<span>Maximum area = sqrt(3)/8
Let's first express the width of the triangle as a function of it's height.
If you draw an equilateral triangle, then a rectangle using one of the triangles edges as the base, you'll see that there's 4 regions created. They are the rectangle, a smaller equilateral triangle above the rectangle, and 2 right triangles with one leg being the height of the rectangle and the other 2 angles being 30 and 60 degrees. Let's call the short leg of that triangle b. And that makes the width of the rectangle equal to 1 minus twice b. So we have
w = 1 - 2b
b = h/sqrt(3)
So
w = 1 - 2*h/sqrt(3)
The area of the rectangle is
A = hw
A = h(1 - 2*h/sqrt(3))
A = h*1 - h*2*h/sqrt(3)
A = h - 2h^2/sqrt(3)
We now have a quadratic equation where A = -2/sqrt(3), b = 1, and c=0.
We can solve the problem by using a bit of calculus and calculating the first derivative, then solving for 0. But since this is a simple quadratic, we could also take advantage that a parabola is symmetrical and that the maximum value will be the midpoint between it's roots. So let's use the quadratic formula and solve it that way. The 2 roots are 0, and 1.5/sqrt(3).
The midpoint is
(0 + 1.5/sqrt(3))/2 = 1.5/sqrt(3) / 2 = 0.75/sqrt(3)
So the desired height is 0.75/sqrt(3).
Now let's calculate the width:
w = 1 - 2*h/sqrt(3)
w = 1 - 2* 0.75/sqrt(3) /sqrt(3)
w = 1 - 2* 0.75/3
w = 1 - 1.5/3
w = 1 - 0.5
w = 0.5
The area is
A = hw
A = 0.75/sqrt(3) * 0.5
A = 0.375/sqrt(3)
Now as I said earlier, we could use the first derivative. Let's do that as well and see what happens.
A = h - 2h^2/sqrt(3)
A' = 1h^0 - 4h/sqrt(3)
A' = 1 - 4h/sqrt(3)
Now solve for 0.
A' = 1 - 4h/sqrt(3)
0 = 1 - 4h/sqrt(3)
4h/sqrt(3) = 1
4h = sqrt(3)
h = sqrt(3)/4
w = 1 - 2*(sqrt(3)/4)/sqrt(3)
w = 1 - 2/4
w = 1 -1/2
w = 1/2
A = wh
A = 1/2 * sqrt(3)/4
A = sqrt(3)/8
And the other method got us 0.375/sqrt(3). Are they the same? Let's see.
0.375/sqrt(3)
Multiply top and bottom by sqrt(3)
0.375*sqrt(3)/3
Multiply top and bottom by 8
3*sqrt(3)/24
Divide top and bottom by 3
sqrt(3)/8
Yep, they're the same.
And since sqrt(3)/8 looks so much nicer than 0.375/sqrt(3), let's use that as the answer.</span>
Answer:
27.5%
Step-by-step explanation:
Given:
- 1 Liter In Teapot
- Alex poured 275 milliliters of the tea into a cup
Review:
- 1,000 Milliliters = 1 Liter
Concepts:
- A liter is a noun meaning a unit of capacity redefined in 1964 by a reduction of 28 parts in a million to be exactly equal to one <em>cubic </em>decimeter. It is equivalent to <em>1.0567</em> U.S. liquid <u>quarts </u>and is equal to the volume of 1 kilogram of distilled <em>water</em> at 4°C.
- A millimeter is a noun meaning a unit of capacity <em>equal</em> to one <u>thousandth </u>of a liter, and equivalent to <em>0.033815</em> fluid<u> </u><u>ounce</u>, or<em> 0.061025</em><em> </em>cubic<u> inch</u>.
Formula:
- The <em>mℓ</em> to <em>ℓ</em> formula is<em> [L] = [mL] / 1000</em>.
Now we must convert <em>275</em> milliliters to liters.
1. Divide the volume by <em>1,000.</em>
275 ÷ 1000 = 0.275
2. Add the Unit Symbol for Liter.
0.275 ⇒ 0.275 L
Now we have to find how much of 1 liter- 0.275 liters as a percentage is.
To convert <em>0.275</em> to percent multiply <em>0.275</em> by <em>100</em>. The result is <em>27.5 </em>percent, or, using the percent sign, <em>27.5 %.</em>
Answer:
1260
Step-by-step explanation:
First you will need to find the amount of sides of the polygon:
(After looking it is 9)
Each angle of the side of a nonagon is 140 degrees.
Multiply by 9 since you can make 9 triangles:
140 x 9 = 1260
Answer:
The second batch
Step-by-step explanation:
ANSWER
The length of a Diameter is 3.714
EXPLANATION
The circumscribed triangle is shown in the attachment.
We use the cosine ratio to find the altitude of the isosceles triangle.
Altitude =8cos(60°)
Altitude=4cm
Let the upper half of the altitude be y cm.
Then the radius of the circle is (y-4)cm
The upper radius meets the tangent at right angles.
From the smaller right triangle,
y=1.857
The diameter is 2y
=2(1.875)
The length of a Diameter is 3.714
