The triangles are similar. what is the value of x? enter your answer in the box. x = two right triangles. the larger triangle has a long leg of 96 units, short leg of 28 units, and the hypotenuse is labeled 6 x plus 28. the smaller triangle has a long leg of 24 units, short leg of 7 units, and hypotenuse of 25 units.

The given right-angled triangles are similar. The value of x is 12 units.

Given information:

The triangles are similar.

There are two right triangles.

The larger triangle has a long leg of 96 units, a short leg of 28 units, and the hypotenuse is labeled .

The smaller triangle has a long leg of 24 units, a short leg of 7 units, and a hypotenuse of 25 units.

See the attached figure.

It is required to calculate the value of x.

Now, the triangles are similar and hence, their sides will be in the same ratio.

Use the ratio to calculate the value of x,

Therefore, the value of x is 12 units.

For more details, refer to the link:

link

Find the scale factor by dividing the long sides and short side:

96 / 24 = 4

28 / 7 = 4

This means the larger triangle is 4 times bigger than the smaller one.

Now use that scale factor to calculate the hypotenuse:

The length of the smaller hypotenuse is 25, scale that by 4 to get 25 * 4  =100

This means the larger hypotenuse needs to equal 100:

6x+28 = 100

Subtract 28 from each side:

6x = 72

Divide both sides by 6:

x = 72/6

x = 12

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)

=====================================

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

=====================================

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).