A piece of wire 23 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle.Required:
a. How much wire must be used for the square in order to maximize the total area?
b. How much wire must be used for the square in order to minimize the total area?
Answers
Answer:
wire for square to maximize total area = 23 m
Wire to minimize total area = 2.019 m
Step-by-step explanation:
For the square, let's say the total length of the square is "y" m.
Thus, length of one side is = y/4
And area of the square = (y/4) = y²/16
Since the wire is 23 m, then total length of equilateral triangle is; 23 - y.
Thus, length of one side of equilateral triangle = (23 - y)/3
Using trigonometric ratio, we can find the height of the triangle and thus area.
Area of triangle = (√3)/4) × ((23 - y)/3)²
Area of triangle = (√3)/36)(23 - y)²
So, total area of square and triangle is;
A_total = (y²/16) + (√3)/36)(23 - y)²
Now, extremizing this function by derivatives, we have;
dA/dy = (y/8) - (√3)/18)(23 - y)
d²A/dy² = ⅛ + (√3)/18)
So, d²A/dy² > 0
Now,let's find the maximum or minimum of the function.
So, we equate dA/dy to zero.
Thus;
(y/8) - (√3)/18)(23 - y) = 0
y/8 = (√3)/18)(23 - y)
(y/8) + (√3)/18)y = 23((√3)/18)
Multiply through by 8 to give;
y + 0.0962y = 2.2132
1.0962y = 2.2132
y = 2.2132/1.0962
y = 2.019 m
2.019 will be a minimum because d²A/dy² > 0
The maximum will occur at a boundary of the allowed values. Thus, the absolute maximum is for y = 23.
Note that a square has more area than a triangle and as such it is normal for the square to get the largest area over the triangle and therefore we will have to use all of the wire to construct the square.
Related Questions
Pls help me with this and show any work
Describe the following Algebraic Expression using the Algebraic Vocabulary words. Use all four words in yourdescription. Use complete sentences: 12y + 6-8x - m Terms, coefficients, variables, constants
The sphere at the right fits snugly inside a cube with 4-in. edges. What is the approximate volume of the space between the sphere and cube?
En una baraja de 52 cartas, donde hay 13 cartas de trébol. 13 cartas decorazones, 13 cartas de espadas y 13 cartas de rombos. Si sacamos doscartas con reemplazo. ¿Cuál es la probabilidad de que sean decorazones?
Answers
Answer:
2683
Step-by-step explanation:
Using the linear regression equation that predict the relationship between the weight of the luggage and the total number of passenger y = 103 + 30x, we can plug in the number of passenger x = 86 to predict the weight of the luggage on a flight:
y = 103 + 30*86 = 2683
rolling a number less than 3 on a six-sided die
spinning a number less than 2 on a spinner that has four equal sections numbered from 1 to 4
choosing an Oregon state quarter out a bag that contains 4 California state quarters, 3 Oregon state quarters, 6 Texas state quarters, and 3 New York state quarters
choosing a spade out of a standard deck of cards that contains 13 hearts, 13 clubs, 13 diamonds, and 13 spades
Answers
Choosing a green jelly bean.
Spinning a number less than 2 on a spinner.
Choosing a spade out of a standard deck of cards.
What are the probabilities?
Probability determines the chances that an event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
Probability of choosing a green jelly bean = number of green jelly bean / total number of beans
2/8 x 100 = 25%
Probability of spinning a number less than 2 on a spinner = number that is less than 2 / total number of sections
1/4 x 100 = 25%
Probability of choosing a choosing a spade= number of spade / total cards in the deck
13/54 x 100 = 25%
To learn more about probability, please check: brainly.com/question/13234031
The three options that have a probability of 25 percent are:
- Choosing a green jelly bean
- Spinning a number less than 2 on a spinner
- Choosing a spade from a standard deck of cards
The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. A probability of 25 percent corresponds to a ratio of 1 out of 4 (since 25% is one-fourth of 100%). Let's analyze the options:
1. Choosing a green jelly bean: There are 2 green jelly beans out of a total of 2 + 1 + 5 = 8 jelly beans. The probability is 2/8, which simplifies to 1/4 or 25%. This option has a probability of 25%.
