A spinner has three sections which are coloured black, red and yellow. Lincoln spun the spinner 100 times in total.
The frequency of spins that landed on red was
35.
The ratio of the frequency of black to the frequency of red was 2: 7.
What is the estimated probability of the spinner landing on yellow?
Give your answer as a decimal.
Answers
Let's use the ratio of black to red frequency to find the frequency of black spins:
Black frequency : Red frequency = 2 : 7
Let x be the frequency of red spins. Then, the frequency of black spins is 2/7 of x, or:
Black frequency = (2/7)x
The total frequency of all three colors is 100, so we can write:
Black frequency + Red frequency + Yellow frequency = 100
Substituting the values we have:
(2/7)x + x + Yellow frequency = 100
Simplifying and solving for x, we get:
(9/7)x = 100 - Yellow frequency
x ≈ 45.22 - (7/9)Yellow frequency
We know that the total number of spins is 100, so we can write:
Black frequency + Red frequency + Yellow frequency = 100
(2/7)x + x + Yellow frequency = 100
Substituting the value we have for x, we get:
(2/7)(45.22 - (7/9)Yellow frequency) + 45.22 - (7/9)Yellow frequency + Yellow frequency = 100
Simplifying and solving for Yellow frequency, we get:
Yellow frequency ≈ 21.08
Therefore, the estimated probability of the spinner landing on yellow is 21.08/100, or approximately 0.2108 as a decimal.
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Related Questions
Esfandairi Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2.18 million. The fixed asset will be depreciated straight-line to zero over its three-year tax life. The project is estimated to generate $1,730,000 in annual sales, with costs of $640,000. The project requires an initial investment in net working capital of $290,000, and the fixed asset will have a market value of $240,000 at the end of the project.
a. If the tax rate is 24 percent, what is the project’s Year 0 net cash flow? Year 1? Year 2? Year 3? (A negative answer should be indicated by a minus sign. Do not round intermediate calculations and enter your answers in dollars, not millions of dollars, e.g., 1,234,567.)
b.
If the required return is 12 percent, what is the project's NPV? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
Answers
Answer:
a
Step-by-step explanation:
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The net present value of the project is $0.15 million.
What is the net present value?Net present value is the present value of after-tax cash flows from an investment less the amount invested.
hee, we have,
The first step is to determine the cash flow of the project.
Cash flow = (sales - cost - depreciation) × (1 - tax) + depreciation
Depreciation = (cost of the asset ) / useful life
Depreciation = $2.18 million / 3
Depreciation = 0.727 million
Cash flow = (1.645 - 0.61 - 0.727) x (1 - 0.21) + 0.727 = $0.97 million
Net present value = -2.18 million + 0.97 / 1.12 + 0.97 / 1.12² + 0.97 / 1.12³
= $0.15 million
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what is 3 plus 3???
as a joke :)
Answers
Answer:6
Step-by-step explanation:biyhfdvsauybi
a 4. In the following diagram it is given that ZABD is congruent to ZEBC. Based on this information, which of the following angle pairs can be proven congruent? (1) ZBDE and ZBCA (2) ZBCE and ZBDA (3) ZBAD and ZREC (4) ZABC
Answers
Based on this information, the angle pairs can be proven congruent is option D. ∠ABC and ∠EBD.
What are congruent angles?Congruent angles are angles that have the same measure or degree. In other words, two angles are congruent if they have the same amount of rotation and are of the same size and shape, regardless of their position or orientation.
Congruent angles are denoted by the symbol ≅, which is read as "congruent to". For example, if angle A and angle B both measure 45 degrees, we can say that angle A ≅ angle B, or that angle A is congruent to angle B.
Therefore, angle ABC and angle EBD shares same measurement or same degrees. Angle ABC is congruent to angle EBD.
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congruent arcs determine ___ central angles
options :
1. complementary
2. congruent
3. infinitely many
4. supplementary
5. three
Answers
Answer:
Congruent arcs determine congruent central angles. This is one of the properties of circles and is true for any two arcs of a circle that have the same measure.
Step-by-step explanation:
Use the graph to answer the questions:
Answers
1) a) slοpe m= 1, y-intercept =0 and equatiοn is y=x.
2) a) slοpe m=-1 , y-intercept = 4 and equatiοn is y=-x+4.
What is equatiοn?The definitiοn οf an equatiοn in algebra is a mathematical statement that demοnstrates the equality οf twο mathematical expressiοns. Fοr instance, the equatiοn 3x + 5 = 14 cοnsists οf the twο equatiοns 3x + 5 and 14, which are separated by the 'equal' sign.
