Answer:
a) Cost (h,x) = 12*x*h + 5*x²
b)
V = V(max) = 355.5 ft³
Dimensions of the hut:
x = 9.48 ft (side of the base square)
h = 3.95 ft ( height of the hut)
Step-by-step explanation:
Let x be the side of the square of the base
h the height of the hut
Then the cost of the hut as a function of "x" and "h" is
Cost of the hut = cost of 4 sides + cost of roof
cost of side = 3* x*h then for four sides cost is 12*x*h
cost of the roof = 5 * x²
Cost(h,x) = 12*x*h + 5*x²
If the troll has only 900 $
900 = 12xh + 5x² ⇒ 900 - 5x² = 12xh ⇒(900-5x²)/12x = h
And the volume of the hut is V = x²*h then
V (h) = x² * [(900-5*x²)]/12x
V(h) = x (900-5x²) /12 ⇒ V(h) = (900*x - 5*x³) /12
Taking derivatives (both sides of the equation):
V´(h) = (900 - 10* x²)/12 V´(h) = 0
900 - 10*x² = 0 ⇒ x² = 90 x =√90
x = 9.48 ft
And h
h = (900-5x²)/12x ⇒ h = [900 - 90(5)]/12*x ⇒ h = 450/113,76
h = 3.95 ft
And finally the volume of the hut is:
V(max) = x²*h ⇒ V(max) = 90*3.95
V(max) = 355.5 ft³
A hut will consist of four walls and one roof. The figures needed evaluates to:
The cost of the hut expressed in terms of its height h and the length x of the side of the square floor is [tex]12hx + 5x^2 \text{\:(in dollars) }[/tex]The biggest volume hut that can be build with 960 dollars at max is 426.67 sq. ft approxHow to find the volume of cuboid?Let the three dimensions(height, length, width) be x, y,z units respectively.
Then the volume of the cuboid is given as
[tex]V = x \times y \times z \: \rm unit^3[/tex]
How to obtain the maximum value of a function?To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
For this case, we're specified that:
Cost of roofing material = $5 / sq. footCost of wall material = $3 / sq. footThe side length of floor = side length of roof = [tex]x \: \rm ft[/tex]The height of the room = [tex]h \: \rm ft[/tex]Four walls are attached to sides of floor. Thus, their one edge's length = length of side of floor = [tex]x \: \rm ft[/tex]
Thus, we get:
Area of four walls = [tex]4 \times (h \times x)[/tex] sq. ftThus, cost of four walls' material = [tex]3 \times 4 \times h \times x = \$ 12hx[/tex]
Area of roof = [tex]side^2 = x^2 \: \rm ft^2[/tex]Thus, cost of roofing material for this roof = [tex]5 \times x^2 = \$5x^2[/tex]
Thus, cost of hut = cost for walls + cost for roof = [tex]12hx + 5x^2 = x(12h+5x) \: \rm (in \: dollars)[/tex]
The volume of the hut is: [tex]x\times x\times h =x^2.h \: \rm ft^3[/tex]
If troll has got only $960, then,
[tex]12hx + 5x^2 \leq 960\\\\\text{Multiplying x on both the sides}\\\\12x^2h + 5x^3 \leq 960x\\\\x^2h \leq \dfrac{960x-5x^3}{12}[/tex]
Let we take [tex]f(x) =\dfrac{960x-5x^3}{12}[/tex]
Then, taking its first and second derivative, we get:
[tex]f(x) =\dfrac{960x-5x^3}{12}\\\\f'(x) = 80 -1.25x^2\\\\f''(x) = -2.5x[/tex]
Putting first derivative = 0, we get critical points as:
[tex]80-1.25x^2 = 0\\\\x = 8 \text{\:(positive root as x denotes side length, thus a non-negative quantity)}[/tex]
At x = 8, the second derivative evaluates to:
[tex]f''(8) = -2.5(8) < 0[/tex]
Thus, we obtain maxima at x = 8.
Thus, we get the maximum value of function when x = 8.
Since we have:
[tex]V = x^2h \leq f(x)[/tex] (V is volume of the hut)
and [tex]max(f(x)) = f(8) = \dfrac{960(8) - 5(8)^3}{12} = 640-213.3\overline{3} \approx 426.67[/tex]
Thus, max(V) = 426.67 sq. ft approximately.
