Question

Find two positive real numbers whose product is a maximum.

The sum of the first and three times the second is 30

Answer 1

To find two positive real numbers whose product is a maximum, we need to use a mathematical technique called optimization. By solving the given equation and substituting the values back, we can find the two positive real numbers with a maximum product.

Step-by-step explanation:

To find two positive real numbers whose product is a maximum, we need to use a mathematical technique called optimization. Let's define the two numbers as x and y.

Since the sum of the first number and three times the second number is 30, we can write the equation:

x + 3y = 30

To find the maximum product, we need to express the product of x and y in terms of a single variable. We can do this by substituting x = 30 - 3y into the product equation xy = (30 - 3y)y.

Now, we have a quadratic equation: y^2 - 10y + 15. Finding the maximum product corresponds to finding the vertex of this quadratic equation, which occurs when y = 5.

Substituting y = 5 back into the equation x + 3y = 30, we can solve for x: x + 3(5) = 30, x + 15 = 30, x = 15. Therefore, the two positive real numbers with a maximum product are x = 15 and y = 5.

Answer 2

3, 9

Product maximum 12
Sum = 3 + (3 x 9) = 30