A piece of wire 19 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle.
How much wire should be used for the square in order to minimize the total area?

Answers

Answer 1

Answer: 8.26 m

Step-by-step explanation:

[tex]$$Let s be the length of the wire used for the square. \\Let $t$ be the length of the wire used for the triangle. \\Let $A_{S}$ be the area of the square. \\Let ${A}_{T}}$ be the area of the triangle. \\One side of the square is $\frac{s}{4}$ \\Therefore,we know that,$$A_{S}=\left(\frac{s}{4}\right)^{2}=\frac{s^{2}}{16}$$[/tex]

[tex]$$The formula for the area of an equilateral triangle is, $A=\frac{\sqrt{3}}{4} a^{2}$ where $a$ is the length of one side,And one side of our triangle is $\frac{t}{3}$So,We know that,$$A_{T}=\frac{\sqrt{3}}{4}\left(\frac{t}{3}\right)^{2}$$We have to find the value of "s" such that,$\mathrm{s}+\mathrm{t}=19$ hence, $\mathrm{t}=19-\mathrm{s}$And$$A_{S}+A_{T}=A_{S+T}$$[/tex]

[tex]$$Therefore,$$\begin{aligned}&A_{T}=\frac{\sqrt{3}}{4}\left(\frac{(19-s)}{3}\right)^{2}=\frac{\sqrt{3}(19-s)^{2}}{36} \\&A_{T+S}=\frac{s^{2}}{16}+\frac{\sqrt{3}(19-s)^{2}}{36}\end{aligned}[/tex]

[tex]$$Differentiating the above equation with respect to s we get,$$A^{\prime}{ }_{T+S}=\frac{s}{8}-\frac{\sqrt{3}(19-s)}{18}$$Now we solve $A_{S+T}^{\prime}=0$$$\begin{aligned}&\Rightarrow \frac{s}{8}-\frac{\sqrt{3}(19-s)}{18}=0 \\&\Rightarrow \frac{s}{8}=\frac{\sqrt{3}(19-s)}{18}\end{aligned}$$Cross multiply,$$\begin{aligned}&18 s=8 \sqrt{3}(19-s) \\&18 s=152 \sqrt{3}-8 \sqrt{3} s \\&(18+8 \sqrt{3}) s=152 \sqrt{3} \\&s=\frac{152 \sqrt{3}}{(18+8 \sqrt{3})} \approx 8.26\end{aligned}$$[/tex]

[tex]$$The domain of $s$ is $[0,19]$.So the endpoints are 0 and 19$$\begin{aligned}&A_{T+S}(0)=\frac{0^{2}}{16}+\frac{\sqrt{3}(19-0)^{2}}{36} \approx 17.36 \\&A_{T+S}(8.26)=\frac{8.26^{2}}{16}+\frac{\sqrt{3}(19-8.26)^{2}}{36} \approx 9.81 \\&A_{T+S}(19)=\frac{19^{2}}{16}+\frac{\sqrt{3}(19-19)^{2}}{36}=22.56\end{aligned}$$[/tex]

[tex]$$Therefore, for the minimum area, $8.26 \mathrm{~m}$ should be used for the square[/tex]


Related Questions

please help me!!

a bag contains 18 marbles. there are 6 red marbles, 6 yellow marbles, and 6 green marbles. a marble is drawn at random and returned. then another marble is drawn at random

which statements are true?

A. the expected probability of drawing a green marble first is the same as the expected probability of drawing a green marble second.

B. there are six possible outcomes

C. the probability of drawing a red marble first is the same as not drawing a red marble second

D. the expected probability of drawing a red marble and then a yellow marble is 1/9

E. there are three possible outcomes from drawing the same color twice

F. the probability of drawing a yellow marble and then a yellow marble is 1/6

Answers

Answer:

Step-by-step explanation:

A. False. The probability of drawing a green marble twice is even smaller (It may be true though depending on what the question is asking, if it is asking the chance of drawing a green marble second is the same as the 1st chance, then yes it is True.

B.False

C. False

D.True, 1/3*1/3 (Simplified version of 6/18*6/18) is 1/9

E. False (Although this may be true, again depending on the type of question)

F. False

On a long hike, Austin drank 5 pints of water. How much did he drink in quarts?