Thus, the figures needed evaluates to:
The cost of the hut expressed in terms of its height h and the length x of the side of the square floor is [tex]12hx + 5x^2 \text{\:(in dollars) }[/tex]The biggest volume hut that can be build with 960 dollars at max is 426.67 sq. ft approxLearn more about maxima and minima here:
https://brainly.com/question/13333267
Two researchers select a sample for a population with a mean of 12.4 and a standard deviation of 9. Researcher A selects a sample of 30 participants. Researcher B selects a sample of 40 participants. Which sample is associated with a smaller standard error?
a. Researcher A's, because the sample size was smaller.
b. Researcher B's, because the sample size was smaller.
c. Researcher A's, because the sample size was larger.
d. Researcher B's, because the sample size was larger.
Answer: Option 'd' is correct.
Step-by-step explanation:
Since we have given that
Researcher A :
Mean = 12.4
Standard deviation = 9
sample size = 30
So, the standard error is given by
[tex]\dfrac{\sigma}{\sqrt{n}}\\\\=\dfrac{9}{\sqrt{30}}\\\\=1.643[/tex]
Researcher B:
Mean = 12.4
Standard deviation = 9
Sample size = 40
So, the standard error is given by
[tex]\dfrac{\sigma}{\sqrt{n}}\\\\=\dfrac{9}{\sqrt{40}}\\\\=1.423[/tex]
Sample B has smaller standard error than sample A because the sample size was larger than A.
Hence, Option 'd' is correct.
explain with words how you find the area of the figure. then find the area.
image attached
Answer:
13x² -3x
Step-by-step explanation:
A horizontal line from the corner of the "notch" will divide the figure into two rectangles whose dimensions are given. The total area is the sum of the areas of those rectangles. Each area is the product of length and width.
A = A1 + A2
= x(3x-7) + 2x(5x+2)
= 3x² -7x +10x² +4x
= (3+10)x² +(-7+4)x
= 13x² -3x
The area is 13x² -3x.
Which of the two painters above can team up to paint the whole room in 3 hours?
Answer:
[tex]\displaystyle Mary\:and\:Neil[/tex]
Step-by-step explanation:
[tex]\displaystyle 2 = \frac{7\frac{1}{2}}{3\frac{3}{4}}[/tex]
Two hours is much closer up three hours, so either way, they would still paint the room in time.
I am joyous to assist you anytime.
Michelle rents a movie for a flat fee of $1.50 plus an additional $1.25 for each night she keeps the movie. Choose the cost function that represents this scenario if x equals the number of nights Michelle has the movie.
A) c(x) = 1.50 + 1.25x
B) c(x) = 1.50x + 1.25
C) c(x) = 2.75
D) c(x) = (1.50 + 1.25)x
Answer:
A) c(x)=1.50+1.25x
Step-by-step explanation:
The fixed rate (constant) is 1.50 and 1.25 (variable) depending on the number of additional nights, that is, c (x) = 1.25 (x) +1.50 =1.50+1.25x
the answer would be A
A farmer has 336 feet of fencing to enclose 2 adjacent rectangular pig pens sharing a common side. What dimensions should be used for each pig pen so that the enclosed area will be a maximum? The two adjacent pens have the same dimensions.
Final answer:
To maximize the enclosed area, each pig pen should have a length of 84 feet and a width of 168 feet.
Explanation:
To find the dimensions of the pig pens that will maximize the enclosed area, we can use the quadratic formula. Let's assume the length of each pen is 'x' feet. Since the two pens share a common side, the combined length of the pens will be '2x' feet. The total length of the fencing, including both sides and the common side, will then be '2x' + 'x' + 'x' = '4x' feet.
According to the problem, the total length of the fencing is 336 feet. Therefore, we can write the equation '4x = 336'. To find the value of 'x', we divide both sides of the equation by 4: 'x = 84'.
So, each pen should have a length of 84 feet and a width of 2x the length, which is '2 × 84 = 168' feet.
Final Answer:
The dimensions for each pig pen that would yield the maximum enclosed area are 84 feet in length and 56 feet in width.
Explanation:
To solve this optimization problem, we can use calculus. Let's denote:
- The length of each pig pen by L
- The width of each pig pen by W
- The total amount of fencing by P, which is 336 feet
Since the two pig pens share a common side, the amount of fencing will be used for 3 widths and 2 lengths. So our perimeter constraint is:
3W + 2L = P
Since we know P is 336 feet, we can write this as:
3W + 2L = 336
We want to maximize the area, A, of the two pens combined. Since the two pens are adjacent and identical, this area can be represented by:
A = 2 * (L * W)
We want to maximize A with respect to our constraint.
