Find the dimensions of the rectangle with largest area that can be inscribed in an equilateral triangle with sides of 1 unit, if one side of the rectangle is on the base of the equilateral triangle.

Answers

Answer 1
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.
Answer 2
The area of the rectangle is equal to L x W where L is the length and W is the width. 

               A = L x W

If we let L be the length and given that the triangle is equilateral with sides equal to 1, the value of W should be (0.5 - L/2)(tan 60)

           W = (0.5 - L/2)(tan 60)

Simplifying the expression for the width,
          W = 0.866 - 0.866L

The area now becomes,
      A = (L)(0.866 - 0.866L)

Simplifying,
    
   A = 0.866L - 0.866L²

Derive and equate to zero for the maximum value of L. 
   dA = 0.866 - 2(0.866)(L) = 0

The value of L from the equation should be 0.5. 
           W = 0.866 - 0.866(0.5) = 0.433

Answer: L = 0.5
             W = 0.433

Related Questions

How is volume calculated given mass and density?

Sum of mass and density
Mass divided by density
Mass multiplied by density
Difference of mass and density

Answers

Answer:

B.

Step-by-step explanation:

when trying to calculate the volume when being given the M or the mass, and the D or density, hears is the calculations needed for that: P=M/V, which also means volume is the mass divided by the volume, or density. Hence, the answer would be mass divided by density. hope this helps!!! (also, please give me brainliest.)

Volume is calculated given mass and density is Mass divided by density.

What is volume?

Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic meter and liter) or by various imperial units (such as the gallon, quart, cubic inch).

Mass is the quantity of matter in a physical body. It is also a measure of the body's inertia, the resistance to acceleration (change of velocity) when a net force is applied.

Density is defined as a measure of how compact the mass in a substance or object is.

Since, density of a substance is measure as the ratio of its mass to its volume,

Density =  Mass/Volume

or, Density x volume =  mass

or, Volume =  Mass/Density

Hence,Volume is calculated given mass and density is Mass divided by density.

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Share the advantages and disadvantages of the different methods for solving quadratic equations that you have learned. Share special techniques that you have developed along the way. For example, explain to your classmates your technique for remembering the quadratic formula (song, rhyme, rhythm, etc.) to help others remember it as well.

Answers

Factoring

The factors of a quadratic equation will be (qx + r) and (sx+t). If the solutions are all integers, you may be able to quickly find q, r, s and t. The advantage of this method is that factoring can be very fast. The disadvantage is that factoring may not work; for instance, factoring will not find solutions that are not integers.

Completing the Square

Completing the square is a multistep process. The main idea is to convert the original equation into one of the form (x + a)^2 = b, where a and b are constants. The advantage of this method are that it always works and that completing the square gives some insight into how algebra works more generally. The disadvantage is that this method is complex.

Quadratic Formula

The quadratic formula is x = (-b +- (b*2 - 4ac)^.5))/2a. The advantages of this method are that the quadratic formula always works and is straightforward. The disadvantages are that the formula provides no insight and can become a rote technique.

Guessing

Sometimes, you can guess an approximate solution. Then, you can increase or decrease your guess, depending on whether the result from your first guess is too big or too small. The advantages of this method are that guessing can be very fast if you guess right, and can get an approximate answer quickly, if that is all you need. The disadvantage is that sometimes you won't be able to make a good guess .

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Which mass is greatest? a. 179,000 mg c. 75 g b. 0.043 kg d. 23,481 cg

Answers

179,000 mg = 0.179 kg
75 g = 0.075 kg
23,481 cg = 0.23481 kg
and 0.043 kg 

so 23,481 cg has greatest mass

answer
d. 23,481 cg

Which phrase best describes a rock? A. a chemical arranged in a lattice B. a gabbro of minerals C. a mixture of crystals D. a fire-formed crystal

Answers

A gabbro of minerals.

