/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 What two positive real numbers w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 11

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

Short Answer

Expert verified
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 = 鈭�50 and y = 鈭�50. The smallest sum is 2鈭�50.

Step by step solution

01

Writing the Product Equation

(Write the given information as an equation, x * y = 50)
02

Express y in terms of x

(From the equation in Step 1, we can express y as y = 50 / x)
03

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))
04

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})
05

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})
06

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}. )

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Even quartics Consider the quartic (fourth-degree) polynomial \(f(x)=x^{4}+b x^{2}+d\) consisting only of even-powered terms. a. Show that the graph of \(f\) is symmetric about the \(y\) -axis. b. Show that if \(b \geq 0\), then \(f\) has one critical point and no inflection points. c. Show that if \(b<0,\) then \(f\) has three critical points and two inflection points. Find the critical points and inflection points, and show that they alternate along the \(x\) -axis. Explain why one critical point is always \(x=0\) d. Prove that the number of distinct real roots of \(f\) depends on the values of the coefficients \(b\) and \(d,\) as shown in the figure. The curve that divides the plane is the parabola \(d=b^{2} / 4\) e. Find the number of real roots when \(b=0\) or \(d=0\) or \(d=b^{2} / 4\)

Suppose you make a deposit of \(S P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\) b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=P e^{n}\)

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\) a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A stone is thrown vertically upward with a velocity of \(30 \mathrm{m} / \mathrm{s}\) from the edge of a cliff \(200 \mathrm{m}\) above a river.

Find the solution of the following initial value problems. $$y^{\prime}(t)=\frac{3}{t}+6 ; y(1)=8$$

Properties of cubics Consider the general cubic polynomial \(f(x)=x^{3}+a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are real numbers. a. Prove that \(f\) has exactly one local maximum and one local minimum provided that \(a^{2}>3 b\) b. Prove that \(f\) has no extreme values if \(a^{2}<3 b\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.