Let's solve both of the problems from the image:

Problem 1:

Joel will fence a rectangular yard measuring 20 yards by 15 yards.

• Perimeter: The perimeter of a rectangle is given by the formula: $P = 2 \times (\text{length} + \text{width})$ Substituting the values: $P = 2 \times (20 + 15) = 2 \times 35 = 70 \, \text{yards}$ So, Joel will need 70 yards of fencing.

• Area: The area of a rectangle is given by the formula: $A = \text{length} \times \text{width}$ Substituting the values: $A = 20 \times 15 = 300 \, \text{square yards}$ So, the area of the fenced yard is 300 square yards.

Problem 2:

Joel fences a larger yard with dimensions that are four times the dimensions in problem 1.

• New dimensions: The original dimensions were 20 yards by 15 yards, so the new dimensions are: $4 \times 20 = 80 \, \text{yards (length)}$ $4 \times 15 = 60 \, \text{yards (width)}$

• Perimeter of the larger yard: Using the formula for the perimeter of a rectangle again: $P = 2 \times (80 + 60) = 2 \times 140 = 280 \, \text{yards}$ So, the perimeter of the larger yard is 280 yards.

• Area of the larger yard: Using the formula for the area: $A = 80 \times 60 = 4800 \, \text{square yards}$ So, the area of the larger yard is 4800 square yards.

Would you like more details on these calculations, or do you have any questions?

Here are 5 related questions to further explore:

1. How much fencing would Joel need if he only fenced two sides of the yard?
2. If Joel increased the length of the original yard by 5 yards, what would the new perimeter and area be?
3. How many times larger is the area of the larger yard compared to the original yard?
4. If the cost of fencing is $10 per yard, how much would it cost to fence the original and the larger yard?
5. If the larger yard is split into four equal parts, what are the dimensions and area of each part?

Tip: Remember, when scaling up dimensions, the perimeter increases linearly, but the area increases quadratically!