What is 65.6 divided by 8

Answers

The correct answer is 8.2
The answer is 8.2  i hope it helped and good luck

Write an equation of the line in slope-intercept form that crosses the y-axis at 3 and increases at a rate of 1?

y = x - 3
y = x + 3
y = -x + 3
y = 3x + 1

Answers

increases at a rate of 1....so ur slope is 1
crosses the y axis at 3...so ur y intercept is 3

in slope intercept form, y = mx + b, the slope will be in the m position and the y intercept will be in the b position

y = mx + b
slope(m) = 1
y int (b) = 3
sub
y = 1x + 3...which can be written as y = x + 3 <=

Use your properties of addition to find 37 + 58 mentally show your steps and identify the properties that you used

Answers

I would use my normal property... no calculator (optional) and i would show my work...
37+58=95 or..........
you can do......

1
37 
+
58
___
9 5

The difference of two numbers is 16 and the sum of their squares is 130. What are the numbers.

Answers

x - y = 16
x^2 + x^2 = 130

----------------
x^2 - y^2 = 256
+(x^2 + y^2 = 130)
2x^2 = 386
x^2 = 193
x = 13.892 (3DP)
----
x - y = 16
13.892 - y = 16
-y = 2.108
y = -2.108
The numbers are: 13.892 and -2.108

need help.... what is half of 1 3/4 IN A FRACTION plzzzzz

Answers

well 1 3/4 as improper fraction is 7/4 so 7/4 / 1/2
7/4 x 2/1 is 14/4 or 3 1/2
first add 1 (= 4/4) to 3/4 which equals 7/4, now to get half of 7/4 you make it easier by changing it to 14/8 (= 7/4) 

now half of 14/8 is easy

Solve the inequality
-8x - 10 > 7x + 20

A.) x < - 30
B.) x > - 2
C.) x > - 30
D.) x < - 2

Answers

Final answer:

To solve the inequality -8x - 10 > 7x + 20, combine like terms, isolate the variable term, and solve for x. The correct answer is D) x < -2.

Explanation:

We need to solve the inequality -8x - 10 > 7x + 20.

First, let's combine like terms by adding 8x to both sides. This gives us -10 > 15x + 20. Next, subtract 20 from both sides to isolate the variable term. This gives us -30 > 15x. Finally, divide both sides by 15 to solve for x. This gives us x < -2.

Therefore, option D.) x < -2 is the correct answer.

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Drinks at the fair cost 2.50$ use an equation to represent the relationship between the numbers of drinks bought ad the cost define your variable and make a table to help write the equation

Answers

C=5/2*d where c-cost. d-drink

Answer:

[tex]C=2.5d[/tex]

Step-by-step explanation:

Your variable is always the number of drinks that you are going to drink, and the independent clause is the cost of the drink, which is $2.5.

Drinks     Cost

1                 2.5

2                  5

3                 7.5

4                  10

5                 12.5

SO now you know that the cos of 3 drinks is 7.5 and that can be escalated to any number that you want just by puttin the number of drinks that are going to be bought and multiply it by the cost.

A packaging plant uses a machine that can bottle 30 fluid ounces of shampoo in 12 seconds. How many fluid ounces of shampoo can be bottled per second?
A) 2/5 fluid ounces
B) 19 fluid ounces
C) 2 ½ fluid ounces
D) 360 fluid ounces

Answers

30 fluid ounces / 12 seconds = 2.5 fluid ounces per second
C) 2.5 fluid ounces. Because 2.5 x 12 = 30, and in 1 second it would be 2.5.

An airplane flies at a constant speed. It can travel a distance (
d. of 1,800 kilometer in 2 hours(t) Write an equation in the form is d=rt that represents the relationship between distance (
d. and time (t), where r is the constant of proportionality

Answers

I will be right back to help

33 is 75% of what number?

Answers

33 is 75% of what number...

33 = 0.75x
33 / 0.75 = x
44 = x <== so 33 is 75% of 44

The area of a garden is 12.9 square feet.  Each square foot of the garden needs to have 2.6 pounds of soil.How many pounds of soil are needed for the garden?

Answers

2.6 pound of soil for each square foot,
so,for 12.9 square
2.6x12.9=33.54 pound of soils

A computer is programmed to scan the digits of the counting numbers. For example, if it scans 
1 2 3 4 5 6 7 8 9 10 11 12 13 
then it scanned 17 digits altogether. If the computer begins it's task and scans the first 1392 digits starting with 1, what is the last counting number scanned?