Here we knοw that, we can sοlve using any twο pοints.
1) Let us take any twο pοints, then
[tex](x_1,y_1)=(3,3) and (x_2,y_2) =(-3,-3)[/tex]
Now using slope formula ,
=> Slope m =[tex]\frac{y_2-y_1}{x_2-x_1}=\frac{-3-3}{-3-3}=\frac{-6}{-6}=1[/tex]
Now using equation formula [tex]y-y_1=m(x-x_1)[/tex]
=> y-3=1(x-3)
=> y-3=x-3
=>y=x-3+3
=> y=x
Hence slope m=1, y-intercept = 0 and equation is y=x.
2) Let us take two points then
[tex](x_1,y_1)=(0,4) and (x_2,y_2)=(4,0)[/tex]
Now using slope formula ,
=> Slope m = [tex]\frac{y_2-y_1}{x_2-x_1}=\frac{0-4}{4-0}=\frac{-4}{4}=-1[/tex]
Now using equation formula then , [tex]y-y_1=m(x-x_1)[/tex]
=> y-4=-1(x-0)
=>y-4=-x
=> y=-x+4
Hence slope = -1 , y-intercept = 4 and the equation is y=-x+4.
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First 500litres=N5
Next 500litres= N6.50k
Next 500 litres =N8.00
Above 1500 litres=N10.
Find the water rate paid for 7500 litres.
Answers
Therefore , the solution of the given problem of unitary method comes out to be N69,750 was given as the water rate for 7500 litres of water.
What is an unitary method?One may use this commonly utilized ease, preexisting variables, or any important elements from the initial Diocesan customizable query to achieve the goal. If so, there will be another chance to engage with the object. If not, every important component that influences the manner in which evidence of an algorithm functions will be lost.
Here,
The price for the first 500 gallons is N5 per litre, so
=> 500 * N5 = N2,500.
The price for the following 500 gallons is N8.00 per litre, or
=> 500 * N8.00 = N4,000.
The expense for this tier is
=> 3000 * N10.00 = N30,000 because the following 3000 litres (from 1501 to 4500 litres) are charged at N10.00 per litre.
The expense for this tier is
=> 3000 * N10.00 = N30,000 because the final 3000 litres (from 4501 to 7500 litres) are also charged at N10.00 per litre.
As a result, the full price for 7500 liters of water is:
The sum of N2,500, N3,250, N4,00, N30,000, and N30,000 is N69,750.
Therefore, N69,750 was given as the water rate for 7500 litres of water.
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What is the meaning of "a group under composition"?
Answers
The resulting set of isometries, together with the operation of composition, forms a group.
Define the term "a group under composition" in given paragraph?In the context of the paragraph, "a group under composition" refers to a group that is formed by combining isometries (permutations that preserve distances) of a geometric configuration through the operation of composition.
In other words, if you have a set of isometries of a geometric configuration (such as translations, rotations about a point, and symmetries about a straight line), you can combine them through composition (applying one after the other) to obtain new isometries. The resulting set of isometries, together with the operation of composition, forms a group, which is called the group of isometries of the configuration.
This group has certain properties, such as closure (meaning that the result of composing two isometries is also an isometry), associativity (meaning that the order of composition doesn't matter), identity (meaning that there is an isometry that leaves all points fixed), and inverses (meaning that for every isometry, there is another isometry that undoes its effect). These properties make the group a useful tool for studying the symmetries and transformations of geometric configurations in fields such as crystallography and quantum mechanics.
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Siobhan rotates a right triangle 90° counterclockwise about the origin.
Is the image of the triangle congruent to the preimage?
Answers
Answer:
yes
Step-by-step explanation:
congruent means similar
Select the correct answer. Which is the correct solution to the expression 3 + 5^2? You can use a calculator to find the answer. A. 10 B. 13 C. 28 D. 64
Answers
The correct solution to the expression 3 + 5^2 is C. 28.
What is an expression?Mathematically, an expression is a combination of variables with numbers, values, or constants.
Mathematical or algebraic expressions use the mathematical operands like addition (+), subtraction (-), division (÷), multiplication (×), exponents (^), etc.
An algebraic expression does not go with the equal symbol (=), unlike an equation.
3 + 5^2
= 3 + 5 x 5
= 3 + 25
= 28
Thus, an evaluation of the algebraic expression 3 + 5^2 gives the solution as Option C.
