The product of 2 positive numbers is 48. find the value of the numbers if the sum of one of the numbers and the cube of the other is a minimum.
a = first number
b = second number
a * b = 48 Divide both sides by a
b = 48 / a
S = the sum of one of the numbers and the cube of the other number
S = a + b ^ 3
S = a + ( 48 / a ) ^ 3
S = a + 110,592 / a ^ 3
S = a + 110,592 * a ^ - 3
First derivaton :
d S / d a = 1 - 3 * 110,592 * a ^ - 4
d S / d a = 1 - 331,776 / a ^ 4
Second derivation :
d ^ 2 S / d a ^ 2 = - 3 * 331,776 ( - 4 ) * a ^ - 5
d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5
A function has minimum or maximum in poit where first derivation = 0
If second derivaton < 0 function has maximum.
If second derivaton > 0 function has minimum.
In this case:
d S / d a = 1 - 331,776 / a ^ 4 = 0
1 = 331,776 / a ^ 4 Multiply both sides by a ^ 4
a ^ 4 = 331,776
a = fourth root of 331,776
a = + OR - 24
For a = - 24
d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5 =
1,327,104 / - 7,962,624 = - -0.166667 < 0
function has maxsimum.
For a = 24
d ^ 2 S / d a ^ 2 = 1,327,104 / a ^ 5 =
1,327,104 / -7,962,624 = 0.166667 > 0
function has minimum.
So a = 24
b = 48 / a = 48 / 24 = 2
The mumbers are a = 24 and b = 2
Local minimum = a + b ^ 3 = 24 + 2 ^ 3 = 24 + 8 = 32
To find the values of the two positive numbers that satisfy the given condition, we can use the concept of optimization.
Let's assume the two numbers as x and y, where x > 0 and y > 0.
According to the problem, the product of the two numbers is 48:
x * y = 48 -------- (Equation 1)
We need to find the minimum value of one of the numbers (x or y) plus the cube of the other number. So, we need to minimize the expression:
f(x, y) = x + y^3
Now, let's solve the problem step-by-step:
Step 1: Solve Equation 1 for one variable in terms of the other:
x * y = 48
Solving for y, we get:
y = 48 / x
Step 2: Substitute the value of y into the expression f(x, y):
f(x) = x + (48 / x)^3
Step 3: To find the minimum value, take the derivative of f(x) with respect to x and set it equal to zero:
f'(x) = 1 - 3 * (48 / x)^2 * (-48 / x^2) = 0
Simplifying further:
1 + 144 / x^4 = 0
144 / x^4 = -1
x^4 / 144 = -1
Taking the fourth root of both sides:
x = (−1)^(1/4) * (144)^(1/4) = ∛(144)(−1)^1/12
Note: (-1)^(1/4) is a complex number, and since we are looking for positive values, let's consider the positive root of x only.
Step 4: Calculate the value of x:
x = ∛144 = 2 * ∛36 ≈ 2 * 3.3 ≈ 6.6
Step 5: Substitute the value of x back into Equation 1 to find y:
y = 48 / x = 48 / 6.6 ≈ 7.27
Therefore, the two positive numbers that satisfy the given condition are approximately x = 6.6 and y = 7.27.
To find the values of the two positive numbers, we need to minimize the sum of one number and the cube of the other. Let's call the two numbers x and y.
Given that the product of the two numbers is 48, we can write the equation: xy = 48.
To minimize the sum, we need to minimize the expression x + y^3.
To proceed, we can use the method of substitution. Since we have an equation in terms of x, we can solve for y in terms of x and substitute it into the expression to get rid of one variable.
From the equation xy = 48, we can isolate y by dividing both sides by x: y = 48/x.
Now substitute this value of y into the expression x + y^3: x + (48/x)^3.
To find the minimum value, differentiate the expression with respect to x and set it equal to zero.
Differentiating, we get d/dx(x + (48/x)^3) = 1 - 144/x^4.
Set the derivative equal to zero: 1 - 144/x^4 = 0.
Solving this equation for x^4 gives: x^4 = 144.
Taking the fourth root of both sides, we get: x = ±2√3.
Since we're looking for positive values, x = 2√3.
Now substitute this value of x back into the equation xy = 48 to find y: (2√3)y = 48.
Solving for y gives: y = 48/(2√3) = 8/√3 = (8/√3) * (√3/√3) = (8√3)/3 = (8/3)√3.
Therefore, the two positive numbers are approximately x ≈ 2√3 and y ≈ (8/3)√3.