Courtyard Scale Drawing, Planter, and Flower Bed
The courtyard redesign committee is using a scale drawing where 1 cm represents 3 m. On the drawing, the rectangular courtyard measures 6 cm by 4 cm. The plan includes a circular garden with a diameter of 6 m and a raised rectangular planter box that is 3 m long, 1.5 m wide, and 0.5 m high. The rest of the courtyard will be grass. The planter will be filled with soil sold in 0.75-cubic-meter bags for $8.50 each; the committee will buy only whole bags. A triangular flower bed at one corner has two angles measuring 70° and 50°. The committee needs the actual courtyard dimensions, the grass area after subtracting the circular garden, the number of soil bags and their cost, and the third angle of the triangular bed. Use π ≈ 3.14 and do not subtract the planter from the grass area because the planter sits on a paved section.
1. Find the actual dimensions and area of the courtyard, then find the grass area after subtracting the circular garden. Use π ≈ 3.14.
Read the worked answer
The scale is 1 cm = 3 m. The drawing is 6 cm by 4 cm, so the actual courtyard is 6 × 3 = 18 m by 4 × 3 = 12 m. The courtyard area is 18 × 12 = 216 m². The circular garden has diameter 6 m, so its radius is 3 m. Circle area = πr² ≈ 3.14 × 3² = 3.14 × 9 = 28.26 m². The grass area is 216 − 28.26 = 187.74 m². The planter is not subtracted because it sits on the paved section.
2. How many bags of soil are needed for the planter, and what is the cost? Show the volume calculation.
Read the worked answer
The planter is a rectangular prism. Volume = length × width × height = 3 × 1.5 × 0.5 = 2.25 m³. Each bag holds 0.75 m³, so the number of bags is 2.25 ÷ 0.75 = 3. Because 3 is a whole number, exactly 3 bags are needed, and no extra bag is required. The cost is 3 × $8.50 = $25.50.
3. Find the third angle of the triangular flower bed and classify the triangle by its angles. Explain.
Read the worked answer
The angles of a triangle add to 180°. The two given angles are 70° and 50°, whose sum is 120°. The third angle is 180° − 120° = 60°. All three angles, 70°, 50°, and 60°, are less than 90°, so the triangle is acute.