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.

Malice

I recently finished a part graphic novel, part comic book called Malice, by Chris Wooding. So basically, this book is a part of a series, so I won't be able to say a whole lot. Malice takes place in England, and there is a comic book of the same name. What's very weird is that it is possible to end up in the comic, if you take certain objects (six items as I recall), and say the words, "Tall Jake take me away" six times, then a tall, dark guy (who is Tall Jake) will then appear when you're alone, and then take you to the world, Malice. Malice is a realm that lots of kids in the real world have heard about, but very, very few survive their first night (what I have read about in Malice shows that this world is much, much more different than our world. For starters, there are two tickets, black and white. Black tickets can take you anywhere throughout parts of Malice (yes, when it comes to "parts of Malice", you may think of that movie Zootopia), and white tickets are able to take you throughout parts of Malice, AND back home. But there are the consequences. Perhaps you may be wondering, all those kids who found white tickets and could get home, why can't they just tell someone about what had happened? As it turns out, the survivors lose memory of everything that had previously occurred. Malice also has its hazards, such as monsters called "Chitters", mechanical fiends that can "drink" time out of you. So by that, I mean make you a whole lot older). Seth and Kady, the main characters of the story think this whole thing is a really stupid myth, but after their friend Luke disappears and suddenly appears in the newest issue of Malice, things don't seem so silly after all. However, after they both wind up in Malice themselves, the two realize they are not alone. There are groups opposing Tall Jake, one of which is called Havoc. Will Seth and Kady be able to survive in Malice? Read to find out.