4. What is wrong with this calculation of the total surface area of a cylinder with height 9 m and a base radius 6 m ? \( \begin{aligned} \text { L.A. } & =h \pi d \\ & =9 \pi(12) \\ & =\pi \cdot \mathrm{r}^{2} \\ & \approx 339.12 \mathrm{~m}^{2}\end{aligned} \) \( \begin{aligned} \text { S.A. } & =339.6^{2} \\ & =\pi \cdot 36\end{aligned} \) \( \begin{aligned} \approx 3.14(36)=113.04 \mathrm{~m}^{2}\end{aligned} \)

Let's analyze the calculation step by step to identify the errors in the calculation of the total surface area of a cylinder. ### Known Conditions - Height \( h = 9 \, \text{m} \) - Base radius \( r = 6 \, \text{m} \) ### Surface Area of a Cylinder The total surface area (S.A.) of a cylinder is given by the formula: \[ \text{S.A.} = 2\pi r(h + r) \] Where: - \( 2\pi r^2 \) is the area of the two circular bases. - \( 2\pi rh \) is the lateral surface area. ### Step 1: Calculate the Lateral Area (L.A.) The lateral area is calculated as: \[ \text{L.A.} = h \cdot \pi d \] Where \( d = 2r \) (the diameter). Thus: \[ d = 2 \cdot 6 = 12 \, \text{m} \] Now substituting the values: \[ \text{L.A.} = 9 \cdot \pi \cdot 12 = 108\pi \, \text{m}^2 \] ### Step 2: Calculate the Area of the Bases The area of one base is: \[ \text{Area of one base} = \pi r^2 = \pi \cdot 6^2 = 36\pi \, \text{m}^2 \] Since there are two bases: \[ \text{Total area of bases} = 2 \cdot 36\pi = 72\pi \, \text{m}^2 \] ### Step 3: Calculate the Total Surface Area Now, we can calculate the total surface area: \[ \text{S.A.} = \text{L.A.} + \text{Total area of bases} = 108\pi + 72\pi = 180\pi \, \text{m}^2 \] ### Step 4: Approximate the Total Surface Area Using \( \pi \approx 3.14 \): \[ \text{S.A.} \approx 180 \cdot 3.14 = 565.2 \, \text{m}^2 \] ### Summary of Errors 1. The lateral area was incorrectly calculated as \( 339.12 \, \text{m}^2 \) instead of \( 108\pi \, \text{m}^2 \). 2. The surface area was incorrectly calculated as \( 339.6^2 \) which is not relevant to the surface area calculation. 3. The area of the bases was incorrectly calculated as \( 113.04 \, \text{m}^2 \) instead of \( 72\pi \, \text{m}^2 \). ### Final Answer The correct total surface area of the cylinder is approximately \( 565.2 \, \text{m}^2 \).