First, let's express L in terms of W using our perimeter constraint:
2L = 336 - 3W
L = (336 - 3W) / 2
Now, we can express the area solely in terms of W:
A = 2 * (L * W)
A = 2 * ((336 - 3W) / 2 * W)
A = (336W - 3W^2)
To maximize A, we take the derivative of A with respect to W and set it to zero:
dA/dW = 336 - 6W
Setting dA/dW to zero gives us:
336 - 6W = 0
6W = 336
W = 336 / 6
W = 56
Now we have the width of each pig pen. Next, we'll use the value of W to find L:
L = (336 - 3W) / 2
L = (336 - 3 * 56) / 2
L = (336 - 168) / 2
L = 168 / 2
L = 84
So the dimensions of each rectangular pen that will maximize the area with 336 feet of fencing are:
- Length (L) = 84 feet
- Width (W) = 56 feet
We should verify this solution is a maximum by checking the second derivative of the area function:
d²A/dW² = -6
Since the second derivative is negative, our critical point W = 56 feet corresponds to a maximum. Therefore, the dimensions for each pig pen that would yield the maximum enclosed area are 84 feet in length and 56 feet in width.
Suppose that Bob places a value of $10 on a movie ticket and that Lisa places a value of $7 on a movie ticket. In addition, suppose the price of a movie ticket is $5. Refer to Scenario 12-2. Suppose the government levies a tax of $1 on each movie ticket and that, as a result, the price of a movie ticket increases to $6.00. If Bob and Lisa both purchase a movie ticket, what is total consumer surplus for Bob and Lisa?
Final answer:
Bob's consumer surplus after a tax is $4, and Lisa's is $1, making the total consumer surplus for both after the tax $5.
Explanation:
Consumer surplus is the difference between the value a consumer places on a good and what they actually pay. Before the government levies a tax, Bob's consumer surplus for a movie ticket is the difference between his valuation of $10 and the market price of $5, which is $5. Lisa's consumer surplus is the difference between her valuation of $7 and the market price of $5, which is $2.
After the government implements a $1 tax on movie tickets, increasing the price to $6, Bob's consumer surplus becomes $4 ($10 - $6), and Lisa's consumer surplus is now $1 ($7 - $6). Thus, the total consumer surplus for Bob and Lisa after the tax is implemented is $5 ($4 for Bob and $1 for Lisa).
A central angle in a circle has a measure of 180 The length of the arc it intercepts is 8 in.
What is the radius of the circle?
**Use 3.14 for π and round your answer to ONE decimal place.
Answer:
Step-by-step explanation:
An arc is the length along the circumference of a circle that is bounded by 2 radii. I
A central angle in a circle has a measure of 180. This means that the arc subtends an angle of 180 degrees at the center of the circle.
The length of the arc is 8 inches
Formula for the length of an arc is expressed as
Length of arc = #/360 × 2πr
r = radius of the circle
Length of arc = 8 inches
# 180 degrees
π = constant = 3.14
Substituting,
8 = 180/360 ×2 × 3.142 × r
8 = 3.14r
r = 8/3.14 = 2.56 inches
A spaceship traveled 3 4 of a light-year and stopped at a space station. Then it traveled 1 12 of a light-year further to a planet. How far did the spaceship travel in all?
Final answer:
The total distance the spaceship traveled is 5/6 of a light-year, which was obtained by adding 3/4 and 1/12 light years together after finding a common denominator for the fractions.
Explanation:
The student's question is about calculating the total distance a spaceship travels based on two separate distances given in light years. To find this total distance, we will perform an addition of the two distances.
First, the spaceship traveled 3/4 of a light-year to a space station. Then, it traveled an additional 1/12 of a light-year to a planet.
To get the total distance traveled, we simply add the two distances:
(3/4) light years + (1/12) light yearsTo add these fractions, we need a common denominator, which is 12 in this case:
(9/12) light years + (1/12) light yearsNow, when we add these fractions, we get:
(9/12 + 1/12) light years = (10/12) light yearsWe can simplify this fraction to:
(5/6) light yearsTherefore, the spaceship traveled a total of 5/6 of a light-year.
Ethan is saving money in his piggy bank for his upcoming trip to Disney world on the first day he put in $12 and plans to add seven more dollars each day write an explicit formula that can be used to find the amount of money saved on any given day
Answer:
S=12+7D
Step-by-step explanation:
Linear relationships.
The initial amount of money Ethan has is $12. Each day, he adds up $7 to his savings. At a given day D after his initial funding, he will have added $7D, and he will have in his piggy bank
S=12+7D
For example, on the day D=30 he will have
S=12+7(30)=$222
A garden center sells a certain grass seed in 5-pound bags at $13.85 per bag, 10-pound bags at $20.43 per bag, and 25-pound bags $32.25 per bag. If a customer is to buy at least 65 pounds of the grass seed, but no more than 80 pounds, what is the least possible cost of the grass seed that the customer will buy?
