The initial amount of sugar in fermenting a batch of beer is 5.5 grams. Over time the sugar decreases at a rate of 0.1grams per day.
a.What is the amount of sugar that remains after 7 days? 23 days?
- I got: 5.5-0.1x= A
- 5.5-0.1(7)= 4.8
- 5.5-0.1(23)= 3.2
b. If A represents the amount of sugar, write an equation for the amount of sugar that remains after x days?
-I got: 5.5-0.1x= A
c. Verify your equation is correct by taking the two ordered pairs from part (a) and finding the equation of the line that contains them
d. graph (don't worry about)
e. Algebraically and graphically, find the time it will take for the amount of sugar to drop below 2.8grams
f. How long will it take for the sugar to be completely gone?
I got a and b but kind of got stuck on the rest, thanks for the help in advance
c) if x=7 , A = 4.8 ---> (7,4.8)
if x = 23, A = 3.2 ---> (23, 2.3)
slope = (2.3 - 4.8)/(23-7)
= -.1
so A = -.1x + b
for one of the points, say (7 , 4.8)
4.8 = -.1(7) + b
b = 4.8 + .1(7) = 55
so A = -.1x + 55
your equation from b) has been verified.
e.
2.8 = -.1x + 55
.1x = 55 - 2.8 = 52.2
x = 522 days
f) you want A = 0
0 = -.1x + 55
.1x = 55
x = 55/.1 = 550 days
Thanks!
Let's break down each part of the question:
a. To find the amount of sugar that remains after a certain number of days, we can use the equation: A = 5.5 - 0.1x, where A is the amount of sugar and x is the number of days.
For 7 days:
A = 5.5 - 0.1(7) = 5.5 - 0.7 = 4.8 grams
For 23 days:
A = 5.5 - 0.1(23) = 5.5 - 2.3 = 3.2 grams
So after 7 days, 4.8 grams of sugar remains, and after 23 days, 3.2 grams of sugar remains.
b. The equation for the amount of sugar that remains after x days is: A = 5.5 - 0.1x, where A is the amount of sugar and x is the number of days. You already obtained this correctly.
c. To verify the equation is correct, we can substitute the two ordered pairs from part (a) into the equation and check if it holds true.
For (7, 4.8):
A = 5.5 - 0.1(7) = 4.8
For (23, 3.2):
A = 5.5 - 0.1(23) = 3.2
In both cases, the equation is true, so it is verified.
d. (Skipping graphing)
e. To find the time it will take for the amount of sugar to drop below 2.8 grams, we can set up the inequality 5.5 - 0.1x < 2.8 and solve for x.
5.5 - 0.1x < 2.8
0.1x > 5.5 - 2.8
0.1x > 2.7
x > 27
Since the rate of decrease is 0.1 grams per day, it will take more than 27 days for the amount of sugar to drop below 2.8 grams.
f. To find how long it will take for the sugar to be completely gone, we need to find when the amount of sugar, A, becomes 0. So we solve the equation:
5.5 - 0.1x = 0
0.1x = 5.5
x = 5.5 / 0.1
x = 55
It will take 55 days for the sugar to be completely gone.