Thursday, July 28, 2016

How to do stuff in Math

Either enter an exact answer in terms of pi or use 3, point, 14 for pi and enter your answer as a decimal.

16π
 unitsstart superscript, 2, end superscript
Hint #1

Getting started

We want to find the area of start fraction, 1, divided by, 4, end fraction of a circle. Let's start by finding the area of the full circle.
Hint #2

Step 1: Find the area of the full circle

πr2π8264π
Hint #3

Step 2: Find the area of start fraction, 1, divided by, 4, end fraction of the circle

141464π644π16π
Hint #4

The answer

The area of the shape is 16, pi unitsstart superscript, 2, end superscript.
(Note that we could also multiply 16 by 3, point, 14 to get  50, point, 24 unitsstart superscript, 2, end superscript.)


Find the arc length of the semicircle.
Either enter an exact answer in terms of pi or use 3, point, 14 for pi and enter your answer as a decimal.

 units

Hint #1

Getting started

We want to find the arc length of start fraction, 1, divided by, 2, end fraction of a circle. Let's start by finding the circumference of the circle.
Hint #2

Step 1: Find the circumference of the circle

2πr2π11112π22π
Hint #3

Step 2: Find the arc length of start fraction, 1, divided by, 2, end fraction of the circle

121222π222π11π
Hint #4

The answer

The arc length of the semicircle is 11, pi units.
(Note that we could also multiply 11 by 3, point, 14 to get 34, point, 54 units.)
Justin lives in Saint Paul and goes to school in Minneapolis. In the morning, he has 3 transportation options (bus, cab, or train) to school, and in the evening he has the same 3 choices for his trip home.
If Justin randomly chooses his ride in the morning and in the evening, what is the probability that he'll use both the bus and the train?



Hint #11 / 4
P, r, o, b, a, b, i, l, i, t, y, equals, start fraction, F, a, v, o, r, a, b, l, e, space, c, o, m, b, i, n, a, t, i, o, n, s, divided by, T, o, t, a, l, space, p, o, s, s, i, b, l, e, space, c, o, m, b, i, n, a, t, i, o, n, s, end fraction
Hint #22 / 4
There are start color blue, 3, end color blue possible choices for each trip, so there are start color blue, 3, end color blue, times, start color blue, 3, end color blue, equals, 9 total possible combinations. If Justin chooses randomly, all combinations are equally likely.
Hint #33 / 4
Each path through the tree represents one possible outcome. The green paths show the start color green, 2, end color green favorable outcomes.

\text{B}\text{C}\text{T}\text{Ride to school}\text{B}\text{C}\text{T}\text{B}\text{C}\text{T}\text{B}\text{C}\text{T}\text{Ride home}
Hint #44 / 4
The probability that Justin will use both the bus and the train is 2 out of 9, or start fraction, 2, divided by, 9, end fraction .


If you flip three fair coins, what is the probability that you'll get heads on the first two flips and tails on the last flip?


Hint #11 / 4
P, r, o, b, a, b, i, l, i, t, y, equals, start fraction, F, a, v, o, r, a, b, l, e, space, o, u, t, c, o, m, e, s, divided by, T, o, t, a, l, space, p, o, s, s, i, b, l, e, space, o, u, t, c, o, m, e, s, end fraction
Hint #22 / 4
If you flip three coins, there are start color blue, 2, end color blue possible outcomes for each individual flip, so there are start color blue, 2, end color blue, times, start color blue, 2, end color blue, times, start color blue, 2, end color blue, equals, 8 total possible outcomes. Since the coin is fair, each outcome is equally likely.
Hint #33 / 4
Each path through the tree represents one outcome. The green path shows the start color green, 1, end color green favorable outcome.

\text{H}\text{T}\text{First}\text{H}\text{T}\text{H}\text{T}\text{Second}\text{H}\text{T}\text{H}\text{T}\text{T}\text{H}\text{T}\text{H}\text{Third}
Hint #44 / 4
The probability of getting heads on the first two flips and tails on the last flip is start fraction, 1, divided by, 8, end fraction .

Josh is hiking Glacier National Park. He has now hiked a total of 17, space, k, m and is 2, space, k, m short of being start fraction, 1, divided by, 2, end fraction of the way done with his hike.
Write an equation to determine the total length in kilometers left parenthesis, h, right parenthesis of Josh's hike.
Find the total length of Josh's hike.
 k, m

Hint #11 / 5
Let h be the total length of Josh's hike.
Hint #22 / 5
start fraction, 1, divided by, 2, end fraction of Josh's hike is equal to start fraction, 1, divided by, 2, end fraction, h, space, k, m. He is 2, space, k, m short of being done with start fraction, 1, divided by, 2, end fraction of his hike.
The distance Josh has hiked already is start fraction, 1, divided by, 2, end fraction, h, minus, 2.
Hint #33 / 5
Since the distance he has already hiked is 17, space, k, m, let's set this equal to 17:
space, start fraction, 1, divided by, 2, end fraction, h, minus, 2, equals, 17
Now, let's solve the equation to find the total length of Josh's hike left parenthesis, h, right parenthesis.
Hint #44 / 5
12h212h2+212h12h12h=17=17+2=19=1912=38add 2 to each sidedivide each side by 12
Hint #55 / 5
The equation is start fraction, 1, divided by, 2, end fraction, h, minus, 2, equals, 17.
The total length of Josh's hike is 38, space, k, m.