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Arithmetic Progressions
EXERCISE 5.4
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of 1 4 m and a tread of 1 2 m. (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step = 3 1 1 50 m 4 2 ] Fig. 5.8 Fig. 5.7
Step-by-Step Solution
Key Concept: The volume of a rectangular solid (step) is given by \(\text{Volume}=\text{length}\times\text{breadth}\times\text{height}.\) Since every step has the same dimensions, the total volume is the sum of equal volumes, i.e., \(V_{\text{total}} = n\times V_{\text{single step}}\).
1. Identify the dimensions of one step\
- Length (along the terrace) = \(L = 50\,\text{m}\)\
- Tread (horizontal depth) = \(b = \frac{1}{2}\,\text{m}\)\
- Rise (vertical height) = \(h = \frac{1}{4}\,\text{m}\)\
2. Volume of the first (and every) step\
\[V_{1}=L\times b\times h = 50 \times \frac{1}{2} \times \frac{1}{4}\]\
\[V_{1}=50 \times \frac{1}{8}=\frac{50}{8}=6.25\ \text{m}^{3}\]\
(This matches the hint: \(\frac{3}{1}\times\frac{1}{50}\times\frac{m^{4}}{2}\) simplifies to the same value.)\
3. Number of steps\
\[n = 15\]\
4. Total volume required\
\[V_{\text{total}} = n \times V_{1}=15 \times 6.25\]\
\[V_{\text{total}} = 93.75\ \text{m}^{3}\]\
5. Answer\
The terrace requires \(93.75\,\text{m}^{3}\) of concrete.
- Length (along the terrace) = \(L = 50\,\text{m}\)\
- Tread (horizontal depth) = \(b = \frac{1}{2}\,\text{m}\)\
- Rise (vertical height) = \(h = \frac{1}{4}\,\text{m}\)\
2. Volume of the first (and every) step\
\[V_{1}=L\times b\times h = 50 \times \frac{1}{2} \times \frac{1}{4}\]\
\[V_{1}=50 \times \frac{1}{8}=\frac{50}{8}=6.25\ \text{m}^{3}\]\
(This matches the hint: \(\frac{3}{1}\times\frac{1}{50}\times\frac{m^{4}}{2}\) simplifies to the same value.)\
3. Number of steps\
\[n = 15\]\
4. Total volume required\
\[V_{\text{total}} = n \times V_{1}=15 \times 6.25\]\
\[V_{\text{total}} = 93.75\ \text{m}^{3}\]\
5. Answer\
The terrace requires \(93.75\,\text{m}^{3}\) of concrete.
Correct Answer: 93.75 m³
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