An experiment involving learning in animals requires placing white mice and rabbits into separate, controlled environments, environment I and environment II. The maximum amount of time available in environment I is 500 minutes, and the maximum amour of time available in environment II is 600 minutes. The white mice must spend 10 minutes in environment I and 25 minutes in environment II, and the rabbits must spend 15 minutes in environment I and 15 minutes in environment II. Find the maximum possible number of animals that can be used in the experiment, and find the number of white mice and the number of rabbits that can be used.
Aubrey and Charlie are driving to a city that is 120 mi from their house. They have already traveled 20 mi, and they are driving at a constant rate of 50 mi/h. Complete the function that models the distance they drive as a function of time. Then complete a reasonable domain for this situation.
To find the maximum possible number of animals that can be used in the experiment, we need to divide the maximum amount of time available in each environment by the time each animal needs to spend in that environment.
For environment I:
Maximum time available: 500 minutes
Time needed for white mice: 10 minutes
Time needed for rabbits: 15 minutes
Maximum number of white mice = 500 / 10 = 50 white mice
Maximum number of rabbits = 500 / 15 = 33.33 rabbits
For environment II:
Maximum time available: 600 minutes
Time needed for white mice: 25 minutes
Time needed for rabbits: 15 minutes
Maximum number of white mice = 600 / 25 = 24 white mice
Maximum number of rabbits = 600 / 15 = 40 rabbits
To determine the maximum possible number of animals, we take the minimum of the calculated maximum numbers in each environment. This is because we need to ensure that the animals can fit within both environments.
The maximum number of white mice that can be used in the experiment is 24, and the maximum number of rabbits is 33.
To find the maximum possible number of animals that can be used in the experiment, we need to determine the limiting factor, which is the environment with less available time.
Let x represent the number of animals, which includes both white mice and rabbits.
For the mice, each mouse needs 10 minutes in environment I and 25 minutes in environment II. So, the total time required for the mice can be calculated as 10x minutes in environment I and 25x minutes in environment II.
For the rabbits, each rabbit needs 15 minutes in environment I and 15 minutes in environment II. So, the total time required for the rabbits can be calculated as 15x minutes in environment I and 15x minutes in environment II.
Since the maximum time available in environment I is 500 minutes and environment II is 600 minutes, we can set up the following inequalities:
10x ≤ 500 (for the mice in environment I)
15x + 10x ≤ 600 (for the mice in environment II and the rabbits in environment I)
Simplifying these inequalities, we get:
10x ≤ 500
25x + 15x ≤ 600
Solving the first inequality, we have:
10x ≤ 500
x ≤ 50 (by dividing both sides by 10)
So, we know that the maximum number of animals, x, is 50.
Next, let's find the number of white mice and rabbits that can be used.
From the second inequality, we have:
25x + 15x ≤ 600
40x ≤ 600
x ≤ 15 (by dividing both sides by 40)
Since x represents the total number of animals, we can subtract the number of white mice from x to find the number of rabbits.
Let's assume y represents the number of white mice.
So, the number of rabbits would be x - y.
Therefore, we have:
x - y = 15
50 - y = 15
y = 50 - 15 = 35
Thus, we can use a maximum of 50 animals in the experiment, including 35 white mice and 15 rabbits.
working all those words into numbers, we have
maximize m+r subject to
10m + 15r <= 500
25m + 15r <= 600
I see that we have 15m <= 100
so m <= 6
Given that, if we choose only 1 mouse, we have
15r <= 490
15r <= 575
so, r <= 32
max achieved at 6 mice and 29 rabbits