Hi,
would you please check to see if I did this problem correctly?
A recipe for 2 dozen corn muffins calls for 3 cups of flour. The number of muffins varies directly with the amount of flour you use.
a. Write a direct variation equation for the relationship between the number of cups of flour and the number of muffins.
m = muffins
f = amount of flour
m = kf
2 = k3
k = 2/3
So, m = 2/3f
b. Estimate how many cups of flour are needed to make 6 dozen muffins.
6 = 2/3f
9 cups of flour are needed to make 6 dozen muffins
a. Your direct variation equation is correct. The relationship between the number of cups of flour (f) and the number of muffins (m) is given by:
m = (2/3)f
b. To estimate how many cups of flour are needed to make 6 dozen muffins, we can substitute m = 6 dozen = 6 x 12 = 72 muffins into the equation and solve for f:
72 = (2/3)f
To find f, we can multiply by the reciprocal of (2/3) to both sides of the equation:
72 × (3/2) = f
108 = f
Therefore, 108 cups of flour are needed to make 6 dozen muffins, or in other words, 9 cups of flour are needed to make 6 dozen muffins. Your estimation is correct.
To check if you did the problem correctly, let's work through it step by step:
a. Write a direct variation equation for the relationship between the number of cups of flour and the number of muffins.
In a direct variation, the two variables (in this case, the number of muffins and the amount of flour) are proportional to each other. We can represent this relationship using the equation:
m = kf
where "m" represents the number of muffins, "f" represents the amount of flour, and "k" is the constant of variation.
To find the value of "k," we can use the given information that for 2 dozen muffins, we need 3 cups of flour. Plugging these values into the equation, we get:
2 = k * 3
Solving for "k," divide both sides by 3:
k = 2/3
Now we can substitute this value back into the original equation:
m = (2/3)f
Therefore, your equation m = 2/3f is correct.
b. Estimate how many cups of flour are needed to make 6 dozen muffins.
To estimate the amount of flour needed to make 6 dozen muffins, we can use the equation we derived in part a:
m = (2/3)f
We know that we want to make 6 dozen muffins, which means we need 6 times the amount of muffins required for 1 dozen. Since 1 dozen is 12 muffins, 6 dozen would be:
6 dozen = 6 * 12 = 72 muffins
Now we can substitute this value into the equation and solve for "f":
72 = (2/3)f
To isolate "f," multiply both sides by 3/2:
72 * 3/2 = f
Simplifying:
108 = f
Therefore, you would need 108 cups of flour to make 6 dozen muffins. It seems like your estimate of 9 cups of flour is incorrect.
Please let me know if you need any further clarification or if you have any other questions!