Four bricks are to be stacked at the edge of a table, each brick overhanging the one below it, so that the top brick extends as far as possible beyond the edge of the table.
*The construction in stable equilibrium, bricks must extend no more than (starting at the top) 1/2, 1/4, 1/6, 1/8 of their length beyond the one below.
A builder wants to build a corbeled arch, based on the principle of stability, what minimum number of bricks, each 0.30 m long, is needed if the arch is to span 1.0 m? Be sure to include in the total number of bricks one brick on top and one brick at the base of each half-span of the arch.
I can't figure out how to approach the above problem. I tried to set up an equation where the distance spanned is equal to the length of the brick divided by two times the number of bricks.
To approach this problem, first let's note that the corbeled arch will be symmetric. Thus, we only need to find out how many bricks are needed to span half of the arch (0.50 m) and then double that number.
Let's denote the number of bricks required to span half of the arch as n. According to the principle of stability given, the length that the top brick can extend is 0.30 m (total length of the top brick) * (1/2 + 1/4 + 1/6 + ... + 1/(2n)), where n is the number of bricks for half of the arch. Notice that we are summing a series of fractions with denominators of even integers.
We want to find the minimum number of bricks such that their extensions can cover half of the arch (0.50 m), so:
0.50 <= 0.30 * (1/2 + 1/4 + 1/6 + ... + 1/(2n))
Divide both sides by 0.30:
5/3 <= 1/2 + 1/4 + 1/6 + ... + 1/(2n)
Now we can find the smallest n that satisfies this inequality by calculating the sum of fractions:
(1/2) = 0.5
(1/2) + (1/4) = 0.75
(1/2) + (1/4) + (1/6) = 0.92
(1/2) + (1/4) + (1/6) + (1/8) = 1.08
We find that n = 4 satisfies the inequality. So we need 4 bricks to span half of the arch. Since the arch is symmetric, we need another 4 bricks to span the other half. Additional bricks are required at the top and the base of each half-span, so we add 4 more bricks (2 for the top and 2 for the base).
Thus, a minimum of 4 + 4 + 4 = 12 bricks is needed to build the corbeled arch.
To approach this problem, we can start by determining the number of bricks needed for each half-span of the arch.
Let's assume "n" as the number of bricks required to span a half-span of the arch. The length of each brick is given as 0.30 m.
In a stable corbeled arch, according to the given condition, the bricks must extend no more than 1/2, 1/4, 1/6, 1/8 of their length beyond the one below. This can be represented as:
0.30m * (1/2) + 0.30m * (1/4) + 0.30m * (1/6) + 0.30m * (1/8) + ... = 1.0m
To simplify the equation, we need to find a common denominator for the fractions. The lowest common multiple (LCM) of 2, 4, 6, and 8 is 24.
Therefore, the equation becomes:
(0.30m * 12/24) + (0.30m * 6/24) + (0.30m * 4/24) + (0.30m * 3/24) + ... = 1.0m
Simplifying further:
0.15m + 0.075m + 0.05m + 0.0375m + ... = 1.0m
To find the minimum number of bricks needed, we have to determine the value of "n" in the equation above. We can do this by trying different values of "n" until the equation is satisfied.
Using trial and error, let's start with "n = 1":
0.15m + 0.075m + 0.05m + 0.0375m + ... = 0.15m + 0.075m = 0.225m
As we can see, with "n = 1", the equation is not satisfied. Therefore, we need more than 1 brick for each half-span of the arch.
Let's try "n = 2":
0.15m + 0.075m + 0.05m + 0.0375m + ... = 0.15m + 0.075m + 0.05m = 0.275m
Again, the equation is not satisfied.
We continue trying different values of "n" until the equation is satisfied. By using trial and error, we find that "n = 5" satisfies the equation:
0.15m + 0.075m + 0.05m + 0.0375m + 0.03m = 1.0m
Therefore, the minimum number of bricks needed for each half-span of the arch is 5.
Since we need bricks on both sides of the arch, the total number of bricks required is twice that number, which is 2 * 5 = 10.
Hence, a minimum of 10 bricks, each measuring 0.30 m in length, is needed to build a corbeled arch spanning 1.0 m.
To solve this problem, let's break it down step by step.
First, let's figure out the number of bricks needed for each half-span of the arch. Each brick is 0.30 m long, and the arch needs to span 1.0 m. Since the bricks need to extend no more than 1/2, 1/4, 1/6, and 1/8 of their length beyond the brick below, we can calculate the number of bricks needed by dividing the total span length by the maximum extension of each brick.
For the top brick, it can extend up to 1/2 of its length, which is 0.30 m/2 = 0.15 m. So, the total span length for the top brick is 0.30 m + 0.15 m = 0.45 m.
For the second brick, it can extend up to 1/4 of its length, which is 0.30 m/4 = 0.075 m. So, the total span length for the second brick is 0.30 m + 0.075 m = 0.375 m.
For the third brick, it can extend up to 1/6 of its length, which is 0.30 m/6 = 0.05 m. So, the total span length for the third brick is 0.30 m + 0.05 m = 0.35 m.
For the fourth brick, it can extend up to 1/8 of its length, which is 0.30 m/8 = 0.0375 m. So the total span length for the fourth brick is 0.30 m + 0.0375 m = 0.3375 m.
Now, let's calculate the minimum number of bricks needed to span 1.0 m. Since each half-span is less than 0.5 m, we can use the concept of division to find the number of bricks. We divide the total span length of 1.0 m by the maximum span length for each brick.
For the top brick, the maximum span length is 0.45 m. So, the number of bricks needed for the top half-span is 1.0 m / 0.45 m = 2.22 (approximated).
For the second brick, the maximum span length is 0.375 m. So, the number of bricks needed for the second half-span is 1.0 m / 0.375 m = 2.67 (approximated).
For the third brick, the maximum span length is 0.35 m. So, the number of bricks needed for the third half-span is 1.0 m / 0.35 m = 2.86 (approximated).
For the fourth brick, the maximum span length is 0.3375 m. So, the number of bricks needed for the fourth half-span is 1.0 m / 0.3375 m = 2.96 (approximated).
Now, let's add up the number of bricks needed for each half-span.
2.22 + 2.67 + 2.86 + 2.96 = 10.71
Since we need to include one brick on top and one brick at the base of each half-span, we add 1 to the total number of bricks for each half-span.
10.71 + 4 = 14.71
Therefore, we need a minimum of 15 bricks to build the corbeled arch based on the principle of stability.