Question

Can someone please help me??? I will fan and medal!!

Answer 1

Wat u need help with

Answer 2

A gardener is mowing a 20-by-40-yd rectangle pasture using a diagonal pattern. He moves from one corner of the pasture to the corner diagonally opposite. What is the length of this pass with the mower? Give your answer in simplified form.

Answer 3

\[A.) 10\sqrt{20}\]
\[B.) 20\sqrt{2}\]
\[C.) 400\sqrt{5}\]
\[D.) 20\sqrt{5}\]

Answer 4

A good starting point would be drawing the picture. Then we know we're looking for that segment running diagonally from one corner to the opposite one, so we could draw that one in too. It gives us an idea of where everything is.

Answer 5

it look like this right?

Answer 6

Yep, that looks good to me.
So, to find the length of the diagonal line segment there, do you have any ideas of how it could be found?

Answer 7

umm \[a^2+b^2=c^2\]

Answer 8

Sounds like a good idea.
Just to fill in some details, we do have a rectangle figure, so those corners create right angles.
The diagonal produces two triangles with the opposite angle being the right angle.
Thus, Pythagorean theorem works!

Answer 9

okay so i would do
\[20^2+40^2\]

Answer 10

\( 20^2 + 40^2 = d^2 \)
This is calling the diagonal length d. Otherwise yes.
I mention this just because it is easy to forget that you need to take the a square root afterwards.

Answer 11

Forgetting to take the square root is probably the most frequent mistake I see with people working with right triangles!

Answer 12

okay so now to ad them
400+1600 = 2000
\[\sqrt{2000}\approx 44.7\]

Answer 13

I agree with that result, but from your answer choices you want to simplify \( \sqrt{2000} \) down in terms of a simplest radical form.

Answer 14

how do i do that?

Answer 15

Basically, you want to take 2000 and pull out any factors squared.
Like, you can see there is a factor of 100 there (10^2) because of the two zeros.
2000 = 20 * 100
So by properties of radicals, sqrt(2000) = sqrt(20 * 100) = sqrt(20) sqrt(100)
sqrt(100) simplifies to 10, and then we have 10 sqrt(20).
There is just one more factor that is squared and can be taken out. Do you see what I'm doing here and can figure the other factor out?
Just think squared numbers. \(3^2 = 9\), \(5^2 = 25\) , ...

Answer 16

ya I see the answer is A
\[10\sqrt{20}\] right?

Answer 17

not quite
that isn't completely simplified.
what are your factors of 20?

Answer 18

4 and 5

Answer 19

but 4 also factors down to 2*2, or 2^2, right?

Answer 20

ya

Answer 21

When we want to take the simplest radical form, we need to basically factor whatever is underneath and pull any expressions out of the radical that have an integer square root... that is, any factor which is one of those square numbers like 100=10^2 and 4=2^2.

2000 = 100 * 20 = 10^2 * 4 * 5 = 10^2 * 2^2 * 5 <-- as you told me was the factorization of 20.
Let's break apart the radical using those factors.

\( \sqrt{2000} = \sqrt{100 \times 20} = \sqrt{10^2} \times \sqrt{2^2 \times 5} = \sqrt{10^2} \times \sqrt{2^2} \times \sqrt{5} \)
The sqrt(10^2) and sqrt(2^2) are just taking inverse operations, thus you get 10 and 2, respectively.

Answer 22

I can't figure out which one it would be :/

Answer 23

Would you agree that \( \sqrt{ a^2} \) = a ?

Answer 24

ya

Answer 25

@JorrdannAshleyy

Answer 26

@Xmoses1 can u help me

Answer 27

It seems that @AccessDenied has this under control

Answer 28

Do you remember your properties of radicals as well?
\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)

Answer 29

no i dont

Answer 30

Ahh. Okay.
The properties of exponents?
\( (a b) ^x = a^x b^x \)
Radicals are just a strange extension of the exponent properties.

Answer 31

okay i remember those

Answer 32

Radicals are literally considered the "fractional" exponents of Math
Like these two expressions are the same value
\( \sqrt{2}\)
and \(\left(2\right)^{1/2} \)
If that doesn't seem obvious, think about how
\( \sqrt{a^2} = a \),
what if we had wrote \( \left(a^2 \right) ^{1/2} \), which is the exponent property power to a power? the powers multiply together...
\( \large a^{1/2 \times 2} = a \)