Question: A box in the shape of a cuboid with a square base is to be made so that the sum of its dimensions (l + b + h) is 20cm. Find its maximum value.
To find the maximum value of the cuboid's volume, we need to use the fact that the sum of the dimensions (l + b + h) is 20 cm.
Let's denote the dimensions of the cuboid as follows:
- Length: l (in cm)
- Breadth: b (in cm)
- Height: h (in cm)
According to the given information, we have the equation:
l + b + h = 20
Since the base of the cuboid is square, we can assume l = b.
Substituting l = b into the equation, we have:
2l + h = 20
Now, we need to express the volume of the cuboid in terms of l and h:
Volume = l * b * h = l² * h
Since l = b, we can rewrite the volume equation as:
Volume = l² * h = l * l * h = l * h²
To find the maximum value of the volume, we need to express it as a function of a single variable. In this case, we will express it as a function of l.
Using the equation 2l + h = 20, we can rearrange it to express h in terms of l:
h = 20 - 2l
Substituting this expression for h into the volume equation, we have:
Volume = l * (20 - 2l)²
To find the maximum value of the volume, we can take the derivative of the volume function with respect to l, set it equal to zero, and solve for l.
I'm assuming you're familiar with derivatives and their application, so I will proceed with finding the maximum value analytically.
Taking the derivative of the volume function with respect to l:
d(Volume) / dl = (20 - 2l)² - 2l * 2(20 - 2l)
Setting the derivative equal to zero:
(20 - 2l)² - 4l(20 - 2l) = 0
Expanding and simplifying the equation:
(20 - 2l)(20 - 2l) - 4l(20 - 2l) = 0
(400 - 40l + 4l²) - (80l - 8l²) = 0
400 - 40l + 4l² - 80l + 8l² = 0
12l² - 120l + 400 = 0
Dividing both sides of the equation by 4:
3l² - 30l + 100 = 0
Now, we can solve this quadratic equation to find the value of l that maximizes the volume.
Using the quadratic formula:
l = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 3, b = -30, and c = 100.
Calculating the discriminant:
Δ = b² - 4ac
Δ = (-30)² - 4(3)(100)
Δ = 900 - 1200
Δ = -300
Since the discriminant is negative (-300), the quadratic equation does not have real solutions.
This means that there is no maximum volume for the given conditions. The volume can be infinitely large as long as the sum of the dimensions (l + b + h) is 20 cm.