A group of student volunteers at the animal shelter is helping feed the dogs. Each day, small dogs eat 1/2 cup less dog food than medium-sized dogs, and large dogs eat 3/4 cup more dog food than the medium-sized dogs. The shelter currently has 22 small dogs, 14 medium-sized dogs, and 8 large dogs.
a. Write an expression for the total amount of dog food the dogs eat each day. Explain what your variable represents and what each term in your expression represents.
b. The dogs eat 72 cups of dog food today. How much food was given to each dog?
(a) Let m = the amount a medium dog eats
so the total amount eaten by all the dogs is
22(m - 1/2) + 14m + 8(m + 3/4)
(b) so now solve for m in
22(m - 1/2) + 14m + 8(m + 3/4) = 72
and then you can find the amounts per dog size
a. Let's define our variables:
- S = number of small dogs
- M = number of medium-sized dogs
- L = number of large dogs
Each small dog eats 1/2 cup less than the medium-sized dogs, so the amount of dog food eaten by the small dogs would be (1/2)M.
Each large dog eats 3/4 cup more than the medium-sized dogs, so the amount of dog food eaten by the large dogs would be M + (3/4)M = (7/4)M.
So, the expression for the total amount of dog food the dogs eat each day is:
Total amount of dog food = (S * (1/2)M) + (M * M) + (L * (7/4)M)
b. We are given that the total amount of dog food eaten today is 72 cups. So we can substitute this value into the expression for the total amount of dog food and solve for M:
72 = (22 * (1/2)M) + (14 * M) + (8 * (7/4)M)
Simplifying the expression:
72 = 11M + 14M + 14M
72 = 39M
Now we can solve for M:
M = 72/39
M = 1.85
To find the amount of food given to each dog, substitute the value of M back into our expressions:
Amount given to small dogs = (1/2) * 1.85 = 0.925 cups
Amount given to medium-sized dogs = 1.85 cups
Amount given to large dogs = 1.85 + (3/4) * 1.85 = 3.209375 cups
a. To write an expression for the total amount of dog food the dogs eat each day, we need to consider the different amounts eaten by small, medium-sized, and large dogs.
Let's use the variable "x" to represent the amount of dog food eaten by medium-sized dogs.
For small dogs, it is given that they eat 1/2 cup less than medium-sized dogs. So the amount of dog food eaten by small dogs would be (x - 1/2).
For large dogs, it is given that they eat 3/4 cup more than medium-sized dogs. So the amount of dog food eaten by large dogs would be (x + 3/4).
Now, we can calculate the total amount of dog food eaten each day by all the dogs by adding together the amount eaten by small, medium-sized, and large dogs:
Total dog food eaten = (amount eaten by small dogs) + (amount eaten by medium-sized dogs) + (amount eaten by large dogs)
= (x - 1/2) + x + (x + 3/4)
In this expression, the variable "x" represents the amount of dog food eaten by the medium-sized dogs. Each term in the expression represents the amount of dog food eaten by each type of dog: (x - 1/2) represents the amount eaten by small dogs, x represents the amount eaten by medium-sized dogs, and (x + 3/4) represents the amount eaten by large dogs.
b. Now, to find how much food was given to each dog when the total dog food eaten is 72 cups, we need to solve the equation:
( x - 1/2 + x + x + 3/4 ) = 72
Combining like terms, we get:
3x + 1/4 = 72
To isolate the variable, subtract 1/4 from both sides:
3x = 72 - 1/4
Simplifying:
3x = 287/4
Divide both sides by 3:
x = (287/4) / 3
Simplifying:
x = 71/4
Now, we can substitute this value of x back into the expression for each type of dog:
Amount eaten by small dogs = x - 1/2 = (71/4) - 1/2 = 69/4
Amount eaten by medium-sized dogs = x = 71/4
Amount eaten by large dogs = x + 3/4 = (71/4) + 3/4 = 74/4
So, each small dog would eat 69/4 cups of dog food, each medium-sized dog would eat 71/4 cups of dog food, and each large dog would eat 74/4 cups of dog food.