the population of a bacteria coleny, p, is doubling every day,d, according to the equation p=2^d. five days later, the population of another bacteria colny, q, is quadrupling everyday,d, according to the equation q=4^d-5. when will the populations of the two bacteria be equal?
what does this question mean explain plz im sooo lost :( do i put 10 for p=2^d --> p=2^10?
I understand that the question might seem confusing at first. Let's break it down step by step.
First, we have two equations that describe the population growth of two different bacteria colonies:
1. For population p: p = 2^d
2. For population q: q = 4^(d-5)
In these equations, the variable "d" represents the number of days. So, when we substitute a specific value for "d," we can calculate the population at that particular day.
Now, the question asks when the populations of the two bacteria colonies will be equal. To find this, we need to set the two equations equal to each other and then solve for "d." It means that we are looking for a specific day when both populations are the same.
Let's set up the equation:
p = q
Substituting the equations for p and q:
2^d = 4^(d-5)
To solve this equation, we need to use the properties of exponents. Here's how we can proceed:
1. Start by converting both sides of the equation to the same base. Let's convert 2^d to exponential form using base 4:
(2^2)^d = 4^(d-5)
2. Next, simplify the left side of the equation:
4^d = 4^(d-5)
3. Since the bases are the same, we can equate the exponents:
d = d - 5
4. Subtract "d" from both sides of the equation:
0 = -5
As you can see, the equation leads to an inconsistency where 0 is equal to -5. Therefore, the populations of the two bacteria colonies will never be equal.
To summarize, no matter what value of "d" we substitute, the populations of the two bacteria colonies described by the equations p = 2^d and q = 4^(d-5) will never be equal.