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Chapter 9: Problem 4

Find two positive numbers satisfying the given requirements. The sum of the first and twice the second is 100 and the product is a maximum.

Short Answer

Expert verified
The two positive numbers satisfying the given requirements are 50 and 25.

Step by step solution

01

Express one variable using the constraint equation

Let's denote the two numbers as \(x\) and \(y\). From the problem, we have \(x + 2y = 100\). This equation can be rearranged to express \(x\) in terms of \(y\). So, we can express \(x\) as \(x = 100 - 2y\).
02

Formulate the objective function

We are asked to maximize the product of the two numbers. The product \(xy\) can be expressed as a function of \(y\) using the substitution \(x = 100 - 2y\). After this substitution we obtain the product function \(P(y)=y(100-2y)= 100y-2y^2\).
03

Differentiate the product function

To set up for finding the maximum value, we need to find the derivative of the resulting function to set up for finding its critical points. The derivative of our function \(P(y)\) is \(P'(y)=100-4y\).
04

Compute the maximum value

We set the derivative equal to zero and solve for \(y\): \(100 - 4y = 0\). Solving for \(y\) we find \(y = 25\).
05

Determine the other variable

Substitute \(y = 25\) into the equation from the first step (\(x = 100 - 2y\)) to get \(x = 100 - 2*25 = 50\).
06

Verification

To confirm if the obtained values are indeed giving the maximum product, one can substitute near values to \(y\) into the function \(P(y)\) and confirm that for no other \(y\) value, \(P(y)\) yields a higher value.

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Most popular questions from this chapter

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x+\frac{32}{x^{2}}\)

Find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at \(f(x+\Delta x)\) and \(y(x+\Delta x)\) for \(\Delta x=-0.01\) and \(0.01\). \(f(x)=\frac{x}{x^{2}+1}\) \((0,0)\)

Compare the values of \(d y\) and \(\Delta y\). \(y=0.5 x^{3} \quad x=2 \quad \Delta x=d x=0.1\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{-1 / 3}\)

The demand function for a product is modeled by \(p=75-0.25 x\) (a) If \(x\) changes from 7 to 8 , what is the corresponding change in \(p\) ? Compare the values of \(\Delta p\) and \(d p\). (b) Repeat part (a) when \(x\) changes from 70 to 71 units.
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