Answers

Answer:

Austin drank 2.5 quarts.

Step-by-step explanation:

There are 2 pints in a quart.

To get from pints to quarts, divide by 2

5/2 = 10

2.5 quarts

mark brainliest please!

Please Help!!!!
Find the slope of the line.

Answers

Answer:

-1/2

Step-by-step explanation:

Use the two points shown.

slope = rise/run

Start at teh left point.

Go down 2 units. That is a rise of -2.

Now go right 4 units. That is a run of 4.

slope = rise/run = -2/4 = -1/2

i need help with this math

Answers

(-5, 1) (-2, 4)
m = y2 - y1
————
x2 - x1
m = 4 - 1 = 3
———
-2 - (-5) = 3
M = 1

What is the solution to the inequality |2x + 3| < 7? 4 < x < 10 –5 < x < 2 x < 4 or x > 10 x < –5 or x > 2

Answers

Answer:

4(−10−9x)

Step-by-step explanation:

Zearnn help... I've been stuck working hard

Answers

Calculate the area of the room and then subtract the area of the tiled floor.


area of room = 11 3/4 x 5 = 58 3/4

Area of tiled floor = 7 1/2 x 3 1/4 = 24 3/8


area of carpet = 58 3/4 - 24 3/8 = 34 3/8


answer: 34 3/8

The number of bats in a colony is growing exponentially. After 1 year, there were 70 bats. After 4 years, there were 560 bats. If the colony continues to grow at the same rate, how many bats are expected to be in the colony after 7 years?

Answers

Answer:

23040 bats

Step-by-step explanation:

Let N(t) be the number of bats at time

We know that exponential function

[tex]y = ab[/tex]

According to question

[tex]N(t) = ab^{2}[/tex]

Where t (in years)

Substitute t=2 and N(2)=180

[tex]180 = ab^{2}[/tex]

Substitute t=5 and N(5)=1440

[tex]1440 = ab^{5}[/tex]

Equation (1) divided by equation (2)

[tex]\frac{180}{1440} = \frac{ab^{2} }{ab^{5}} = \frac{1}{b\frac{5-2}{}}[/tex]

By using the property

Substitute the values of b in equation (1)

Substitute t=9

Hence, after 9 years the expected bats in the colony=23040 bats

I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER

Answers

Step-by-step explanation:

z = 180° - 113° = 67°

sum of angles of quadrilateral shape = 360°

y = 360° - ( 67° + 81° + 106° ) = 106°

Find the area of the triangle.
Height = 4 ft
Base = 12 1/2 ft
Area = ? square feet

Answers

Answer:
25 square feet

Explanation:
Area of triangle= 1/2 * height * base
1/2 * 4ft * 12 1/2 ft
2ft * 12 1/2 ft
25 square feet

Hope this helps you :)

Can someone help me find the volume of this?

Answers

The answer is 194.4 cubic units

Answer:

  97.2 m³

Step-by-step explanation:

The volume of a triangular prism is half the volume of a rectangular prism with the same length, width, and height.

  V = 1/2LWH

  V = 1/2(9 m)(7.2 m)(3 m) = 97.2 m³

_____

Additional comment

The relation we used above comes from the more general formula for the volume of a prism:

  V = Bh

where B is the area of the base.

You know a rectangular base will have an area of ...

  A = bh

and a triangle with the same base and height will have an area of ...

  A = 1/2bh

That is, the base area of a triangle is half that of a rectangle with the same dimensions. When we use this fact in the volume formula, we find the volume of a triangular prism is 1/2 that of a rectangular prism with the same length, width, and height.

Evaluate 27+ x when x = 3.
What is the value of the expression

Answers

Answer:

30

Step-by-step explanation:

Evaluating an expression when given a value of an unknown means to substitute the unknown value with a certain value given and simplify fully:

27 + (3)  

Simplify by doing simple addition :

27 + 3 = 30

Therefore our answer is 30.

Hope this helped and brainliest please.

Replace the variable x with 3 in the expression.

27 + 3

Remove parentheses.

27 + 3

Add 27 and 3.

30

What is the factorization of the polynomial graphed below? Assume it has no
constant factor. Write each factor as a polynomial in descending order.

