Problem 11 - Calculus Early Transcendentals

Question

What two positive real numbers whose product is 50 have the smallest possible sum?

Step-by-Step Solution

Answer

Question: Find the two positive real numbers whose product is 50 and have the smallest possible sum. Also, find the smallest sum. Answer: The two positive real numbers that have a product of 50 and the smallest possible sum are x = √50 and y = √50. The smallest sum is 2√50.

Step 1: Writing the Product Equation

(Write the given information as an equation, x * y = 50)

Step 2: Express y in terms of x

(From the equation in Step 1, we can express y as y = 50 / x)

Step 3: Write the Sum Equation

(Now express the sum of x and y as S(x) = x + y and substitute the expression from Step 2: S(x) = x + (50 / x))

Step 4: Find the Minimum Sum

(To find the minimum sum, we must differentiate S(x) with respect to x: S'(x) = \frac{d}{dx}(x + \frac{50}{x}) = 1 - \frac{50}{x^2}) Now we set S'(x) = 0 and solve for x: 1 - \frac{50}{x^2} = 0 \Rightarrow x^2 = 50 \Rightarrow x = \sqrt{50})

Step 5: Find the Corresponding Value of y

(Now use the value of x found in Step 4 and the equation from Step 2 to find the corresponding value of y: y = \frac{50}{\sqrt{50}} = \sqrt{50})

Step 6: Write the Answer

( The two positive real numbers whose product is 50 and have the smallest possible sum are x = \sqrt{50} and y = \sqrt{50}. The smallest sum is S(\sqrt{50}) = \sqrt{50} + \sqrt{50} = 2\sqrt{50}. )