A rectangular piece canvass with dimensions 10m by 6m is used to make a pool.Equal sizes squares are to be cut from each corner and remaining will folded up around some plastic tubing.what is the dimension of the pool so the water volume is maximum?

Question

Aurora Alvarado · Accepted Answer

Dimensions:
Length = 7.58m
Width = 3.58m
Height = 1.21m

I hope this helps, let me know if you have any questions :)

Axel Alvarez · Answer

undefined To maximize the water volume in the pool, we need to maximize the dimensions of the pool itself.

Let's denote the side length of the squares cut from each corner as $ x $. After cutting out squares from each corner, the new dimensions of the rectangular piece of canvas will be $ (10 - 2x) $ meters by $ (6 - 2x) $ meters.

When folding up the sides to form the pool, the depth of the pool will be equal to the side length of the squares cut out, which is $ x $ meters.

The volume $ V $ of the pool is given by the formula: 
$$ V = x \times (10 - 2x) \times (6 - 2x) $$

To find the maximum volume, we need to find the critical points of the volume function $ V(x) $ and determine whether they correspond to a maximum. We can do this by finding the derivative of $ V(x) $ with respect to $ x $ and setting it equal to zero.

After finding the critical points, we can use the second derivative test or evaluate $ V(x) $ at the critical points to determine whether we have a maximum or minimum.

Once we find the value of $ x $ that maximizes the volume, we can substitute it back into the expressions for the dimensions of the pool to find its dimensions.