Otto used 5.5 cups of whole wheat flour and x cups of white flour in the recipe. What is the equation that can be used to find t
1 answer:
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Answer:
5a. approximately 6 grams remain after 2 seconds
5b. The graph shown cannot be a solution. The solution has negative slope everywhere.
5c. y = 50/(t+5)
5d. The amount is changing at a decreasing rate. (As y gets smaller, so does the magnitude of dy/dt.)
Step-by-step explanation:
5a. The tangent line has the equation ...
y = f'(0)t +f(0)
Here, that is
y = -0.02·10²·t +10 = 10 -2t
Then at t=2, the value is ...
y = 10 -2·2 = 6 . . . . grams remaining
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5b. y² is always positive (or zero), so -0.02y² will be negative. This is dy/dt, the slope of the curve with respect to time, so any curve with positive slope somewhere cannot be a solution.
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5c. The equation is separable so can be solved by integrating ...
∫y^-2·dy = -0.02∫dt
-y^-1 = -0.02t +c . . . . for some arbitrary constant c
Multiplying by -50 gives ...
50/y = t + c . . . . for some constant c
We can find the value of c by invoking the initial condition. At t=0, y=10, so we have ...
50/10 = 0 +c = 5
Then, solving for y, we get ...
y = 50/(t+5)
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5d. As noted above (and as described by the differential equation), the magnitude of the rate of change is proportional to the square of y. As y decreases, its rate of change will also decrease (faster). You can see that the curve for y flattens out as t increases. The amount of the substance is changing at a decreasing rate.
To figure out which part has the higher concentration of almonds, we want to see how many almonds are there for every part in each recipe. For the first recpe, we have 2 parts almond for ever 5 total parts, and in the second part, we have 3 parts almonds for every 7 total parts, so we can represent them as the fractions:
So, which one is greater? Well, let's say they both make 35 total batches (36 is the least commun multiple of the two denominators) that would give them:
So we have Deon's recipe as the large concentration of raisins.