Question

Malik purchased 7,500 square meters of riverfront property. He gets to determine the dimensions of his property, with the condition that his property boundaries form a rectangle whose area is 7,500 square meters, and the river determines one side of Malik's rectangular property boundaries.

a. Which of the following quantities are varying in this situation? Select all that apply.
the area of Malik's property
the width of Malik's property
the length of the river
the perimeter of Malik's property
the length of Malik's property
b. Define the length of Malik's property boundary, l, in terms of the width of Malik's property boundary,
c. Define the total length of fencing, F, Malik would need to enclose his property (in meters) in terms of the width of Malik's property boundary, w. (Note that Malik does not need fencing on the side of his property against the river).

Answer 1

The varying quantities for Malik's riverfront property are the width, perimeter, and length. The length (l) is defined as 7500 divided by the width (w) and the fencing needed (F) is calculated using the formula F = 2w + (7500/w), excluding the river side.

Step-by-step explanation:

Regarding the quantities that are varying for Malik's riverfront property, we can list them as follows:

• The width of Malik's property (c or w)
• The perimeter of Malik's property (F)
• The length of Malik's property (l)

It is given that the area of Malik's property is a constant at 7,500 square meters, and the length of the river corresponds with one side of the property, which is fixed by the riverfront and not variable. However, the perimeter of Malik's property varies because as the width and length change, so does the perimeter.

Now, to define the length (l) of Malik's property in terms of the width (w), we use the formula for the area of a rectangle (A = l × w), where A is the area:

l = A/w

Since we know the area (A) is 7,500 square meters, we get:

l = 7,500/w

For the total length of fencing required (F), we do not include the river side, so the perimeter formula P = 2(l + w) modifies to:

F = 2w + l (since the river side is not fenced)

Substituting the value of l, we get:

F = 2w + (7,500/w)

Malik can use this formula to determine how much fencing he'll need for any given width he chooses for his property, with the understanding that the length will adjust accordingly to maintain the property's area.