Andrea is buying some new shirts and sweaters. she is able to buy 3 shirts and 2 sweaters for $114, or she is able to buy 2 shirts and 4 sweaters for $164. how much does a shirt cost? how much does a sweater cost?

Shirt $ 16

Sweater $ 33

Step-by-step explanation:

Let the cost of each shirt is $ x, and the cost of each sweater is $ y.

Since, cost of each shirt is $ x, the cost of 3 shirts will be $ 3x and cost of 2 shorts will be $ 2x.

Likewise, cost of each sweater is $ y, so the cost of 2 sweaters will be $ 2y and cost of 4 sweaters will be $ 4y.

Andrea was able to buy 3 shirts and 2 sweaters for $114. This means:

Cost of 3 shirts + Cost of 2 sweaters  = 114

3x + 2y = 114                                            Equation 1

She was able to buy 2 shirts and 4 sweaters for $164. This means:

2x + 4y = 164

Dividing both sides by 2, we get:

x + 2y = 82                                              Equation 2

Subtracting Equation 2 from Equation 1, we get:

3x + 2y - (x + 2y) = 114 - 82

3x + 2y - x - 2y = 32

2x = 32

x = 16

Using the value of x, in Equation 2, we get:

16 + 2y = 82

2y = 66

y = 33

Since, x represents the cost of shirt, the cost of each shirt is $ 16.

y represents the cost of sweater, so the cost of each sweater is $ 33.

for b its (t) and c its (d)

step-by-step explanation:

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).