Answers

[tex]\bold{ANSWER:}[/tex]
(x - 5)(x - 3)(x - 2)

[tex]\bold{EXPLANATIONS:}[/tex]

The track is 9.76 m wide all the way around. What is the distance around the outside of the track? Explain or show your reasoning.

Answers

Answer:

The distance around the outside of the track is 4183.58m².

Step-by-step explanation:

The area of running track that goes around the field is 9.76m

To proof :

Formula -

(Area of rectangle = length × breadth)

(Area of running track = outside area - inside area)

First case (outside),

length = 73 +2× 9.76

= 92.52m

Breadth = 84.39 m

radius of outside semi circle,

= 46.26m

outside area = area of rectangle + 2 × area of the semi-circle

= 92.52 x 84.39 + 3.14 x 46.26 × 46 × 26

= 7807.76+6719.56

Second case,

radius of the semi-circle,

=36.5m

inside area = area of rectangle + 2x area of the semi-circle

=73x 84.39+ 3.14x 36.5×36.5

= 6160.47 +4183.27

= 10343.74m²

Now,

Area of running track = 14527.32 m²- 10343.74m²

= 4183.58m²

The distance around the outside of track is 4183.58m².

Solve the equation. Write your answer as an integer or simplified fraction.
5n−6 n−2 =19
n = _____

Answers

Explaination :

The given equation,

[tex] \boxed{\sf{5n \: - 6(n - 2) \: = \: 19}} \: \bigstar[/tex]

Lets start solving it..!!

★ Multiplying -6 by (n -2),

[tex] \implies \: \sf{5n \: - 6 \times (n - 2) \: = \: 19} [/tex]

[tex] \implies \: \sf{5n \: - 6n + 12 \: = \: 19} [/tex]

★ Subtracting 5n from 6n,

[tex]\implies \: \sf{ - n + 12 \: = \: 19} [/tex]

★ Transposing 12 from L.H.S. to R.H.S. it would be in negative,

[tex]\implies \: \sf{ - n \: = \: 19 - 12} [/tex]

★ On subtracting 12 from 19 we gets,

[tex] \implies \: \sf{ - n \: = \: 7} [/tex]

★ Now,

[tex] \implies \: \red{\bf{ n \: = \: - 7} }[/tex]

Therefore,

Value of n is -7

Find the probability that a point chosen randomly inside the rectangle is in each given shape. Round to the nearest hundredth.

Answers

its a tringale

Step-by-step explanation:

Please help me GENIUS Rank

Directions: Complete the following frequency distribution table, then compute the mean, median, and mode of the distribution. Show your complete solutions. (25 points)

Class interval] frequency (f)] fxm
99-101 ] 17 ].
102-104 ] 25 ]
105-107 ] 13 ]
108-110 ]. 16 ]
111-113 ]. 29 ]
total ]N= ]. Efxm=

less than cumutalti frequency (<cf)
[. ]
[. ]
[. ]
[. ]
[. ]
[. ]​

Answers

Hey there!

Here's your required mode

Refer to the image above

And btw,in the earlier question I've done for mean also you can check that again which I've answered earlier.

Hope it helps

Pls help giving 20 points

Answers

Answer:

220.5 ft

Step-by-step explanation:

suppose the integers from 1 to 25 are arranged in consecutive order. How many different pairs of integers may be selected if there are at least four integers between the selected integers?

Answers

there is 5 different pairs

If two lines intersect, their intersection is___________
O many points
O one plane
O many planes
O one point

Answers

Answer:

one point and one plane

Step-by-step explanation:

Any two line if they intersects... they will intersects in one point..

And to have an intersection they always need to be on one plan..

What is the answer to this?…

Answers

[tex]~~~~~\dfrac 34n = 9\\\\\\\implies n = \dfrac{9 \times 4} 3\\\\\\\implies n = \dfrac{36}3\\\\\\\implies n = 12[/tex]

Which of the following is equivalent to
?
60 ^1/2

Answers

Answer:

60^1/2 = 60^0.5 = sqrt 60 = 7.75

good luck :))) <3

8x = 2/3 ; x =
1 1/2 = 2x; x =
5x = 2/7; x =
1/4x = 5; x =
1/5 = 2/3x; x =​

Answers

We can simply solve the equations by just dividing by the coefficient value of x. For example, we would divide both sides of the first equation by 8 to make the equation into x = 2/24 = 1/12

8x=2/3 --> x=1/12
1.5=2x --> x=0.75
5x=2/7 --> x= 2/35
1/4x=5 --> x=20
1/5=2/3x --> 3/5=2x --> x=3/10

a 23 ө C С B 2
help me??​

Answers

a = 30 degrees or pi/6 radian
o = 60 degrees or pi/3 radian

Sorry man, it’s a tough time for figure out

Use this paper and examine the steps. This will help you understand the trigs formula to solve for each.

