Each second after shut-off, the speed of the blade is 2/3 of the speed in the previous second. After the first 8 s, the saw has turned 258 times. What was the speed of the saw before the motor shut off, to the nearest tenth of a turn per second?
I get 805.4 but I don't think I'm right.
i wanna answer this inequality of this graph f(x)= x -3 if x >= 3
You want the sum of the first 8 terms:
S8 = a(1-r^8)/(1-r) = a(1-(2/3)^8)/(1/3) = 2.883a
So, a = 258/2.883 = 89.49
Your answer is totally unreal. It would have turned 805.4 times in the 1st second, which is way more than the total turns after 8 seconds. I hope that is why you doubted it, and not just a vague uneasiness.
Thanks Steve, turns out my calculator is a bit sensitive..
To solve this problem, let's break it down step by step:
1. We know that after the first 8 seconds, the saw has turned 258 times. This means that the speed of the blade decreases each second after the shut-off.
2. Let's assume the speed of the blade before the motor shut off is "x" turns per second. Since the speed decreases by 2/3 each second, the speed in the first second after shut-off would be (2/3)x, the speed in the second second after shut-off would be (2/3)(2/3)x = (2/3)^2x, and so on.
3. As the problem states, after 8 seconds, the saw has turned 258 times. So, we need to figure out the speed of the blade in the first 8 seconds.
4. The total number of turns in the first 8 seconds can be calculated by summing up the geometric sequence: x + (2/3)x + (2/3)^2x + ... + (2/3)^7x = 258.
5. We can use the formula for the sum of a geometric series to solve for x:
S = a * (1 - r^n) / (1 - r),
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = x, r = 2/3, and n = 8. So we have:
258 = x * (1 - (2/3)^8) / (1 - 2/3).
6. Solving this equation will give us the value of x, which represents the speed of the blade before the motor shut off.
Let's do the math:
258 = x * (1 - (2/3)^8) / (1 - 2/3)
258 = x * (1 - (256/6561)) / (1/3)
258 = x * (1 - 256/6561) * 3
258 = x * (1 - 0.0391) * 3
258 = x * (0.9609) * 3
258 = x * 2.8827
x ≈ 89.493
Therefore, the speed of the saw before the motor shut off, to the nearest tenth of a turn per second, is approximately 89.5 turns per second.