Math Is Fun Forum

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Shortest Radius Setting

An adjustable water sprinkler that sprays water in a circular pattern is placed at the center of a square field whose area is 1250 square feet. What is the shortest radius setting that can be used if the field is to be completely enclosed within the circle?

I will need the area of a square formula: A = (side)^2.
I know that A = 1250 (feet)^2.

1250 = s^2

sqrt{1250} = sqrt{s^2}

35.3553 = s

So, the side of the square is about 35 feet.

I need the hypotenuse of the square.

a^2 + b^2 = c^2

(35)^2 + (35)^2 = c^2

1225 + 1225 = c^2

2450 = c^2

sqrt{2450} = sqrt{c^2}

49.49747468306 = c

Let c = hypotenuse = diagonal of square.

To find the radius distance, I divide c by 2.

So, c/2 = 24.7487. If I round this decimal number to the nearest first place, I get 25 feet.

I say the shortest radius setting is about 25 feet.

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Shortest Radius Setting

Yes, that's right.

At this point

35.3553 = s

So, the side of the square is about 35 feet.

you make an approximation and then use this to continue.

At the end you make another to reach the answer.

It's better to avoid approximations like this:

Say s =  √ 1250 and leave it like this.

Then c^2 = ( √ 1250 )^2 + ( √ 1250 )^2 = 1250 + 1250 exactly.  So there's no need to make the two approximations.  The answer is exactly 25.

Your method is OK but I used a quicker method that led to 25 directly:

Sketch the square and draw its diagonals, splitting the square into four triangles. The distance from the centre of the square to any vertex is the radius we want so call it x.

area of a triangle = half base times height = 0.5 times x^2

Four triangles makes the square so we have the equation

If you want to really 'show off' you can say that -25 also fits the equation but is not an answer as a positive answer is needed.

This business of rounding part way through a problem can lead to very inaccurate answers. here's an example:

Given x^2 = 56 work out x^5

On a calculator x = 7.483314774...

If I call that 7.5 and work out x^5 I get 23730.46875

If I use the full value I get 23467.67513.....

You can see that using the rounded up number and multiplying five times leads to an answer that is quite a bit too big.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

e_jane_aran

Member

Registered: 2023-09-15

Posts: 10

Re: Shortest Radius Setting

This business of rounding part way through a problem can lead to very inaccurate answers. here's an example:

Given x^2 = 56 work out x^5

On a calculator x = 7.483314774...

If I call that 7.5 and work out x^5 I get 23730.46875

If I use the full value I get 23467.67513.....

You can see that using the rounded up number and multiplying five times leads to an answer that is quite a bit too big.

Yes! Amen! Wait to round until the end!

---

"They are fast. Faster than you can believe. Don't turn your back. Don't look away. And most of all, don't blink." -the 10th Doctor

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Shortest Radius Setting

Yes, that's right.

At this point

35.3553 = s

So, the side of the square is about 35 feet.

you make an approximation and then use this to continue.

At the end you make another to reach the answer.

It's better to avoid approximations like this:

Say s =  √ 1250 and leave it like this.

Then c^2 = ( √ 1250 )^2 + ( √ 1250 )^2 = 1250 + 1250 exactly.  So there's no need to make the two approximations.  The answer is exactly 25.

Your method is OK but I used a quicker method that led to 25 directly:

Sketch the square and draw its diagonals, splitting the square into four triangles. The distance from the centre of the square to any vertex is the radius we want so call it x.

area of a triangle = half base times height = 0.5 times x^2

Four triangles makes the square so we have the equation

If you want to really 'show off' you can say that -25 also fits the equation but is not an answer as a positive answer is needed.

This business of rounding part way through a problem can lead to very inaccurate answers. here's an example:

Given x^2 = 56 work out x^5

On a calculator x = 7.483314774...

If I call that 7.5 and work out x^5 I get 23730.46875

If I use the full value I get 23467.67513.....

You can see that using the rounded up number and multiplying five times leads to an answer that is quite a bit too big.

Bob

I didn't think I would get the right answer. I simply tried to do my best work. Your method leading to 25 feet is pretty cool.

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Shortest Radius Setting

This business of rounding part way through a problem can lead to very inaccurate answers. here's an example:

Given x^2 = 56 work out x^5

On a calculator x = 7.483314774...

If I call that 7.5 and work out x^5 I get 23730.46875

If I use the full value I get 23467.67513.....

You can see that using the rounded up number and multiplying five times leads to an answer that is quite a bit too big.

Yes! Amen! Wait to round until the end!

I get it now. I found this problem to be uniquely interesting.