Answers

Number of 1-digit numbers scanned (i.e. 1 - 9) = 9 digits

Number of 2-digits numbers scanned (i.e. 10 - 99) = 2 x 90 = 180 digits

Let the number of 3-digits numbers scanned be x (i.e. 100 - y), then 3 x (y - 99) = (1392 - 180 - 9)
3y - 297 = 1203
3y = 1203 + 297 = 1500
y = 1500 / 3 = 500

Therefore, the last counting number scanned is 500.

Please help me with both questions thank you.

Answers

The First question would be, Transitive Property.
if RS = ST and ST = TU, then RS = TU
this is the transitive property which states : if a = b, and b = c, then a = c
==================
there is a pattern....light, medium, dark, light, medium, dark....etc
so the 10th item is : the lightest triangle...1st answer choice

What is the value of x? 3x-3

Answers

i think its written     3x-3=0?
if it is then, you move the -3 over the equals
now 3x=3
then x=3/3   which is 1

The value of x in the given figure is 19.

What are Vertical angles?

Vertical angles are the angles formed when two lines intersect each other. The opposite angles formed by these lines are called vertically opposite angles.

Given that, a figure, we need to find the value of x,

Here, angles, 3x-3 and [6(x-10)] are vertical angles,

We know that,

Vertical angles are equal in measures,

Therefore,

3x-3 = [6(x-10)]

3x-3 = 6x-60

3x = 57

x = 19

Hence, the value of x is 19.

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The complete question is attached

(-11/2x+3)-2(-11/4x-5/2)=expression in simplest form needed!!

Answers

(-11/2x+3)-2(-11/4x-5/2)
= -11/2x + 3 + 11/2x + 5
= 8

answer
8

The given expression in its simplest form is 8.

The given expression is (-11x/2+3)-2(-11x/4-5/2).

We need to write the given expression in the simplest form.

What is an expression in its simplest form?

A simplified expression is one in which each base is written only once, there are no negative exponents and no parentheses. Simplifying an expression makes it easier to solve complex problems, and using the rules for exponents makes simplifying expressions easier.

Now, the given expression can be simplified as follows:

(-11x/2+3)-2(-11x/4-5/2)

=-11x/2+3+22x/4+10/2

=-11x/2+22x/4+3+5

=-22x/4+22x/4+8

=8

Therefore, the given expression in its simplest form is 8.

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Can someone please please help me.

Answers

Cd || gh #2 is given look at the bottom of the paper
Check out the attached image for the answers all on one sheet.

------------------------------------------------------------------------------

Label each row 1, 2, ..., all the way up to row 7. I'll refer to each statement and reason by the row number. For example, in row 2 we have "statement 2" and "reason 2". Statement 2 is already filled in for us while Reason 2 is blank. 

Statement 1 has "CD perpendicular to AE". This is given to us to be true as shown in the "given" portion at the very top of the problem (just under the diagram).

What's also given to us is the fact that "line segment CD is parallel to line segment GH". So we'll write "given" for reason 2.

------------------------------------------------------------------------------

Next up is reason 3, which is "definition of perpendicular". Why is this? Because the statement "angle ABD is a right angle" means angle ABD is 90 degrees. The term "perpendicular" means two lines or line segments or rays cross/meet at a 90 degree angle. 

Now onto reason 5. Note how angle ABD and angle GEF are alternate exterior angles. They are outside of the train tracks (CD and GH) and lay on alternate sides of the line FA. Since CD is parallel to GH (see line 2), this means the alternate exterior angles are congruent. For reason 5, we'll put "alternate exterior angles theorem"

------------------------------------------------------------------------------

Statement 7 is the the thing we want to prove. For any proof, the last statement is ALWAYS what you want to prove. If there's anything after this statement then we still have work to do and the proof isn't done. This is the last stop so to speak. So we simply write exactly what we want to prove (next to the "prove:" portion just below the diagram)

Reason 7 is "definition of perpendicular" because GEF is a right angle so GH is perpendicular to line FA. Like with reason 3, a 90 degree angle forms. 

------------------------------------------------------------------------------

Once again, check out the attached image to see all of the answers in the right spots.

