Prism Volume Calculation: Layered Height
The problem asks for the total volume of a prism constructed in layers. The prism has a square base and its height increases with each layer following an arithmetic progression.
Determining Layer Heights
The height follows an arithmetic progression (AP). Let the height of the first layer be '$a$' cm and the common difference be '$d$' cm.
- Given common difference, $d = 4$ cm.
- The height reaches 16 cm in the fourth layer. Using the AP formula $a_n = a + (n-1)d$: $a_4 = a + (4-1)d = a + 3d$ $16 = a + 3(4)$ $16 = a + 12$ $a = 16 - 12 = 4$ cm.
- The heights of the four layers are:
- Layer 1: $h_1 = a = 4$ cm
- Layer 2: $h_2 = a + d = 4 + 4 = 8$ cm
- Layer 3: $h_3 = a + 2d = 4 + 2(4) = 12$ cm
- Layer 4: $h_4 = a + 3d = 4 + 3(4) = 16$ cm
- The total height ($H$) of the prism is the sum of the heights of all layers: $H = h_1 + h_2 + h_3 + h_4 = 4 + 8 + 12 + 16 = 40$ cm.
Calculating Total Volume
The prism has a square base with a side length of 5 cm.
- Base Area ($A$): $A = (\text{side length})^2 = (5 \text{ cm})^2 = 25 \text{ cm}^2$.
- Total Volume ($V$): The volume of a prism is given by the formula $V = \text{Base Area} \times \text{Total Height}$. $V = A \times H$ $V = 25 \text{ cm}^2 \times 40 \text{ cm}$ $V = 1000 \text{ cm}^3$.
The total volume of the prism is $1000 \text{ cm}^3$.
Similar Questions
- What is the 12th term of
5, 10, 15, ...?
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