10 A scale diagram of a recreation park is shown alongside, What is the scale factor for this diagram? b Find: I the distance between the water fountains If the dimensions of the picnic area Iili the dimensions of the playground. e The perimeter fence of the park needs to be replaced. What length of fencing is required? Assume each gate is 2 m wide, d I Find the area of the playground. II The playground is to be covered with a layer of bark chips. Each tonne of bark chips covers \( 20 \mathrm{~m}^{2} \) of ground to the required depth. How many tonnes of bark chips will be needed?

**Step 1. Establishing the Scale Factor** In scale‐drawing problems the “scale factor” tells us how many units in reality correspond to one unit in the drawing. In this problem the diagram comes with its own scale so that, for example, if \[ 1\,\text{cm}\text{ on the diagram} = 5\,\text{m}\text{ in reality} \] then the scale factor expressed as a ratio is \[ 1:500 \] since \(5\,\text{m} = 500\,\text{cm}\). (Any answer equivalent to “1 cm : 5 m” or “1:500” is acceptable.) --- **Step 2. Converting Diagram Measurements to Actual Measurements** In every case the actual (real‐life) measurement is obtained by multiplying the measurement on the diagram (in centimeters) by \(5\,\text{m}\) (the “real‐life value” per centimetre). For instance, if a feature measures \(x\) cm on the diagram then its actual size is \[ x\,\text{cm} \times 5\,\text{m/cm} = 5x\,\text{m}. \] The following answers assume that the diagram provides these measurements: - The distance between the water fountains is given as \(4\,\text{cm}\) on the diagram. - The picnic area is shown with dimensions \(6\,\text{cm}\) by \(4\,\text{cm}\). - The playground is shown with dimensions \(8\,\text{cm}\) by \(5\,\text{cm}\). - The complete outline of the park (its “perimeter”) is measured around on the diagram; in our example, suppose the total length of the perimeter is \(40\,\text{cm}\) when you “walk” around the diagram. Use the scale factor to convert these. --- **Step 3. Solving Each Part** **(a) The Scale Factor** As explained above, if \[ 1\,\text{cm} = 5\,\text{m}, \] the scale factor is \[ 1:500. \] --- **(b)(i) Distance Between the Water Fountains** If the diagram shows the fountains are \(4\,\text{cm}\) apart, then \[ \text{Actual distance} = 4\,\text{cm} \times 5\,\text{m/cm} = 20\,\text{m}. \] --- **(b)(ii) The Dimensions of the Picnic Area** If the diagram’s picnic area measures \(6\,\text{cm}\) by \(4\,\text{cm}\), then the actual dimensions are calculated by multiplying by \(5\,\text{m}\) per cm. - Length: \[ 6\,\text{cm} \times 5\,\text{m/cm} = 30\,\text{m}. \] - Width: \[ 4\,\text{cm} \times 5\,\text{m/cm} = 20\,\text{m}. \] Thus the picnic area is \(30\,\text{m}\) by \(20\,\text{m}\). --- **(b)(iii) The Dimensions of the Playground** If the playground measures \(8\,\text{cm}\) by \(5\,\text{cm}\) on the diagram, then its actual dimensions are: - Length: \[ 8\,\text{cm} \times 5\,\text{m/cm} = 40\,\text{m}. \] - Width: \[ 5\,\text{cm} \times 5\,\text{m/cm} = 25\,\text{m}. \] So the playground is \(40\,\text{m}\) by \(25\,\text{m}\). --- **(e) Perimeter Fencing Required** Suppose the total perimeter of the park as measured on the diagram is \(40\,\text{cm}\). Converting this gives: \[ 40\,\text{cm} \times 5\,\text{m/cm} = 200\,\text{m}. \] If the park has several gates built into the perimeter (and these gaps do not need fencing) and each gate is \(2\,\text{m}\) wide, then you must subtract the total width of the gates from the full perimeter measured. For example, if there are 3 gates: - Total gate gap = \[ 3 \times 2\,\text{m} = 6\,\text{m}. \