The standard error of the mean is calculated using which two quantities:

Answers

If a random variable has a standard deviation of σ and a sample of size n is collected for the variable, then the mean has a standard error given by σ/√n. So the standard error of the mean is calculated using the two quantities: standard deviation σ and the sample size n. 

Strawberry chews cost $0.03 more than lollipops. Sarah pays $0.54 for 2 strawberry chews and 4 lollipops. What is the price of a strawberry chew?

Answers

We will use this system of equations to solve this:

l = lollipop
s = strawberry chew

[tex] \left \{ {{4l + 2s = 0.54} \atop {s = l + 0.03}} \right. [/tex]

So first of all, let me explain these equations.

4l + 2s = 0.54 means that 4 lollipops and 2 strawberry chews cost $0.54
s = l + 0.03 means a lollipop plus 0.03 equals the cost of a strawberry chew.

Now, let's solve for s.

4l + 2s = 0.54
Subtract 4l from boths sides.
2s = 0.54 - 4l
Divide by 2 on both sides.
s = -2l + 0.27

Substitute.

-2l + 0.27 = l + 0.03
Subtract l from both sides.
-3l + 0.27 = 0.03
Subtract 0.27 from both sides.
-3l = -0.24
Divide by -3 on both sides.
l = .08

.08 + .03 = .11

A strawberry chew costs $0.11

Sweet T purchased a new pair of pants to wear to the battle of the bands on sale at 40 % off . He saved $20 . What was the regular price of the jeans ?

Answers

Hey there!

The answer is $50 because $20 is 40%, so times that by 2 which you get $20 and 80%. Then you divide 20 by 2 then you get 10 which is 20% so you add 10 and you get $50 which is 100%

I have a Brainliest for math

Answers

The answer is D. 280.50 
Hoped this helped
Feel free to ask anymore questions here at brainly.com
Just find ten percent of 3298.

3298 * .10 = 329.8 

Hope this helps. (:

NEED HELP ASAP 30 POINTS WILL BE GIVEN TO BRAINLIEST
The coordinates of the vertices of △GHJ are  G(−1, 3),  H(1, 2), and  J(−3, −1)
Drag and drop the choices into each box to correctly complete the sentences


been trying to solve this but just don't get it I keep getting it wrong on my homework

Answers

we know that

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

we have

[tex]G(-1,3)\ H(1,2)\ J(-3,-1)[/tex]

Step 1

Find the slope GH

[tex]G(-1,3)\ H(1,2)[/tex]

substitute in the formula

[tex]m=\frac{2-3}{1+1}[/tex]

[tex]m=\frac{-1}{2}[/tex]

[tex]mGH=-\frac{1}{2}[/tex]

Step 2

Find the slope HJ

[tex]H(1,2)\ J(-3,-1)[/tex]

substitute in the formula

[tex]m=\frac{-1-2}{-3-1}[/tex]

[tex]m=\frac{-3}{-4}[/tex]

[tex]mHJ=\frac{3}{4}[/tex]

Step 3

Find the slope JG

[tex]J(-3,-1)\ G(-1,3)[/tex]

substitute in the formula

[tex]m=\frac{3+1}{-1+3}[/tex]

[tex]m=\frac{4}{2}[/tex]

[tex]mJG=2[/tex]

Step 4

Verify if triangle GHJ is a right triangle

we know that

if two lines are perpendicular, then the product of their slopes is equal to minus one

so

[tex]m1*m2=-1[/tex]

compare slope GH with slope JG

we have

[tex]mGH=-\frac{1}{2}[/tex]

[tex]mJG=2[/tex]

[tex]-\frac{1}{2}*2=-1[/tex] -----> side GH and side JG are perpendicular

The triangle GHJ is a right triangle

The answer in the attached figure


Answer:

- 1/2, 3/4, 2, is, two of these slopes have a product of -1

Step-by-step explanation:

How do I work out 5x+2+3x

Answers

5x + 2 + 3x

add same types of numbers

5x + 3x + 2

8x + 2

find x

8x + 2 = 0
8x + 2 (-2) = 0 (-2)
8x = -2
8x/8 = -2/8

x = -1/4

hope this helps
5x + 3x = 8x

8x + 2

FINAL ANSWER: 8x + 2

You would add 5x and 3x because they are like terms. The 2 is just a regular number, it doesn’t have a variable along with it.

