-Find two positive numbers satisfying the given requirements:
The sum of the first and twice the second is 80 and the product is a maximum.
a +2 b = 80 so a = 80 - 2 b
y = a b so y = (80-2 b)b = 80 b - 2 b^2
dy/db = 80 - 4 b
= 0 for max or min
4 b = 80
b = 20
then a = 40
To find two positive numbers that satisfy the given requirements, we can use algebraic representation and optimization techniques.
Let's assume the two numbers as x and y, where x is the first number and y is the second number.
According to the requirements:
1. The sum of the first and twice the second is 80:
x + 2y = 80
2. The product is a maximum:
We need to find the maximum value of the product, which means maximizing the function xy.
To solve this problem, we can use the concept of simultaneous equations. We have one equation (x + 2y = 80) and the equation for the product (xy). We can substitute the value of x from the first equation into the second equation and maximize the resulting equation.
Let's solve these equations step-by-step:
Step 1: Solve the first equation for x (x + 2y = 80):
x = 80 - 2y
Step 2: Substitute the value of x in the equation for the product (xy):
P = (80 - 2y)y
Step 3: Simplify the equation for the product:
P = 80y - 2y^2
Step 4: To find the maximum value of the product, we need to maximize the function P = 80y - 2y^2. We can do this by finding the vertex of the parabola represented by this equation.
Step 5: The vertex of the parabola can be found using the formula: y = -b/2a, where a = -2, and b = 80. Substituting the values in the formula:
y = -80/2(-2)
y = -80/-4
y = 20
Step 6: Substitute the value of y back into the first equation to find x:
x = 80 - 2(20)
x = 80 - 40
x = 40
Therefore, the two positive numbers that satisfy the given requirements are x = 40 and y = 20.
To find two positive numbers that satisfy the given requirements, let's denote the first number as 'x' and the second number as 'y'. We are given two conditions:
1. The sum of the first number and twice the second number is 80: x + 2y = 80
2. The product of the two numbers should be a maximum.
To find the numbers that maximize their product, we can use the concept of derivatives. The product of the two numbers, xy, can be expressed as a function P(x) = xy.
To maximize P(x), we need to find the critical points where the derivative of P(x) is equal to zero. So, let's take the derivative of P(x) with respect to x and set it equal to zero:
dP(x)/dx = y + x(dy/dx) = 0
Since we are given x + 2y = 80, we can express y in terms of x: y = (80 - x)/2
Now, substitute the value of y into the derivative equation:
(80 - x)/2 + x(dy/dx) = 0
Simplify and solve for the derivative (dy/dx):
(80 - x)/2 + (x/2)*(dy/dx) = 0
(dy/dx) = (x - 80)/x
Setting dy/dx to zero to find the critical points:
(x - 80)/x = 0
Since x cannot be equal to zero, we can multiply both sides of the equation by 'x':
x - 80 = 0
x = 80
So, we have found one critical point: x = 80.
Now, let's find the corresponding value of y using the equation x + 2y = 80:
80 + 2y = 80
2y = 0
y = 0
However, we are looking for positive numbers, so this critical point does not satisfy our requirements.
To find the second critical point, let's analyze the endpoints. Since we are searching for positive numbers, we should consider limiting x to be positive.
When x is at its maximum value (80), the other number y will be 0. On the other hand, when x is at its minimum value (0), the other number y will be 40.
Therefore, to satisfy the given requirements, the two positive numbers that maximize their product are: x = 80, y = 0 and x = 0, y = 40.