2. Rolling a number less than 3 on a six-sided die: There are 2 favorable outcomes (1 and 2) out of 6 possible outcomes (1 through 6). The probability is 2/6, which simplifies to 1/3 or approximately 33.33%. This option does not have a probability of 25%.
3. Spinning a number less than 2 on a spinner: There is 1 favorable outcome (1) out of 4 possible outcomes (1 through 4). The probability is 1/4 or 25%. This option has a probability of 25%.
4. Choosing an Oregon state quarter: There are 3 Oregon state quarters out of a total of 4 + 3 + 6 + 3 = 16 state quarters. The probability is 3/16, which is not equal to 25%.
5. Choosing a spade from a standard deck of cards: There are 13 spades out of a total of 13 + 13 + 13 + 13 = 52 cards. The probability is 13/52, which simplifies to 1/4 or 25%. This option has a probability of 25%.
Learn more about probability here
#SPJ3
0.15x + 200 650
Answers
Answer:
13.3
Step-by-step explanation:
Answers
10.5
inches
width is
6.5
inches
Explanation:
Let length be
l
Let width be
w
Let perimeter be
P
First, we must construct an equation for these variables:
l
=
w
+
4
P
=
34
But, Perimeter of a rectangle
=
l
+
w
+
l
+
w
=
2
l
+
2
w
So:
34
=
2
l
+
2
w
But, since
l
=
w
+
4
, we can substitute for
l
, having only the
w
variable:
34
=
2
(
w
+
4
)
+
2
w
34
=
2
w
+
8
+
2
w
34
=
4
w
+
8
Solve for
w
:
4
w
=
34
−
8
4
w
=
26
w
=
26
4
w
=
6.5
inches
Now, we can substitute
6.5
for
w
in the Perimeter Equation:
34
=
2
l
+
2
w
becomes:
34
=
2
l
+
2
⋅
6.5
34
=
2
l
+
13
Solve for
l
:
2
l
=
34
−
13
2
l
=
21
l
=
21
2
l
=
10.5
inches
Thus, length is
10.5
inches
Thus, width is
6.5
inches
Answer: L = 4 + w
A = 2P - 4
lw = 2(2l +2w) - 4
lw = 4(l + w) - 4
(w+4)w = 4 ( w+4+w) -4
(w +4)w = 4(2w + 4) - 4
w^2 + 4w = 8w + 16 - 4
w^2 + 4w = 8w + 12
w^2 - 4w - 12 = 0
( w - 6 )( w + 2 ) = 0
w - 6 = 0 ----> w = 6 ----> L=10 ---> P = 32 and A = 60
w + 2 = 0 ---> w = -2 <--- width cannot be negative; disqualified/rejected
Step-by-step explanation:
Answers
Answer:
n = 15
Step-by-step explanation:
(x^n)(x^5)^3 = x^30
(x^n)(x^15) = x^30
x ^ (n+15) = x^30
n + 15 = 30
n = 15
To do the question, you need to know the exponent rules, which are short cuts to be able to do exponent work.
If you raise a power to a power, then times the exponents.
If you are multiplying two terms with the same base (in this case, x; the base is the large font bottom number) then ADD the exponents.
Answers
The simplest form of the given expression is,
.
Here, given expression is,
What is simplest form of equation?
The simplest form is the smallest possible equivalent fraction of the number.
Now,
Simplest form of expression,
Hence, The simplest form of the given expression is,
.
Learn more about the simplest form of the expression visit:
https://brainly.in/question/19257508
#SPJ2
Answer:
6x^3-5x^2
Step-by-step explanation:
(4x^3+x^2)+2(x^3-3x^2)
4x^3+x^2+2x^3-6x^2
4x^3+2x^3+x^2-6x^2
6x^3+x^2-6x^2
6x^3-5x^2