How to do stuff in Math

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

16π

 units${}^2$start superscript, 2, end superscript

Hint #1

Getting started

We want to find the area of $\frac{1}{4}$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

$\begin{array}{cc}& \pi r^2\\ \\ & \pi 8^2\\ \\ & 64\pi \\ \\ \end{array}$

Hint #3

Step 2: Find the area of $\frac{1}{4}$start fraction, 1, divided by, 4, end fraction of the circle

$\begin{array}{cc}\frac{1}{4}& \frac{1}{4}64\pi \\ \\ & \frac{64}{4}\pi \\ \\ & 16\pi \\ \\ \end{array}$

Hint #4

The answer

The area of the shape is $16\pi$16, pi units${}^2$start superscript, 2, end superscript.

(Note that we could also multiply $16$16 by $3\mathrm{.}14$3, point, 14 to get  $50\mathrm{.}24$50, point, 24 units${}^2$start superscript, 2, end superscript.)

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

 units

Hint #1

Getting started

We want to find the arc length of $\frac{1}{2}$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

$\begin{array}{cc}& 2\pi r\\ \\ & 2\pi 11\\ \\ & 112\pi \\ \\ & 22\pi \\ \\ \end{array}$

Hint #3

Step 2: Find the arc length of $\frac{1}{2}$start fraction, 1, divided by, 2, end fraction of the circle

$\begin{array}{cc}\frac{1}{2}& \frac{1}{2}22\pi \\ \\ & \frac{22}{2}\pi \\ \\ & 11\pi \\ \\ \end{array}$

Hint #4

The answer

The arc length of the semicircle is $11\pi$11, pi units.

(Note that we could also multiply $11$11 by $3\mathrm{.}14$3, point, 14 to get $34\mathrm{.}54$34, point, 54 units.)

Justin lives in Saint Paul and goes to school in Minneapolis. In the morning, he has $3$3 transportation options (bus, cab, or train) to school, and in the evening he has the same $3$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

$\text{Probability} = \dfrac{\text{Favorable combinations}}{\text{Total possible combinations}}$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 $\blue3$start color blue, 3, end color blue possible choices for each trip, so there are $\blue3\times\blue3=9$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 $\green2$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$2 out of $9$9, or $\dfrac29$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

$\text{Probability} = \dfrac{\text{Favorable outcomes}}{\text{Total possible outcomes}}$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 $\blue2$start color blue, 2, end color blue possible outcomes for each individual flip, so there are $\blue2\times\blue2\times\blue2=8$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 $\green1$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 $\dfrac18$start fraction, 1, divided by, 8, end fraction .

Josh is hiking Glacier National Park. He has now hiked a total of $17 \text{ km}$17, space, k, m and is $2 \text{ km}$2, space, k, m short of being $\dfrac12$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 $(h)$left parenthesis, h, right parenthesis of Josh's hike.

Find the total length of Josh's hike.

 $\text{km}$k, m

Hint #11 / 5

Let $h$h be the total length of Josh's hike.

Hint #22 / 5

$\dfrac12$start fraction, 1, divided by, 2, end fraction of Josh's hike is equal to $\dfrac12h \text{ km}$start fraction, 1, divided by, 2, end fraction, h, space, k, m. He is $2\text{ km}$2, space, k, m short of being done with $\dfrac12$start fraction, 1, divided by, 2, end fraction of his hike.

The distance Josh has hiked already is $\dfrac12h - 2$start fraction, 1, divided by, 2, end fraction, h, minus, 2.

Hint #33 / 5

Since the distance he has already hiked is $17 \text{ km}$17, space, k, m, let's set this equal to $17$17:

$\qquad \dfrac12h - 2=17$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 $(h)$left parenthesis, h, right parenthesis.

Hint #44 / 5

12h−212h−2+212h12h12h=17=17+2=19=1912=38add 2 to each sidedivide each side by 12

Hint #55 / 5

The equation is $\dfrac12h - 2=17$start fraction, 1, divided by, 2, end fraction, h, minus, 2, equals, 17.

The total length of Josh's hike is $38 \text{ km}$38, space, k, m.