Among all pairs of numbers with a sum of 34, find the pair whose product is maximum. Write your answers as fractions reduced to lowest terms.
Not really. I don't understand what the problem is wanting me to do exactly. Could you explain further?
You need to find the pairs numbers that add up to 34.
17 + 17
16 + 18
15 + 19
and so on
Then you're to find which of these pairs when multiplied together has the biggest product.
To find the pair of numbers with a sum of 34 and whose product is maximum, we can use a simple approach.
Let's assume the pair of numbers are x and y (where x + y = 34). We need to maximize the product xy.
To start, let's express one of the variables in terms of the other variable. From x + y = 34, we can rearrange it to y = 34 - x.
Now, the product of x and y is given by xy = x(34 - x) = 34x - x².
We can see that we have a quadratic equation, which is a downward-opening parabola.
To find the maximum product, we need to find the vertex of this parabola since the vertex represents the highest point.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -1 and b = 34 in our case.
x = -34 / (2 * -1) = 34 / 2 = 17.
To get the y-coordinate of the vertex, substitute x = 17 into the equation y = 34 - x.
y = 34 - 17 = 17.
Therefore, the pair with the maximum product is (17, 17).
To write this as a fraction reduced to lowest terms, we can see that both numbers are equal, so the fraction is 17/17.
However, 17/17 can be further reduced to 1/1, which is the final answer.
Hence, the pair of numbers with a sum of 34 and maximum product is (1, 1).
17 + 17; 17^2 = 289
16 * 18; 16 * 18 = 288
15 + 19; 15 * 19 = 285
Do you see the pattern?