a)
Simplifying this equation, we get:
51.2 = 2 * (b + a)
Dividing both sides by 2, we get:
(b + a) = 25.6
Since we have two equations (16 = a * b and (b + a) = 25.6), we can solve for a and b by substituting the value of (b + a) from the second equation into the first equation:
16 = a * (25.6 - a)
Expanding the equation, we get:
16 = 25.6 * a - a^2
Rearranging the equation to form a quadratic, we get:
a^2 - 25.6 * a + 16 = 0
Using the quadratic formula to solve for a, we get two solutions:
a = 20 or a = 0.8
Substituting these values back into the equation (16 = a * b), we can solve for b:
b = 16 / a
For the first solution, a = 20, we get:
b = 16 / 20
b = 0.8
So the numbers are a = 20 and b = 0.8.