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When a specific variety of radish is grown without
fertilizer, the weights of the radishes produced are
, normally distributed with mean 40 g and standard
deviation 10 g.
Determine the proportion of radishes grown:
a) Without fertilizer with weights less than 50 grams
C
b) Without fertilizer with weights between 20 and 60 g
c) Without fertilizer with weights greater than 60 g
0
Submit
When the same variety of radish is grown in the same
way but with fertilizer added, the weights of the
radishes produced are also normally distributed, but
with mean 140 g and standard deviation 40g.
Determine the proportion of radishes grown:
d) With fertilizer with weights less than 60 grams
e) With fertilizer with weights between 20 and 60 g
f) With fertilizer with weights greater than 60 g
Answers
Answer:
Step-by-step explanation:
a) Without fertilizer with weights less than 50 grams:
Let X be the weight of radishes produced without fertilizer. Then, X ~ N(40, 10^2).
We need to find P(X < 50).
Using the standard normal distribution, we have:
Z = (X - 40)/10
P(X < 50) = P(Z < (50-40)/10) = P(Z < 1) = 0.8413 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown without fertilizer with weights less than 50 grams is 0.8413.
b) Without fertilizer with weights between 20 and 60 g:
We need to find P(20 < X < 60).
Using the standard normal distribution, we have:
Z1 = (20 - 40)/10 = -2
Z2 = (60 - 40)/10 = 2
P(20 < X < 60) = P(-2 < Z < 2) = 0.9544 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown without fertilizer with weights between 20 and 60 grams is 0.9544.
c) Without fertilizer with weights greater than 60 g:
We need to find P(X > 60).
Using the standard normal distribution, we have:
Z = (60 - 40)/10 = 2
P(X > 60) = P(Z > 2) = 0.0228 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown without fertilizer with weights greater than 60 grams is 0.0228.
d) With fertilizer with weights less than 60 grams:
Let Y be the weight of radishes produced with fertilizer. Then, Y ~ N(140, 40^2).
We need to find P(Y < 60).
Using the standard normal distribution, we have:
Z = (60 - 140)/40 = -2
P(Y < 60) = P(Z < -2) = 0.0228 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown with fertilizer with weights less than 60 grams is 0.0228.
e) With fertilizer with weights between 20 and 60 g:
We need to find P(20 < Y < 60).
Using the standard normal distribution, we have:
Z1 = (20 - 140)/40 = -3
Z2 = (60 - 140)/40 = -2
P(20 < Y < 60) = P(-3 < Z < -2) = 0.0668 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown with fertilizer with weights between 20 and 60 grams is 0.0668.
f) With fertilizer with weights greater than 60 g:
We need to find P(Y > 60).
Using the standard normal distribution, we have:
Z = (60 - 140)/40 = -2
P(Y > 60) = P(Z > -2) = 0.9772 (using a standard normal table or calculator)
Therefore, the proportion of radishes grown with fertilizer with weights greater than 60 grams is 0.9772.
You can afford a $350 per month car payment. You've found a 5 year loan at 6% interest. How big of a loan can you afford?
Answers
Find the interest on $4000 at 3.5% annual interest for 1 year 6 months
Answers
the interest for 1 year:
[tex]3.5\% \cdot 4000 = 140[/tex]
the interest for 6 months:
[tex]\dfrac{140}{2} = 70[/tex]
the interest for 1 year 6 months:
[tex]140+70 = \bf \$ 210[/tex]
Kareem read 36 pages in his book in two days how many pages can you read in seven days
Answers
I attached my work in the image below because it involves too many fractions to type out. Essentially, you’re going to want to set up a proportion to find the amount of pages he can read in one day (which ends up being 18.) Once you have the number of pages he can read in one day, multiply that by the number of days in the question (7.) 7 x 18 = 126.
Hope this is helpful & accurate!
Use a calculator to find the measure of
Answers
Answer:
The answer is 78.5° .................
Answer:
78.5°
Step-by-step explanation:
For <A, AC is the adjacent leg.
AB is the hypotenuse.
The trig function that relates the adjacent leg to the hypotenuse is the cosine.
cos A = adj/hyp
cos A = 3/15
cos A = 1/5
A = cos^-1 0.2
A = 78.5°
A car uses data to help determine the color of cars to order for this year. The data displays how many cars were sold the previous year Each dot represents 10 cars. Use the dot plot to answer the question
Number of Cars Per Color to Stock
O 100
O 190
019
025
Answers
Therefore , the solution of the given problem of dot plot comes out to be on the dot plot, 19 vehicles were sold that were not white.