A) $94.03
B) $96.75
C) $98.78
D) $102.07
E) $105.3
Answer:
B)$96.75
Step-by-step explanation:
5pound bag cost = $13.85
10pound bag cost= $20.43
25pound bag cost = $32.25
The least quantity of bag the customer can by is 65 pounds = (2*25 pound)+ 10 pound + 5pound = $98.78
But careful examination will show actually that since the most the customer can buy is 80 pound, then buying 25pound in three places give him even a lot cheaper than buying the least amount
75 pound = 3*25 pound = 32.25*3 = $96.75.
Therefore the least amount in cost the customer can buy is $96.75
Which relations are functions? List the relation number(s) that are function(s) in the answer bank
1. (3, 2), (9, 1), (-4, 7), and (0, -2)
2. (7, 1), (-5, 2), (1, 0), and (-5, 3)
3. (-2, -4), (2, 4), (6, 8), and (-6, -8)
4. (1, 3), (-1, 3), (2, 3), and (-2, 3)
Answer:
options 1,3,4 are functions.
Step-by-step explanation:
RULE: a relation is said to be a function if every element in the domain ( the numbers in the left side in the below sets) is related to only one number ( number on the right side in the below sets).
Let us check each option one by one:
1. 3 2
9 1
-4 7
0 -2
here each number on the left side is mapped to or is related to one number only.
so this relation is a function
2. 7 1
-5 2,3
1 0
here, "-5" is mapped to two different numbers. so this relation is not a function.
3. -2 -4
2 4
6 8
-6 -8
here each number on the left side is mapped to or is related to one number only.
so this relation is a function
4. 1 3
-1 3
2 3
-2 3
here each number on the left side is mapped to or is related to one number only.
so this relation is a function.
even if it is related to the same number, it doesn't matter.
it should follow the above given rule that's it.
A group of 68 friends meets for lunch. They greet each other by exchanging fist bumps. How many fist bumps are exchanged if each friend must bump with each of the 67 others? The total number of fist bumps exchanged is nothing .
Answer: 68C2 = 2278
Step-by-step explanation:
The midpoint of the segment connecting the points labeled K and N has coordinates (5, -4). If the coordinates of point K are (7, -2) then what are the coordinates of point N?
Answer:
Co-ordinates of point N is (3,-6)
Step-by-step explanation:
Given point:
Endpoint K(7,-2)
Mid-point of segment KN (5,-4)
Let endpoint [tex]N[/tex] have co-ordinates [tex](x_2,y_2)[/tex]
Using midpoint formula:
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are co-ordinates of the endpoint of the segment.
Plugging in values to find the midpoint of segment KN.
[tex]M=(\frac{7+x_2}{2},\frac{-2+y_2}{2})[/tex]
We know [tex]M(5,-4)[/tex]
So, we have
[tex](5,-4)=(\frac{7+x_2}{2},\frac{-2+y_2}{2})[/tex]
Solving for [tex]x_2[/tex]
[tex]\frac{7+x_2}{2}=5[/tex]
Multiplying both sides by 2.
[tex]\frac{7+x_2}{2}\times 2=5\times 2[/tex]
[tex]7+x_2=10[/tex]
Subtracting both sides by 7.
[tex]7+x_2-7=10-7[/tex]
∴ [tex]x_2=3[/tex]
Solving for [tex]y_2[/tex]
[tex]\frac{-2+y_2}{2}=-4[/tex]
Multiplying both sides by 2.
[tex]\frac{-2+y_2}{2}\times 2=-4\times 2[/tex]
[tex]-2+y_2=-8[/tex]
Adding both sides by 2.
[tex]-2+y_2+2=-8+2[/tex]
∴ [tex]y_2=-6[/tex]
Thus co-ordinates of point N is (3,-6)
Suppose that f(x)=14x−6x3. (A) Find the average of the x values of all local maxima of f. Note: If there are no local maxima, enter -1000.
