find the two numbers whose sum is 10, if the product of one by the cube of other is a maximum.
x+y = 10, so y = 10-x.
p = xy^3 = x(10-x)^3
dp/dx = -2(10-x)^2(2x-5)
p is max when x = 5/2
Well, finding the two numbers that add up to 10 is the easy part. They would be 5 and 5. However, if you want to maximize the product of one number and the cube of the other, things get a bit more interesting.
Let's call one number "x" and the other "y". We want to maximize the product, so we can say that y = 10 - x. Now, we want to maximize the product of x and the cube of y, which is x * (10 - x)^3.
To find the maximum, we can take the derivative of this expression with respect to x and set it equal to zero. Then we solve for x.
However, solving this mathematically would take some time and effort. So, instead, I'll tell you a joke while you ponder this problem:
Why don't mathematicians like solving problems on the beach?
Because they have too many "sine" and "cosine" waves crashing in their heads!
So, take your time, work on the math, and remember that the answer isn't as important as the laughter along the way.
To find the two numbers whose sum is 10, and the product of one by the cube of the other is a maximum, we can use the concept of optimization.
Let's assume the two numbers are x and 10 - x. We want to maximize the value of x multiplied by (10 - x)^3.
We can find the maximum value by taking the derivative of this expression with respect to x and setting it equal to zero.
Let's go step by step to find the maximum value:
Step 1: Write the expression for the product: f(x) = x * (10 - x)^3.
Step 2: Take the derivative of f(x) with respect to x, using the chain rule:
f'(x) = (10 - x)^3 + 3x(10 - x)^2.
Step 3: Set the derivative equal to zero and solve for x:
(10 - x)^3 + 3x(10 - x)^2 = 0.
Step 4: Simplify the equation:
(10 - x)^3 = -3x(10 - x)^2.
Step 5: Divide both sides by (10 - x)^2 (assuming x is not equal to 10):
10 - x = -3x.
Step 6: Solve for x:
10 = 2x.
x = 5.
Step 7: Calculate the two numbers:
The two numbers are x = 5 and 10 - x = 10 - 5 = 5.
Therefore, the two numbers whose sum is 10, with the product of one by the cube of the other being a maximum, are 5 and 5.
To find the two numbers whose sum is 10, we can assign one of the numbers as 'x' and the other as '10 - x' (since their sum is 10).
Now, we need to maximize the product of one number and the cube of the other. This means we need to find the values of 'x' and '10 - x' that will give us the maximum product.
Let's denote the product as P.
P = x * (10 - x)^3
To find the maximum value of P, we can take the derivative of P with respect to x, set it equal to zero, and solve for x.
dP/dx = (10 - x)^3 - 3x(10 - x)^2
Setting dP/dx equal to zero:
(10 - x)^3 - 3x(10 - x)^2 = 0
Now, solve this equation to find the value of x. Once we find x, we can substitute it into '10 - x' to get the other number.