Dot plot: What is it?The most typical way to display data in a graph is to demonstrate how two additional factors relate to one another. Digital and hand-drawn sketches are both acceptable. From the beginning position, move two or more pieces to the correct and three pieces up. The reference system must display the numerals for positions 2, 3, and 4. In your points, be very clear.
Here,
The following numbers are shown as follows based on the dot plot:
Number of scarlet vehicles: 5
=> 8 black vehicles total.
=> 6 white vehicles total.
=> One silver vehicle is equal to four grey cars.
=> Blue vehicle count equals one.
The following vehicles are not white vehicles based on the aforementioned criteria.
Thus, we have the depiction shown below.
=> Number of scarlet vehicles: 5
=> 8 black vehicles total.
=> number of silver vehicles Equals 1
=> 4 grey vehicles totaled
=> Blue vehicle count equals one.
These wavelengths combine to give us the following
=> Total = 5 + 8 + 1 + 4 + 1
Analyze the amount
=> Overall = 19
Therefore, there are 19 vehicles.
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Answer: Its 190. The problem states each dot represents 10 cars. 19 out of 25 dots weren't white. 19*10 is 190.
Step-by-step explanation:
In the figure, XYZ and MNL
Find m
Answers
Answer:
∠ Y = 124°
Step-by-step explanation:
given the triangles are congruent , then corresponding angles are congruent, that is
∠ Y = ∠ N = 124°
Perform the operation. ( 8 � 2 + 3 � − 4 ) + ( − 2 � 2 + 8 � ) (8x 2 +3x−4)+(−2x 2 +8x)
Answers
Therefore, the simplified expression is 6x² + 11x - 4.
What is expression?In mathematics, an expression is a combination of numbers, symbols, and operators (such as +, −, ×, ÷, ^) that represents a quantity or a mathematical relationship between quantities. Expressions can include variables, which are letters that represent unknown or varying values. Expressions can be simple or complex, and they can be used to represent mathematical formulas, equations, inequalities, and functions. Expressions can be evaluated by replacing the variables with specific values and performing the arithmetic operations according to the rules of algebra. Expressions can also be simplified by combining like terms, distributing multiplication, and factoring. Simplifying expressions can help us to better understand their structure and relationships, and it can make them easier to work with in calculations and problem-solving.
Here,
To perform the operation (8x² + 3x - 4) + (-2x² + 8x), we need to combine like terms, i.e., terms that have the same variable and exponent.
First, let's combine the x² terms:
8x² - 2x² = 6x²
Next, let's combine the x terms:
3x + 8x = 11x
Finally, let's combine the constants:
-4 + 0 = -4
Putting it all together, we have:
(8x² + 3x - 4) + (-2x² + 8x) = 6x² + 11x - 4
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At the hardware store, a zinc washer has an inner radius of r inches and a width of 1 inch from inner edge to outer edge. Write a polynomial equation for the area A of one side of the zinc washer.
Answers
A polynomial equation for the area of one side of the zinc washer is given by A = pi*(r+1)² - pi*r².
What is polynomial?Polynomial is a mathematical expression containing one or more terms, each of which is composed of a constant coefficient and one or more variables raised to a power. It can be used to represent a variety of functions, such as linear, quadratic, and cubic functions.
This equation is derived from the area formula for a circle, which states that the area of a circle is equal to pi multiplied by the square of the radius. In this case, the inner radius of the zinc washer is r and the outer radius is r + 1, so the area of one side of the zinc washer is equal to the area of a circle with a radius of r + 1 minus the area of a circle with a radius of r. This can be expressed mathematically as pi*(r+1)² - pi*r², which is a polynomial equation for the area of one side of the zinc washer.
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Consider the following function.
g(x)=3−1/2x
Step 2 of 2 : Find two points on the line to graph the function.
Answers
Answer:(0,3) (2,2)
Step-by-step explanation:
You survey customers at a restaurant who have contracts with three different wireless telecommunication providers. The survey asks whether they have cell phone service at the restaurant. The results, given as joint relative frequencies, are shown in the two-way table. Find the probability that a randomly selected customer has a contract with provider B. Then find the probability that a randomly selected customer who does not have cell phone service has a contract with provider B. Write your answers as decimals rounded to the nearest hundredth. Use your answers to determine whether having a contract with provider B and not having cell phone service are independent events.