Answer:
Maximum at [tex]x =\frac{\sqrt{7}}{3}[/tex]
Step-by-step explanation:
Given function,
[tex]f(x) = 14x - 6x^3[/tex]
Differentiating with respect to x,
[tex]f'(x) = 14 - 18x^2----(1)[/tex]
For critical values :
[tex]f'(x) = 0[/tex]
[tex]14 - 18x^2 =0[/tex]
[tex]14 = 18x^2[/tex]
[tex]x^2 = \frac{14}{18}[/tex]
[tex]x^2=\frac{7}{9}[/tex]
[tex]x = \pm \frac{\sqrt{7}}{3}[/tex]
Now, differentiating equation (1) again with respect to x,
[tex]f''(x) = -36x[/tex]
Since,
[tex]f''(\frac{\sqrt{7}}{3}) = -36(\frac{\sqrt{7}}{3}) < 0[/tex]
This means that the function is maximum at [tex]x=\frac{\sqrt{7}}{3}[/tex]
While,
[tex]f''(-\frac{\sqrt{7}}{3}) = 36(\frac{\sqrt{7}}{3}) > 0[/tex]
This means that the function is minimum at [tex]x=-\frac{\sqrt{7}}{3}[/tex]
A company that manufactures small canoes has a fixed cost of $20,000. It cost $40 to produce each canoe. The selling price is $80 per canoe. (In solving this exercise, let x represent the number of canoes produced and sold.)
1. Write the cost function.
2. Write the revenue function.
3. Determine the break-even point. Make sure your answer is an ordered pair.
4. This means that when the company produces and sells the break-even number of
canoes: a. there is less money coming in than going out b. the money coming in
equals the money going out c. there is more money coming in than going out d.
there is not enough information
Please explain how to work all of this out.
Answer:1) $40x 2)$80x 3) 500units 4)b
Step-by-step explanation:
For the cost function, which is the amount used for production, we are told to use x and number of canoes produced, and canoe is produced at $40 per canoe, multiplying both
So production cost is $40x
And each canoe is sold at $80 per canoe, multiplying with no of canoes
so revenue is $80x
The break even cost happens when the amount of money put into the business equals the amount of revenue got, so total amount of money put into the business is the addition of the fixed cost and production cost of the canoes which is $20,000 + $40x (1)
And the revenue cost is 80x (2)
So equating (1) and (2) together, we find the value of x to reach the break even point
20000 + 40x = 80x
20000 = 80x - 40x
20000 = 40x
20000/40 = x
x = 500 units
I've already explained the answer to 4 being option b, because that's the fact we used to solve the amount of units to produce and sell to reach the break even point
Anthony purchases two bags. The price of all bags is $5.20. Anthony purchases one school bag and one hand bag. Write an expression that represents the total cost,T, of the bag if s represents the number of school bags and h represents the number of hand bags
Answer:
t=5.20s
Step-by-step explanation:
I think that is the expression because it say he purchases two bags, all the bags are 5.20 so he brought two bags which are school bag and one hand bag the formula I used was y=mx+b but in these case I had to put t=5.20x. I hope this really helped you..
Answer:
Step-by-step explanation:
Anthony purchases two bags. The price of all bags is $5.20
Anthony purchases one school bag and one hand bag. It means that he purchased one handbag and one school bag for a total cost of $5.20
Let s represent the number of school bags and
Let h represent the number of hand bags.
An expression that represents the total cost,T, of the bag will be
T = s + h
Since total cost = $5.20
Then,
5.20 = s + h
Find the y-intercept of the line on the graph. HELP PLEASE !!!!
Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is:
a. reflexive and transitive
b. symmetric and transitive
c. reflexive, symmetric, and transitive.
Answer:
Remember, if B is a set, R is a relation in B and a is related with b (aRb or (a,b))
1. R is reflexive if for each element a∈B, aRa.
2. R is symmetric if satisfies that if aRb then bRa.
3. R is transitive if satisfies that if aRb and bRc then aRc.
Then, our set B is [tex]\{1,2,3,4\}[/tex].
a) We need to find a relation R reflexive and transitive that contain the relation [tex]R1=\{(1, 2), (1, 4), (3, 3), (4, 1)\}[/tex]
Then, we need:
1. That 1R1, 2R2, 3R3, 4R4 to the relation be reflexive and,
2. Observe that
1R4 and 4R1, then 1 must be related with itself.4R1 and 1R4, then 4 must be related with itself.4R1 and 1R2, then 4 must be related with 2.Therefore [tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(4,1),(4,2)\}[/tex] is the smallest relation containing the relation R1.
b) We need a new relation symmetric and transitive, then
since 1R2, then 2 must be related with 1.since 1R4, 4 must be related with 1.and the analysis for be transitive is the same that we did in a).
Observe that
1R2 and 2R1, then 1 must be related with itself.4R1 and 1R4, then 4 must be related with itself.2R1 and 1R4, then 2 must be related with 4.4R1 and 1R2, then 4 must be related with 2.2R4 and 4R2, then 2 must be related with itselfTherefore, the smallest relation containing R1 that is symmetric and transitive is
[tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}[/tex]
c) We need a new relation reflexive, symmetric and transitive containing R1.
