Question

how many square feet of outdoor carpet will we need for this hole? 6 ft 3 ft a₂ = 9 ft² a₁ = 72 ft² 6 ft 12 ft

Explanation:

Step1: Identify the area of the hole

The hole is a triangle with area \( A_2 = 9 \, \text{ft}^2 \). Wait, no—actually, the total area of the rectangle (the green part plus the hole?) Wait, no, the problem is to find the area of the carpet needed, which is the area of the large rectangle minus the area of the hole (the triangle). Wait, the large rectangle \( A_1 = 72 \, \text{ft}^2 \), and the hole (the triangle) has area \( 9 \, \text{ft}^2 \)? Wait, no, let's check again. Wait, the large rectangle: length 12 ft, width 6 ft? Wait, \( 12 \times 6 = 72 \), which matches \( A_1 = 72 \, \text{ft}^2 \). The hole is the triangle with area \( 9 \, \text{ft}^2 \)? Wait, no, maybe the hole is the triangle, and we need to find the area of the carpet, which is the area of the rectangle minus the area of the hole? Wait, no, the question is "how many square feet of outdoor carpet will we need for this hole?" Wait, maybe the hole is the triangle, and we need to cover it? Wait, the triangle has area \( 9 \, \text{ft}^2 \)? Wait, the triangle's base is 6 ft (the vertical side) and height? Wait, the triangle's area is given as \( A_2 = 9 \, \text{ft}^2 \). Wait, maybe the problem is that the carpet needed is the area of the hole, which is the triangle. Wait, the triangle's area is already given as \( 9 \, \text{ft}^2 \)? Wait, no, maybe I misread. Wait, the large rectangle is 12 ft by 6 ft, area 72. The hole is a triangle with base 3 ft? Wait, no, the triangle has height 6 ft? Wait, the vertical side is 3 ft? Wait, the triangle's area is \( \frac{1}{2} \times \text{base} \times \text{height} \). If the height is 6 ft (the vertical side) and base is 3 ft? Wait, \( \frac{1}{2} \times 3 \times 6 = 9 \), which matches \( A_2 = 9 \, \text{ft}^2 \). So the area of the hole (the triangle) is 9 square feet. Wait, but the question is "how many square feet of outdoor carpet will we need for this hole?" So we need to cover the hole, which is the triangle. So the area needed is the area of the triangle, which is 9? Wait, no, maybe the hole is the triangle, and we need to cover it, so the carpet area is the area of the triangle. Wait, but let's check again. The diagram shows a large rectangle (12 ft by 6 ft, area 72) with a triangular hole (area 9). Wait, maybe the question is to find the area of the carpet needed to cover the hole, which is the area of the triangle. So the answer is 9? Wait, no, maybe I'm wrong. Wait, the problem says "how many square feet of outdoor carpet will we need for this hole?" So the hole is the triangle, so we need the area of the triangle. The triangle's area is given as \( A_2 = 9 \, \text{ft}^2 \). So the carpet needed is 9 square feet? Wait, but let's confirm. The triangle: base 6 ft (vertical) and height 3 ft? Wait, no, the horizontal side is 6 ft? Wait, the triangle's area is \( \frac{1}{2} \times 6 \times 3 = 9 \), which matches. So the area of the hole (the triangle) is 9 square feet, so we need 9 square feet of carpet? Wait, but maybe the hole is the area to be covered, so the carpet area is the area of the hole, which is 9. Wait, but let's check the numbers again. The large rectangle is 12x6=72. The triangle: if we calculate its area, base 6 ft (the vertical side) and height 3 ft (the horizontal segment? Wait, no, the horizontal segment is 6 ft? Wait, the triangle is inside the rectangle. Wait, maybe the hole is the triangle, and we need to cover it, so the carpet area is the area of the triangle, which is 9. So the answer is 9? Wait, but maybe I made a mistake. Wait, the problem says "how many square feet of outdoor carpet will we need for this hole?" So the hole is the triangle, so the area needed is the area of the triangle, which is 9. So the answer is 9.

Wait, no, maybe the hole is the area that's missing, and we need to cover the entire area except the hole? No, the question is "for this hole", so we need to cover the hole. So the area of the hole is 9 square feet, so we need 9 square feet of carpet.

Step2: Confirm the area of the hole

The triangle's area is given as \( A_2 = 9 \, \text{ft}^2 \), so the carpet needed for the hole is 9 square feet.

Answer:

\( 9 \)