Math / Algebra

QuestionAmong all pairs of numbers whose sum is 22 , find a pair whose product is as large as possible. What is the maximum product?

Studdy Solution

STEP 1

1. We are given two numbers, say $x$ and $y$ , such that their sum is 22.
2. We need to maximize the product $x \cdot y$ .
3. The problem can be approached using algebraic manipulation and calculus or by recognizing a pattern.

STEP 2

1. Express one variable in terms of the other.
2. Formulate the product function.
3. Use calculus to find the maximum product.
4. Verify the solution and calculate the maximum product.

STEP 3

Express one variable in terms of the other using the given sum condition:
Let $x$ and $y$ be the two numbers. We know:
$x + y = 22$
Express $y$ in terms of $x$ :
$y = 22 - x$

STEP 4

Formulate the product function $P(x)$ using the expression for $y$ :
The product $P$ is given by:
$P(x) = x \cdot y = x \cdot (22 - x)$
Simplify the expression:
$P(x) = 22x - x^2$

STEP 5

To find the maximum product, we take the derivative of $P(x)$ with respect to $x$ and set it to zero to find critical points:
$P'(x) = \frac{d}{dx}(22x - x^2) = 22 - 2x$
Set the derivative equal to zero:
$22 - 2x = 0$
Solve for $x$ :
$2x = 22$ $x = 11$

STEP 6

Verify that this critical point is a maximum by checking the second derivative:
$P''(x) = \frac{d^2}{dx^2}(22x - x^2) = -2$
Since $P''(x) = -2 < 0$ , the function $P(x)$ is concave down, indicating a maximum at $x = 11$ .

STEP 7

Calculate the maximum product using the value of $x$ found:
Substitute $x = 11$ back into the expression for $y$ :
$y = 22 - x = 22 - 11 = 11$
Calculate the product:
$P(11) = 11 \cdot 11 = 121$
The maximum product is:
$\boxed{121}$

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QuestionAmong all pairs of numbers whose sum is 22 , find a pair whose product is as large as possible. What is the maximum product?

Studdy Solution

STEP 1

1. We are given two numbers, say $x$ and $y$ , such that their sum is 22.
2. We need to maximize the product $x \cdot y$ .
3. The problem can be approached using algebraic manipulation and calculus or by recognizing a pattern.

STEP 2

1. Express one variable in terms of the other.
2. Formulate the product function.
3. Use calculus to find the maximum product.
4. Verify the solution and calculate the maximum product.

STEP 3

Express one variable in terms of the other using the given sum condition:
Let $x$ and $y$ be the two numbers. We know:
$x + y = 22$
Express $y$ in terms of $x$ :
$y = 22 - x$

STEP 4

Formulate the product function $P(x)$ using the expression for $y$ :
The product $P$ is given by:
$P(x) = x \cdot y = x \cdot (22 - x)$
Simplify the expression:
$P(x) = 22x - x^2$

STEP 5

To find the maximum product, we take the derivative of $P(x)$ with respect to $x$ and set it to zero to find critical points:
$P'(x) = \frac{d}{dx}(22x - x^2) = 22 - 2x$
Set the derivative equal to zero:
$22 - 2x = 0$
Solve for $x$ :
$2x = 22$ $x = 11$

STEP 6

Verify that this critical point is a maximum by checking the second derivative:
$P''(x) = \frac{d^2}{dx^2}(22x - x^2) = -2$
Since $P''(x) = -2 < 0$ , the function $P(x)$ is concave down, indicating a maximum at $x = 11$ .

STEP 7

Calculate the maximum product using the value of $x$ found:
Substitute $x = 11$ back into the expression for $y$ :
$y = 22 - x = 22 - 11 = 11$
Calculate the product:
$P(11) = 11 \cdot 11 = 121$
The maximum product is:
$\boxed{121}$

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QuestionAmong all pairs of numbers whose sum is 22 , find a pair whose product is as large as possible. What is the maximum product?

Studdy Solution

STEP 1

1. We are given two numbers, say x x x and y y y, such that their sum is 22.2. We need to maximize the product x⋅y x \cdot y x⋅y.3. The problem can be approached using algebraic manipulation and calculus or by recognizing a pattern.

STEP 2

1. Express one variable in terms of the other.2. Formulate the product function.3. Use calculus to find the maximum product.4. Verify the solution and calculate the maximum product.

STEP 3

Express one variable in terms of the other using the given sum condition:Let x x x and y y y be the two numbers. We know:x+y=22 x + y = 22 x+y=22Express y y y in terms of x x x:y=22−x y = 22 - x y=22−x

STEP 4

Formulate the product function P(x) P(x) P(x) using the expression for y y y:The product P P P is given by:P(x)=x⋅y=x⋅(22−x) P(x) = x \cdot y = x \cdot (22 - x) P(x)=x⋅y=x⋅(22−x)Simplify the expression:P(x)=22x−x2 P(x) = 22x - x^2 P(x)=22x−x2

STEP 5

STEP 6

STEP 7

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1. We are given two numbers, say $x$ and $y$ , such that their sum is 22.
2. We need to maximize the product $x \cdot y$ .
3. The problem can be approached using algebraic manipulation and calculus or by recognizing a pattern.

1. Express one variable in terms of the other.
2. Formulate the product function.
3. Use calculus to find the maximum product.
4. Verify the solution and calculate the maximum product.

Express one variable in terms of the other using the given sum condition:
Let $x$ and $y$ be the two numbers. We know:
$x + y = 22$
Express $y$ in terms of $x$ :
$y = 22 - x$

Formulate the product function $P(x)$ using the expression for $y$ :
The product $P$ is given by:
$P(x) = x \cdot y = x \cdot (22 - x)$
Simplify the expression:
$P(x) = 22x - x^2$