Question 1170245
Let's break down this problem step by step.

**1.a) Deduce the Total Surface Area of a Closed Cylinder**

* **Area of the Top and Bottom Circles:** Each circle has an area of πR². Since there are two circles (top and bottom), their combined area is 2πR².
* **Lateral Surface Area (Curved Surface):** Imagine unrolling the side of the cylinder. This forms a rectangle with height H and width equal to the circumference of the base (2πR). Therefore, the lateral surface area is 2πRH.
* **Total Surface Area (S):** To find the total surface area, add the areas of the top, bottom, and lateral surfaces:
    S = 2πR² + 2πRH
    S = 2πR(R + H)

**1.b) Find the Volume of the Cylinder**

* **Volume Formula:** The volume (V) of a cylinder is given by V = πR²H.
* **Given Values:** H = 15.0 m and R = 5.0 m.
* **Calculation:**
    V = π(5.0 m)²(15.0 m)
    V = π(25 m²)(15.0 m)
    V = 375π m³

**1.c) Calculate the Amount of Water the Tank Can Hold**

The amount of water the tank can hold is equal to its volume.

* **Answer:** The tank can hold 375π m³ of water.

**1.d) Calculate the Height of a Cuboid Tank with the Same Capacity**

* **Cuboid Volume Formula:** The volume of a cuboid is given by V = Area of base × height.
* **Given Information:**
    * Volume of the cuboid = Volume of the cylinder = 375π m³
    * Cross-sectional area (base area) of the cuboid = 25 m²
* **Calculation:**
    375π m³ = 25 m² × h
    h = (375π m³) / (25 m²)
    h = 15π m

**Answers:**

a)  S = 2πR(R + H)
b)  V = 375π m³
c)  375π m³
d)  h = 15π m