The following section is a statement from the rental agreement Tim signed when he rented his car this past weekend. "Upon checkout, the fuel level of the vehicle will be determined by turning the vehicle on and visually inspecting the fuel gauge. The approximate fuel level will be recorded on the Check-Out sheet and verified with initials by the vehicle Renter. One copy of the Check-Out sheet will be given to the customer. Another copy will be kept with the on-site records of the vehicle. "The rented vehicle must be returned with a minimum fuel level the same as that indicated on the Check-Out sheet. A vehicle returned with a fuel level less than the approximate level indicated on the Check-Out sheet will be completely refueled with on-site pumps. The price of the fuel used to refuel the vehicle will be added to the Renter's total charge at a cost of $4.50 per gallon plus a $5.00 refueling charge." Tim is running extremely late and has just 5 minutes to return his rental car to the agency. The fuel tank in the car is half empty. It was full when he picked it up. (The car has a 12-gallon fuel tank.) He doesn't think he can get gas and get to the agency to return the car in time. Returning the car a day late will result in an extra daily rental charge of $32.00 and refueling the car on his own will cost $2.59 per gallon at the local gas station. Which of the following statements is true?
a. It would be cheaper to refuel the car and return it a day late.
b. It would be cheaper to return the car on time and pay the refueling costs.
c. It would cost about the same to return the car on time without gas or one day late with gas.
d. Tim is not responsible for refueling the car when he returns it.

Answers

It is cheaper for Tim to return the car on time and pay the refueling costs. If the gas tank on the car is half empty and the rental agency charges $4.50 per gallon the costs to refill for him would be. $27.00 plus the $5.00 refueling fee. This would equal $32.00. If Tim returns the car late, he faces a $32.00 charge including the cost to refuel the car on his own. The cost of returning the car a day late is not the same costs as returning the car on time. Choice b is the better option and the best choice.

Part A: Solve –vp + 40 < 95 for v. Show your work. (4 points) Part B: Solve 8w – 3r = 45 for r. Show your work. (6 points)

Answers

Answer:

A)[tex]v>-\frac{55}{p}[/tex]

B)[tex]r=\frac{8}{3}w-15[/tex]

Step-by-step explanation:

A)

The inequality is

–vp + 40 < 95

Subtract 40 to both sides

–vp  < 95 - 40

-vp< 55

Now, divide both sides by -p. Whenever, we divide by a negative number inequality sign changes

[tex]v>\frac{55}{-p}\\\\

B)

The given equation is

[tex]8w-3r=45[/tex]

Subtract 8w to both sides

[tex]-3r=45-8w[/tex]

Divide both sides by -3

[tex]r=-15+\frac{8w}{3}\\\\r=\frac{8}{3}w-15[/tex]

Answer:

Step-by-step explanation:

Part A: 

The inequality is–vp + 40 < 95.

Work:

We subtract 40 to both sides = –vp  < 55. Now, we divide both sides by -p. = v > 55/-p

Part B: Inequality is 8w − 3r = 45. 

Work:

We Subtract 8w to both sides = 3r = 45 - 8w. Now, we divide both sides by -3 = r = -15 + 8w and then r = 8/3w - 15. 

10 1/2 - 5 2/3 pleaseee

Answers

10 1/2 - 5 2/3 =
21/2 - 17/3 =
63/6 - 34/6 =
29/6 or 4 5/6 <==
First you convert both of the fractions into improper fractions:
10 1/2 = 21/2
5 2/3 = 17/3

Then you times the fractions by the other fractions denominator:
(21*3)/(2*3)= 63/6
(17*2)/(3*2)=34/6

The you can take the numerators away from eachother and place them over the denominator:
(64-34)/6 = 29/6

Then you take out as many 6's as possible from the 29 and you have that number before the fraction and the remainder as the numerator:
4 5/6

Hope this helps! :)

Write an equivalent expression using the multiplicative identity property

Answers

A possible answer could be 1x9= 9 which is itself.

For an expression equivalent to, let's say, -8, you could do: -2 + -6.

Other Questions
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