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
If the ratio of the cost price and the selling price of an article be 4:5, then the percentage of profit is:
A and B invest Rs. 3000 and Rs. 2400 respectively in a business. If after one year there is a loss of Rs. 720, how much loss will B bear? (Loss or Profit is in proportion to their investments)
To cover a certain distance with a speed of 60 km/hr, a train takes 15 hours. If it covers the same distance in 12 hours, what will be its speed?
If the speed of a boat in still water is 20 km/hr and the speed of the current is 5 km/hr, then the time taken by the boat to travel 100 km with the current is
A man bought 30 defective machines for Rs. 1000. He repaired and sold them @ Rs. 300 per machine. He got a profit of Rs. 150 per machine. How much did he spend on repairs?
The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19, find the first term.
What is the sum of the first 11 terms of an arithmetic progression if the first term is -31 and last term is 29?
An article was sold at Rs. 950 allowing 5% discount on the marked price. The marked price of the article is
What will be the percentage of increase in the area of a square when each of its sides is increased by 10%?
Average of two numbers is 8 and average of other three numbers is 3; the average of the five numbers is
Important Questions from Arithmetic Progression
If the arithmetic mean of a, b, c is \(\rm \frac M 3\) and \(\rm \frac{1}{a} + \frac{1}{b} = -\frac{1}{c} \) , then the arithmetic mean of a 2, b 2, c 2 is
How many two-digit numbers are divisible by 3 ?
A person saves Rs. 1000 more than he did the previous year. If he saves Rs. 2000 in the first year, in how many years will he save Rs. 170000?
A car starts with a speed of 60 km/h with its speed increasing every one hour by 5 km/h. In how many hours will it cover 435 kms?
How many natural numbers lie between 3 and 200 which are divisible by 7?