Provider A Provider B Provider C
Yes 0.31 0.43 .12
No 0.04 0.07 .03
Answers
Having a contract with provider B and not having cell phone are independent events.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
If two events are independent, the multiplication of the probabilities of each event is equals to the probability of both events happening.
The probabilities for this problem are given from the table as follows:
Provider B: 0.43 + 0.07 = 0.5.No cell phone: 0.04 + 0.07 + 0.3 = 0.14.Provider B and no cell phone: 0.07.The multiplication of the probabilities of each event is:
0.5 x 0.14 = 0.07 -> independent events = Provider B and no cell phone: 0.07.
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create a hanger to represent 7 = 4x + 2
Answers
4x + 2 = 7 or, 4x = 7 - 2 or, 4x = 5 or, x = 5/4 = 1.25
The time it takes to cover the distance between two cities by car varies inversely with the speed of the car. The trip takes 9 hours for a car moving at 35 mph.
How long does the trip take for a car moving at 21 mph ?
Answers
time = k / speed
where k is a constant of proportionality. We can find the value of k by using the information given in the problem.
When the car is moving at 35 mph, the time it takes to complete the trip is 9 hours. So we can write:
9 = k / 35
Multiplying both sides by 35, we get:
k = 315
Now we can use this value of k to find the time it takes for a car moving at 21 mph:
time = 315 / 21
time = 15 hours
Therefore, it would take a car moving at 21 mph a total of 15 hours to cover the same distance that a car moving at 35 mph covers in 9 hours.
a) -7 + 8 = 1
b) 13 + -9 = 4
c) 23 + -9 = 14
d) -17 + 18 = 1
e) 14 + -26 = -12
f) -8 - 7 = -15
g) 13 - (+9) = 4
h) 23 – (9) = 20
i) 18 - -7 = -25
j) 26 - -14 = -40
are these correct?
Answers
Answer:
a) 1
b) 4
c) 14
d) 1
e) 12
f) 15
h) 20
i) 25
j) 40
Step-by-step explanation:
Some mistakes have been made.
Correct answers so far:
a, b, c, d, e, f are correct
h, maybe
Corrections:
g)
13 - ( + 9) can be rewritten as 13 - 9, which is 4
h) I assume the 9 is supposed to be a [tex]\sqrt{ 9}[/tex] ? If so, then the answer is correct
i) 18 - -7 can be rewritten as 18 + 7 = 25. The two negatives turn into a positive.
j)26 - -14 = 40. The two negatives turn into a positive.
A skating rink has the dimensions shown in the figure below. What is the area of the skating rink? Use π ≈ 3.14 and round to the nearest tenth. 35 m 12 m
Answers
Answer: The area of the skating rink can be found by multiplying its length and width:
Area = length × width
In this case, the length is 35 m and the width is 12 m. Therefore, we have:
Area = 35 m × 12 m
Area = 420 m^2
Using π ≈ 3.14 to round to the nearest tenth, we get:
Area ≈ 420.0 m^2
Therefore, the area of the skating rink is approximately 420.0 square meters.
Step-by-step explanation:
In the figure below, find the exact value of . (Do not approximate your answer.)
Answers
The value for x is 4√3. The solution has been obtained by using the Pythagoras theorem.
What is Pythagoras theorem?
Pythagoras' Theorem states that the square of a right - angled triangle's hypotenuse side is equal to the sum of the squares of its other two sides.
We are given the length of the hypotenuse to be 8.
Let the base be 'y'.
So, using Pythagoras theorem,
[tex]x^{2} + y^{2} = 8^{2}[/tex]
From this, we get
[tex]y^{2} = 8^{2} - x^{2}[/tex] ...(1)
Also,
[tex]2^{2} + Perpendicular^{2} = y^{2}[/tex] ...(2)
From the figure, we also know that
[tex]6^{2} + Perpendicular^{2} = x^{2}[/tex]
So,
[tex]Perpendicular^{2} = x^{2} - 6^{2}[/tex]
On substituting this in (2), we get
[tex]2^{2} + x^{2} - 6^{2} = y^{2}[/tex] ...(3)
On equating (2) and (3), we get
⇒[tex]2^{2} + x^{2} - 6^{2} = 8^{2}-x^{2}[/tex]
⇒ 4 + [tex]x^{2}[/tex] - 36 = 64 - [tex]x^{2}[/tex]
⇒ [tex]2x^{2}[/tex] - 32 = 64
⇒ [tex]2x^{2}[/tex] = 96
⇒ [tex]x^{2}[/tex] = 48
⇒ x = 4√3
Hence, the value for x is 4√3.