You mix the letters S,E,M,I,T,R,O,P,I,C,A, and L throughly.Without looking,you draw one letter.Find the probability that you select a vowel.Write your answer in simplest form.

A. 12/5

B. 5/12

C. 1/3

D. 7/12

I need this answer in 5 MINUTES!

Answers

Answer:

B. 5/12

Explanation:

Total Letters mixed: {S, E, M, I, T, R, O, P, I, C, A, L} = 12 Letters

There are 5 vowels and they are: 'a', 'e', 'i', ' o', and 'u'

In here the vowels are: E, I, O, I, A = 5 vowels here

So,

probability: number of favourable outcomes/total number of outcomes

probability: 5/12

Resuelve el siguiente problema aplicando las estrategias de solución de problemas:
• La suma de dos números es 20 y su diferencia es de 4. Halla los números.

Answers

Los numeros :
Numero mayor = x
Numero menor = 20 - x
x -( 20 - x) = 4
x - 20 + x = 4
2x = 4 +20
x = 24/2
x = 12
El numero mayor es 12
El numero menor es = 20 - 12 = 8

El producto es = 12*8 = 96

Una persona camina desde su casa hasta el parque la distancia de 100 M y el tiempo recorrido fue de 10 minutos cuál fue su velocidad

Answers

Usando la relación entre la velocidad, la distancia y el tiempo, se encuentra que su velocidad fue de 10 m/min.

¿Cuál es la relación entre la velocidad, la distancia y el tiempo?

Velocidad es distancia dividida por el tiempo, o sea:

[tex]v = \frac{d}{t}[/tex]

En este problema, hay que:

La distancia es de d = 100 m.El tiempo es de t = 10 min.

Por eso:

v = 100 m/10 min = 10 m/min.

Su velocidad fue de 10 m/min.

Puede-se aprender más a cerca de la relación entre la velocidad, la distancia y el tiempo en https://brainly.com/question/24558377

#SPJ1

Solve for x in the given equation: ax - b = c

Answers

An equation is formed of two equal expressions. The value of x is the equation ax-b=c is (c+b)/(a).

What is an equation?

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.

The solution of x in the given equation can be written as,

[tex]ax - b = c\\\\ax = c+b\\\\x = \dfrac{c+b}{a}[/tex]

Hence, the value of x is the equation ax-b=c is (c+b)/(a).

Learn more about Equation:

https://brainly.com/question/2263981

#SPJ1

Evaluate and simplify

Answers

Answer:

[tex]\frac{-3}{-26}[/tex]

Step-by-step explanation:

[tex]\frac{-x-8}{x^{2}-4x-21}[/tex] is what you are looking for.

So with x = -5, it would look like this

[tex]\frac{-(-5)-8}{-5^{2}-(-5)4-21}[/tex]

Simple math now.

-(-5) is 5

[tex]-5^{2} = -25[/tex]

-5 * -4 = 20.

So it should be [tex]\frac{5-8}{-25+20-21}[/tex]

5-8=-3

-25+20-21 = -26

So the answer is[tex]\frac{-3}{-26}[/tex]

6. Annie has a cube that has a side length of 9 cm. What is the surface area of the cube?​

Answers

Answer:

486

Step-by-step explanation:

Surface area means the area of each side added upIn a cube, every side is the samethe area of one side = 9 * 9 = 81, since it is a square (width*height) and width and height are the same. in a cube there are 6 sidesso there are 6 sides with area 8181 * 6 = 486

The bacteria in a petri dish doubles every hour. If there are 3 present, how many will there be in 15 hours?

Answers

Answer:

98304 bacteria

Step-by-step explanation:

3x2^1 = 6

3x2^2 = 12

3x2^3 = 24

3x2^15 = 98304

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