Kylie wanted to make bread pudding with lemon sauce for dessert for her family. however, she couldn't find a recipe that would make four servings. she decided to use a recipe for eight servings. the recipe for eight called for 1 1/3 cups of raisins. how many cups of raisins will she need for four people?

First, we need to figure out what portion of the recipe we should make. The recipe makes 8 servings, but we only want 4 servings. 4/8 simplifies to 1/2 (if you divide both the numerator and the denominator by the GCF of 4), so we need to halve the recipe. Now, we must find 1/2 of 1 1/3, which is 1/2 * 1 1/3 (because the word "of" refers to multiplication in math). However, we cannot just multiply a fraction by a mixed number, we must first convert the mixed number to an improper fraction. To do this, we multiply the denominator by the whole number and add the numerator, while keeping the same denominator. This is modeled below:

1 1/3 = (3*1 + 1)/3 = (3+1)/3 = 4/3

If we substitute in our improper fraction value, we get:

1/2 * 1 1/3

1/2 * 4/3

Now, we are ready to multiply. We just multiply the two numerators together and the two denominators together.

(1 *4) / (3*2) = 4/6

To simplify, we must divide both the numerator and the denominator by their shared greatest common factor, or GCF. The greatest number that both 4 and 6 can be divided by is 2, so we divide both the numerator and denominator by this number, as follows:

(4/2) / (6/2) = 2/3

Therefore, your answer is 2/3 cups of raisins.

Hope this helps!

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

r and t.

both are 64 degrees.