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easy question for people, btw the question is not for me lol
Maya spent 2 hours on her phone, then she was left with 38% battery left.
then after another hour passed by and now it is 9%.
How many hours and minutes did she spend on her phone
Answers
Now we can use this rate to figure out how much battery Maya used in the third hour. If she was left with 38% battery after 2 hours, then she had 100% - 38% = 62% of her battery left. If another hour passed by and she was left with 9% battery, that means she used 62% - 9% = 53% of her battery in that hour. Since we know she uses 5.17% of her battery per hour, we can divide 53% by 5.17% to get the total number of hours she spent on her phone:
53% / 5.17% = 10.24 hours
So Maya spent 2 hours + 1 hour + 10.24 hours
what is another name for line qr?
Answers
Answer: Another name would be Line RQ
Not sure if this is what you're looking for but hope it helps!
e) What is the expected number of 6's if you roll 9 dice?
Expected number is
Round answer to two decimal places
Answers
Answer:
approximately 1.5 times but round up to 2
Answer
if you were to roll a dice 9 times you would have the chance to roll 6, 1.5 times.
1/6 and 0.5/3
Step-by-step explanation:
for every number on a dice there is a chance that you'll get that number 1.5 times
How to find a surface area with 2in 6in 10in 8in 2in
Answers
The surface area of the rectangular prism is 376 square inches.
What is surface area?Surface area is the total area that the surface of a 3-dimensional object covers, including all of its faces and sides. It is measured in square units.
To find the surface area of a 3-dimensional object with dimensions of 2in, 6in, 10in, 8in, and 2in, you will need to add up the areas of each of its faces.
Assuming that the object is a rectangular prism, here are the steps to find its surface area:
Identify the dimensions of the rectangular prism: length, width, and height.
Length: 10in
Width: 8in
Height: 6in
Calculate the area of each face:
Front and back faces: length x height = 10in x 6in = 60 sq.in x 2 = 120 sq.in
Top and bottom faces: width x length = 8in x 10in = 80 sq.in x 2 = 160 sq.in
Side faces: height x width = 6in x 8in = 48 sq.in x 2 = 96 sq.in
Add up the areas of each face to get the total surface area:
Total surface area = 120 sq.in + 160 sq.in + 96 sq.in = 376 sq.in
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Straight Line Graphs Question. I tried to solve the section (a) part of the question correctly I followed the gradient method plugging in the values and got [tex]y = \frac{1}{6} x + 1.6 (recurring). However the answer is y = \frac{1}{2} x + 3.[/tex]. Where am I going wrong? Can someone explain step by step how to solve this please.
Answers
The equation of the straight line joining A to B is y = (1/2)x + 3, the equation of the straight line joining B to C is y = (1/2)x + 3, and A, B, and C are collinear.
What are collinear points?
Collinear points are three or more points that lie on the same straight line. In other words, if three points A, B, and C are collinear, then there exists a straight line that passes through all three points. To determine if three or more points are collinear, we can find the slope of the line that passes through any two of the points. If the slope is the same for all possible pairs of points, then the points are collinear. Otherwise, they are not collinear.
a) To find the equation of the straight line joining A to B, we can use the slope-intercept form of the equation of a line:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
The slope of the line passing through A and B is:
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) = (-8, -1) and (x₂, y₂) = (-4, 1).
m = (1 - (-1)) / (-4 - (-8)) = 2/4 = 1/2
To find the y-intercept, we can use the point-slope form of the equation of a line:
y - y₁ = m(x - x₁)
where (x₁, y₁) = (-8, -1) and m = 1/2.
y - (-1) = (1/2)(x - (-8))
y + 1 = (1/2)(x + 8)
y = (1/2)x + 3
Therefore, the equation of the straight line joining A to B is y = (1/2)x + 3.
b) To find the equation of the straight line joining B to C, we can use the same method:
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) = (-4, 1) and (x₂, y₂) = (12, 9).
m = (9 - 1) / (12 - (-4)) = 8/16 = 1/2
Using point-slope form with the point (x₁, y₁) = (-4, 1) and slope m = 1/2:
y - 1 = (1/2)(x - (-4))
y - 1 = (1/2)x + 2
y = (1/2)x + 3
Therefore, the equation of the straight line joining B to C is y = (1/2)x + 3.
c) From the equations of the lines joining A to B and B to C, we can see that both lines have the same slope of 1/2. This implies that the three points A, B, and C are collinear, i.e., they lie on the same straight line.
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