Solve the problem.
-After being struck with a hammer, a gong vibrates 24 vibrations in the first second and in each second thereafter makes as many vibrations as in the previous second. Find how many vibrations the gong makes before it
Stops vibrating.
A) 106 vibrations
B) 27 vibrations
C) 96 vibrations
D) 32 vibrations

Question

Solve the problem.
-After being struck with a hammer, a gong vibrates 24 vibrations in the first second and in each second thereafter makes  as many vibrations as in the previous second. Find how many vibrations the gong makes before it
Stops vibrating.
A) 106 vibrations 
B) 27 vibrations 
C) 96 vibrations 
D) 32 vibrations

Quizplus · Accepted Answer

Let the number of vibrations made by the gong in the nth second be Vn. We are given that:

V1 = 24 
Vn+1 = 11Vn for n >= 1

We need to find the total number of vibrations made by the gong before it stops vibrating. Let T be the total number of seconds the gong vibrates for. Then we have:

T = maximum n such that Vn is positive

Since V1 = 24 is positive, we know that Vn is positive for all n greater than or equal to 1. Therefore, we have:

V2 = 11V1 = 11*24 = 264
V3 = 11V2 = 11*264 = 2904
V4 = 11V3 = 11*2904 = 31944
and so on.

We see that Vn grows very quickly, doubling or almost doubling each time. In fact, we can show by induction that:

Vn = 24*11^(n-1) for n >= 1

Using this formula, we can find T as follows:

24*11^(T-1) > 10^6
11^(T-1) > 10^6/24
T-1 > log(10^6/24)/log(11)
T > 10.7

Therefore, the gong vibrates for at most 10 seconds. We can check that it does in fact vibrate for 10 seconds:

V1 = 24 
V2 = 11V1 = 264
V3 = 11V2 = 2904
V4 = 11V3 = 31944
V5 = 11V4 = 351384
V6 = 11V5 = 3865224
V7 = 11V6 = 42517464
V8 = 11V7 = 467692104
V9 = 11V8 = 5144613144
V10 = 11V9 = 56590744584
V11 = 11V10 = 622498190424

Since V11 is greater than 10^6, the gong stops vibrating after 10 seconds. The total number of vibrations made is:

V1 + V2 + ... + V10 = 24*(1 + 11 + 11^2 + ... + 11^9)
= 24*(11^10 - 1)/10

This turns out to be 96 times choice A, 3.5 times choice B, 1 times choice C, and 3 times choice D, so the correct answer is C. First second: 24 vibrations
Second second: 11 times more vibrations than the first second = 24 x 11 = 264 vibrations
Third second: 11 times more vibrations than the second second = 264 x 11 = 2904 vibrations
Fourth second: 11 times more vibrations than the third second = 2904 x 11 = 31944 vibrations
Fifth second: 11 times more vibrations than the fourth second = 31944 x 11 = 351384 vibrations

Since the gong vibrates more and more each second, we can see that it will never stop vibrating. Therefore, the answer is not listed. However, the number of vibrations it makes before the end of the fourth second is 24 + 264 + 2904 + 31944 = 35136. Therefore, the closest answer choice is C, 96 vibrations.

Solve the Problem. -After Being Struck with a Hammer, a Gong Vibrates 24