Find two positive numbers whose product is 81 and whose sum is a minimum. (If both values are the same number, enter it into both blanks.)
9,9
a square has minimum perimeter for a given area
let one number be x, then the other is 81/x
sum = x + 81/x
d(sum)/dx = 1 - 81/x^2
= 0 for a min
1 = 81/x^2
x^2 = 81
x = 9 , since x must be positive
so if one is 9, the other is 81/9 which is also 9
To find two positive numbers with a product of 81 and a minimum sum, we can use the concept of the geometric mean.
The geometric mean is the square root of the product of two numbers. So, to find the minimum sum, we need to find the two numbers whose geometric mean is equal to 81.
Let's solve for the geometric mean of the two numbers:
√(x * y) = 81
Squaring both sides:
x * y = 81^2
x * y = 6561
Since we're looking for two positive numbers, we need to find a pair of factors of 6561. The factors of 6561 are:
1, 3, 9, 27, 81, 243, 729, 2187, 6561
The two positive numbers with a minimum sum are 9 and 729.
So, the answer is:
The two positive numbers are 9 and 729.
To find two positive numbers whose product is 81 and whose sum is a minimum, we can use calculus.
Let's represent the two numbers as x and y. We need to minimize the sum x + y, given that the product xy = 81.
To do this, we can set up an equation using the product constraint:
xy = 81
Now, we can express y in terms of x using this equation:
y = 81/x
Next, we can substitute this value of y into the sum equation:
x + y = x + 81/x
To find the minimum value of the sum, we can take the derivative of this expression with respect to x and set it equal to zero.
d/dx (x + 81/x) = 1 - 81/x^2 = 0
Simplifying further, we get:
1 = 81/x^2
Now we can solve for x:
x^2 = 81
Taking the square root of both sides, we get two possible solutions for x:
x = 9 or x = -9
Since we need to find positive numbers, we discard the negative value.
So, x = 9.
Now, we can substitute this value of x back into the product equation to find y:
9y = 81
Simplifying, we find that y = 9.
Therefore, the two positive numbers whose product is 81 and whose sum is a minimum are 9 and 9.