For be reflexive
1 must be related with 1,2 must be related with 2,3 must be related with 3,4 must be related with 4For be symmetric
since 1R2, 2 must be related with 1,since 1R4, 4 must be related with 1.For be transitive
Since 4R1 and 1R2, 4 must be related with 2,since 2R1 and 1R4, 2 must be related with 4.Then, the smallest relation reflexive, symmetric and transitive containing R1 is
[tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}[/tex]
To find the smallest relation containing given pairs with specific properties, we add missing pairs and ensure all existing pairs satisfy the required properties.
Explanation:a. To find the smallest relation that is reflexive and transitive, we need to add any missing pairs that would make the relation reflexive and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it reflexive, we add (2, 2) and (4, 4). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is reflexive and transitive is {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
b. To find the smallest relation that is symmetric and transitive, we need to add any missing pairs that would make the relation symmetric and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it symmetric, we need to add (2, 1) and (4, 4). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is symmetric and transitive is {(1, 1), (1, 2), (1, 4), (2, 1), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
c. To find the smallest relation that is reflexive, symmetric, and transitive, we need to add any missing pairs that would make the relation reflexive, symmetric, and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it reflexive, we add (2, 2) and (4, 4). To make it symmetric, we need to add (2, 1). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is reflexive, symmetric, and transitive is {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
Learn more about Sets and Relations here:https://brainly.com/question/34619284
#SPJ11
I really need help with this, I got 13.7 yards as the altitude by using the law of cosines and I found the area of the triangle by using the formula 1/2(side)(side)cos
Answer:
The length of the altitude is 9.3 yards and
The area of the triangle Δ UVW is 139.3 yd².
Step-by-step explanation:
Given
WU = 22 yd
WV = 30 yd
∠ UWV = 25°
To Find:
Altitude, UM = ?
area of the Δ UVW = ?
Construction:
Draw UM perpendicular to WV, that is altitude UM to WV.
Solution:
In right triangle Δ UWM if we apply Sine to angle W we get
[tex]\sin W = \frac{\textrm{side opposite to angle W}}{Hypotenuse}\\ \sin W=\frac{UM}{UW} \\[/tex]
substituting the values we get
[tex]\sin 25 = \frac{UM}{22}\\0.422 = \frac{UM}{22} \\UM = 0.422\times 22\\UM = 9.284\ yd[/tex]
Therefore, the altitude from U to WV is UM = 9.3 yd.(rounded to nearest tenth)
Now for area we have formula
[tex]\textrm{area of the triangle UVW} = \frac{1}{2}\times Base\times Altitude \\\textrm{area of the triangle UVW} = \frac{1}{2}\times VW \times UM\\=\frac{1}{2}\times 30\times 9.284\\ =139.26\ yd^{2}[/tex]
The area of the triangle Δ UVW is 139.3 yd². (rounded to nearest tenth)
Assume that John Smith is a salesperson employed by McCrackin Company. Smith's regular rate of pay is $36 per hour, and any hours worked in excess of 40 hours per week are paid at 1½ times the regular rate. Smith worked 42 hours for the week ended October 27. What are his total earnings for the week?
Answer: $1548
Step-by-step explanation:
We are told the normal rate of payment is $36 per hour
and with an excess of 40 hours the pay will be 1 and a half the normal rate(1.5)
And John works for 42hours
For first we know John worked for an excess of 2 hours
And calculating his pay for 40hours of the normal rate that week, we multiply $36 by 40 which will give $1440
Then the extra 2 hours, the new pay rate will be $36 multiplied by 1.5 which will give $54 per hour
And for the extra 2 hours, John will get extra $54 multiplied by 2 which will give $108
Adding both $1440 and $108, we will get $1548
true or false, and explain: (a) If a die is rolled three times, the chance of getting at least one ace is 1/6 + 1/6 + 1/6 = 1/2. (b) If a coin is tossed twice, the chance of getting at least one head is 100%.
Answer:
a.False
b.False
Step-by-step explanation:
a.Total possible outcomes of a die=1,2,3,4,5,6=6
Probability of getting an ace=[tex]\frac{favorable\;cases}{total\;number\;of\;cases}[/tex]
Favorable cases=1
Probability of getting an ace=[tex]\frac{1}{6}[/tex]
A die is rolled three times .
We are given that the probability of getting at least one ace is
[tex]\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}[/tex]
There are using addition rule but it is not correct because addition rule used when the events are mutually exclusive .
The events are mutually exclusive when the events cannot occur at the same time.
Since it is possible to obtain one ace on more than one roll of a die.
Therefore, the events are not mutually exclusive.
