Question

Ari is filling a one-man hot-air balloon. He starts with only 3 feet3 of air in the balloon and observes that the volume doubles every minute. What equation can be used to find the number of minutes, m, it will take for the volume to reach a capacity of 24,576 feet3 of air?

Answer 1

`24576=3(2)^(m-1)`

Explanation:

Given : Ari is filling a one-man hot-air balloon. He starts with only 3 cubic feet of air in the balloon and observes that the volume doubles every minute.

To Find: What equation can be used to find the number of minutes, m, it will take for the volume to reach a capacity of 24,576 cubic feet of air?

Solution:

Ari starts with only 3 cubic feet of air in the balloon and observes that the volume doubles every minute.

Let m be the number of minutes

The given situation forms G.P.

Formula of nth term in G.P. =
`a_n=ar^(m-1)`

a is the first term = 3 cubic feet

r is the common ratio =2

We are required to find equation can be used to find the number of minutes, m, it will take for the volume to reach a capacity of 24,576 feet3 of air

So,
`24576=3(2)^(m-1)`

Hence equation can be used to find the number of minutes, m, it will take for the volume to reach a capacity of 24,576 cubic feet of air is
`24576=3(2)^(m-1)`

Answer 2

13

Explanation:

Let's start by creating the beginning of the data sample, knowing the volume doubles every minute.

Min 0 = 3

Min 1 = 6

Min 2 = 12

Min 3 = 24

Min 4 = 48

If we look at the numbers, we see they are all divided by 3:

Min 0 = 3 / 3 = 1

Min 1 = 6 / 3 = 2

Min 2 = 12 / 3 = 4

Min 3 = 24 / 3 = 8

Min 4 = 48 / 3 = 16

We then see the result of the division by 3 is in fact the table of multiplication by 2.

So, by inverting the operation, we can see the volume of air for any given minute starting from now is 3 times 2 at the power of the minute (m).

`x = 3 * 2^(m)`

We then have to isolate m knowing x.

`24576 = 3 * 2^(m)`

`8192= 2^(m)`

Which power of 2 = 8192? 13 (0=1,2,4,8,16,32,64,128,256,512,1024,2048,4096,13=8192)