number of times a light bulb blinks from 12.00 to 12.08 if it blinks every 4 seconds

[siaaia88]

number of times a light bulb blinks from 12.00 to 12.08

A light bulb blinks at 12 noon. It then blinks after 4 seconds, then after 8 seconds, then after 12 seconds and so on. How many times shall it have already blinked until it blinks at12.08pm?s

 

[MarkFL]

How many terms does the sequence 0,1,2,3,...,n have?

 

[HallsofIvy]

Also there are 8(60)= 480 seconds in 8 minutes.

 

[soroban]

Hello, siaaia88!

The wording of the problem is misleading.

A light bulb blinks at 12 noon.
It then blinks after 4 seconds, then after 8 seconds, then after 12 seconds and so on.
How many times shall it have already blinked until it blinks at 12.08 pm?

If I understand the problem, the light blinks at t = 0.
Four seconds later, it blinks at t = 4.
Eight seconds later, it blinks at t = 12.
Twelve seconds later, it blinks at t = 24.
Sixteen seconds later, it blinks at t = 40.
. . And so on.

We find that \(\displaystyle t\) is a quadratic function of \(\displaystyle n\),
. . the number of blinks.
Specifically: .\(\displaystyle f(n) \,=\,2n(n+1)\)

We have: .\(\displaystyle 2n(n+1) \:=\:480 \quad\Rightarrow\quad n^2 + n -240\:=\:0\)

. . . . . . . . \(\displaystyle (n-15)(n+16) \:=\:0 \quad\Rightarrow\quad n \:=\:15,\,\color{red}{\rlap{///}}\text{-}16\)

It reaches 480 seconds at the 15th time interval.
Therefore, the light blinks 16 times.


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Now I see that the title reads "every 4 seconds".

Then the answer is: .\(\displaystyle \dfrac{480}{4} + 1 \:=\:121\)

 

[lookagain]

How many times shall it have already blinked until it blinks at12.08pm?s

That sentence is not clear. I interpreted it to mean how many times shall it have blinked
before it finally blinks at 12:08 p.m.?



Then the answer would be 120 if that were to be the case.

 

[dsk2]

A light bulb blinks at 12 noon. It then blinks after 4 seconds, then after 8 seconds, then after 12 seconds and so on. How many times shall it have already blinked until it blinks at12.08pm?s

A. If the light blinks after every 4 seconds:

From 12 noon until 12:08 pm, there are 8 min = 8*60=480 seconds. If the light blinks every 4 seconds including at 12 noon, it would blink a total of 1+480/4 times=121 times.

B. If the light blinks after 4 sec, 8 sec, 12 sec..etc, it means the light blinks at

t=0, 0+4, 0+4+8, 0+4+8+12..etc..

At some point, the sum should exceed 480 seconds, and the blinking would stop. The number of terms in the sum is the number of times it blinks. Let n+1 be the number of terms in the sum.

Thus, 0+4+8+12+...=0+4(1+2+3+...n)=0+4*n(n+1)/2=0+2*n*(n+1)=480=>n^2+n=240=>n^2+n-240=0. Solving the quadratic equation gives two values, n=(-1+-sqrt(1+4*240))/(2)=15 and another negative value for n. Negative doesn't make sense. Hence the bulb will blink 15+1=16 times.

Note: 1+2+3+4+...+n=n*(n+1)/2. This is a standard formula. Just like 1^2+2^2+3^2+4^2+....n^2=n*(n+1)*(2n+1)/6. You donot need the second formula, we just used the first one.

Cheers,
Sai.