Hence, the given statement is false.
b.Total cases in one coin=2(H,T)
Number of cases in favor of head=1
The probability of getting on head=[tex]\frac{1}{2}[/tex]
The coin is tossed twice.
We are given that
If a coin is tossed twice ,the chance of getting at least on head is 100%.
There are using addition rule but it is not correct because addition rule used when the events are mutually exclusive .
The events are mutually exclusive when the events cannot occur at the same time.
Since it is possible to obtain head on both tosses of coin.
Therefore, the events are not mutually exclusive.
Hence, the given statement is false.
Both statements are false. For the dice rolls, the chance of getting at least one ace is calculated by finding the complementary probability. For the coin tosses, the chance of getting at least one head is 3 in 4, not 100%.
Explanation:The question revolves around basic probability concepts applied to dice and coin tossing. Firstly, part (a) of the question is false. When a die is rolled three times, the chances of getting at least one ace (or a one) are not simply the sum of the individual probabilities. Events are independent, meaning the outcome of one roll doesn't affect the other.
The correct approach is to calculate the probability of not getting an ace in all three rolls (5/6 * 5/6 * 5/6) and subtract this from 1 to get the complementary probability of at least one ace.
For part (b), the statement is also false. When a coin is tossed twice, the chance of getting at least one head is not 100%. To find the correct probability, we can list all possible outcomes (HH, HT, TH, TT) and calculate that there is a 3 in 4 chance of getting at least one head.
A system of inequalities is shown:
y ≤ x + 6
y ≥ 9x − 9
Which point is in the solution set of the system of inequalities shown? Explain your answer.
Question 5 options:
(3, 10) because it lies below the boundary line y=9x−9 and above the boundary line y=x+6.
(-1, 7) because it lies below the boundary line y=x+6.
(3, 2) because it lies above the boundary line y=9x−9.
(-2, 2) because it lies above the boundary line y=9x−9 and below the boundary line y=x+6.
Answer:
(-2, 2)
Step-by-step explanation:
Well first you would have to graph both inequalities.
y = mx + b
b is the y-axis intercept
m is the slope
Of the 36 students in a certain class, 10 are in the chess club and 13 are in the bridge club. If 20 of the students are not in either club, how many of the students are in only one of the two clubs?A. 7B. 9C. 14D. 16E. 23
There are 9 students in only one of the two clubs.
Step-by-step explanation:
Since we have given that
Number of students = 36
Number of students are in chess club = 10
Number of students are in bridge club = 13
Number of students are not in either club = 20
So, Number of students in both the club is given by
[tex]Total=n(chess)+n(bridge)-n(both)+n(neither)\\\\36=10+13-n(both)+20\\\\36=43-n(both)\\\\36-43=n(both)\\\\-7=-n(both)\\\\n(both)=7[/tex]
Number of students only in chess = 10-7 =3
Number of students only in bridge = 13-7=6
Hence, there are 3+6=9 students in only one of the two clubs.
A truck driver drives from Chicago to Cincinnati in 14 hours. The distance traveled is 840 miles. Write the average speed as a unit rate in fraction form
Answer:
60 miles/hour
Step-by-step explanation:
840 miles divided by 14 hours
840/14=60 miles per hour
Murphy loves pistachio nuts. Every saturday morning, she walks to the market and buy some. Last week, she bought two pounds and paid $7.96, and this week she bought only one half pound and paid $1.99
Question is Incomplete, Complete question is given below,
Murphy loves pistachio nuts. Every Saturday morning, she walks to the market and buys some. Last week, she bought two pounds and paid $7.96, and this week she bought only one-half pound and paid $1.99. What the unit rate for pistachio nuts?
Answer:
The unit rate for pistachio nuts is $3.98.
Step-by-step explanation:
Given;
Price of 2 pounds of pistachio nuts = $7.96
We need to find the price of 1 pound of pistachio nuts.
To find the same we will use the unitary method,
Hence ,
Price of 1 pound of pistachio nuts = [tex]\frac{\$7.96}{2} = \$3.98[/tex]
Also given:
Price of half pound of pistachio nuts = $1.99
We need to find the price of 1 pound of pistachio nuts.
To find the same we will use the unitary method,
Hence ,
Price of 1 pound of pistachio nuts = [tex]\$1.99\times 2 = \$3.98[/tex]
Hence, The unit rate for pistachio nuts is $3.98
A plumber charges $45 per hour, plus a one-time fee for making a house call. The total fee for 3 hours of service is $285. Write the point-slope form of an equation to find the total fee y for any mumber if hours
Answer:
y - 285 = 45(x - 3)
Step-by-step explanation:
The given point is (hours, fee) = (3, 285), and the slope is given as 45 per hour.
