Two Boys are building a skateboard ramp. They want the ramp to be 3 feet tall and 5 feet long. One of them thinks that they can use a piece of plywood that is 4 feet long to cover the top of the frame. This is one of theirs work.

a2 + b2 = c2
32 + b2 = 52
9 + b2 = 25
b2 = 16
b = 16−−√
b = 4

Why is his reasoning incorrect? Explain.

How long does the piece of plywood have to be to cover the top of the frame of the skateboard ramp? Round your answer to the nearest tenth of a foot. Show your work.

Another boys parents bought a 10 foot piece of plywood for him and his friend to use to cover the top of the frame. Draw and label a ramp they could create. Justify your work using the Pythagorean Theorem.
Draw

The answer is below

Step-by-step explanation:

A triangle is a polygon with three sides and three angles. Types of triangles are acute, obtuse, scalene, isosceles, equilateral and right angled triangle.

Pythagoras theorem states that for a right angled triangle, the square of the hypotenuse is equal to the sum of the square of the remaining two sides.

The ramp has the form of a triangle. The height of the ramp is the vertical side of a triangle, the top of the frame is the hypotenuse while the length of the ramp is the base side of the triangle.

Let the hypotenuse be b Hence:

b² = 3² + 5²

b² = 9 + 25

b² = 34

b = 5.83 feet

step-by-step explanation:

milk=16 large

soda=280 kids

juice box=82 regular

63 soda +63 regular

124 large total

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).