The point-slope form of the equation for a line is ...
y - k = m(x - h) . . . . . . . for a slope m and a point (h, k)
Using the given values, and letting x stand for the number of hours, the equation is ...
y - 285 = 45(x - 3)
Answer:
y - 285 = 45(x - 3)
Step-by-step explanation:
Find the inverse function of
(Show work)
f(x)=x^2-4
Answer:
The answer to your question is [tex]f(x) = \sqrt{x+ 4}[/tex]
Step-by-step explanation:
f(x) = x² - 4
Process
1.- Change f(x) for y
y = x² - 4
2.- Change "x" for "y" and "y" for "x".
x = y² - 4
3.- Make "y" the object of the equation
y² = x + 4
[tex]y = \sqrt{x+ 4}[/tex]
4.- Change "y" for f(x)
[tex]f(x) = \sqrt{x+ 4}[/tex]
In the first equation in the system of equations, y represents the money collected from selling sweatshirts. In the second equation, y represents the money spent to produce x sweatshirts with team logos on them for a professional sports league.
y=35x; y=-0.05(x-400)^2+9,492
What does the solution of the system represent in this context?
The solution to the system of equations represents the break-even point for the production and sale of sweatshirts, indicating the number of units that must be sold to cover production costs.
Explanation:The solution of the system of equations y=35x and y=-0.05(x-400)^2+9,492 represents the break-even point where the money collected from selling x number of sweatshirts equals the money spent to produce those sweatshirts. In this context, solving the system means finding the number of sweatshirts (x) that need to be sold at $35 each to exactly cover the production costs described by the second equation.
This involves first substituting the expression for y from the first equation into the second equation, then solving for x to find the exact point where income equals expenses. Once x is found, it can be plugged back into either equation to verify the break-even amount of money (y).
A processor of carrots cuts the green top off each carrot, washes the carrots, and inserts six to a package. Twenty packages are inserted in a box for shipment. Each box of carrots should weigh 20.4 pounds. The processor knows that the standard deviation of box weight is 0.5 pound. The processor wants to know if the current packing process meets the 20.4 weight standard. How many boxes must the processor sample to be 95% confident that the estimate of the population mean is within 0.2 pound?
Answer:
24 boxes
Step-by-step explanation:
The processor knows that the standard deviation of box weight is 0.5 pound
[tex]\sigma = 0.5[/tex]
We are supposed to find How many boxes must the processor sample to be 95% confident that the estimate of the population mean is within 0.2 pound
Formula of Error=[tex]z \times \frac{\sigma}{\sqrt{n}}[/tex]
Since we are given that The estimate of the population mean is within 0.2 pound
So, [tex]z \times \frac{\sigma}{\sqrt{n}}=0.2[/tex]
z at 95% confidence level is 1.96
[tex]1.96 \times \frac{0.5}{\sqrt{n}}=0.2[/tex]
[tex]1.96 \times \frac{0.5}{0.2}=\sqrt{n}[/tex]
[tex]4.9=\sqrt{n}[/tex]
[tex](4.9)^2=n[/tex]
[tex]24.01=n[/tex]
Hence the processor must sample 24 boxes to be 95% confident that the estimate of the population mean is within 0.2 pound
To be 95% confident that the estimate of the population mean is within 0.2 pound, the processor must sample approximately 25 boxes, as the calculation using the sample size estimation formula indicates.
To determine how many boxes must be sampled to be 95% confident that the estimate of the population mean is within 0.2 pound, we use the formula for the sample size in estimation:
n = (Z·σ/E)^2
Where:
n is the sample sizeZ is the z-score corresponding to the desired confidence levelσ is the population standard deviationE is the margin of errorFor a 95% confidence level, the z-score (Z) is approximately 1.96. Given that the population standard deviation (σ) is 0.5 pound and the desired margin of error (E) is 0.2 pound, the formula becomes:
n = (1.96· 0.5/0.2)^2
Calculating:
n = (1.96· 2.5)^2
n = (4.9)^2
n = 24.01
The processor must sample approximately 25 boxes (since we round up to the nearest whole number when it comes to sample size) to be 95% confident that the estimate of the population mean is within 0.2 pound.
In a simple linear regression model, the slope term is the change in the mean value of y associated with _____________ in x.
A) a variable change
B) a corresponding increase
C) a one-unit increase
D) no change
Answer:
a one-unit increase
Step-by-step explanation:
In a simple regression model, the relationship between x and y can be represented by the equation y = ax+b, where
a is the slopeb is the y-interceptThe slope term is the change in the mean value of y associated with a one-unit increase in x.