The area of triangle TUV with vertices T(-8,2), U(-2,8), and V(-9,9) are 13.4 units.
What is triangle?A triangle is a closed two-dimensional plane figure that has three sides, three angles, and three vertices. The sum of the angles of a triangle is always 180 degrees. Triangles can be classified based on the length of their sides and the size of their angles. Some common types of triangles include equilateral, isosceles, scalene, acute, obtuse, and right triangles. Triangles are a fundamental concept in geometry and are used in many areas of mathematics and science.
Here,
To find the area of triangle TUV, we can use the formula:
Area = 1/2 * base * height
We can choose any two sides of the triangle as the base and the corresponding height. Let's choose TU as the base and the perpendicular distance from V to TU as the height.
First, let's find the length of TU:
TU = √[(8 - 2)² + (-2 - (-8))²]
= √[6² + 6²]
= 6√(2)
Next, let's find the slope of TU:
mTU = (8 - 2) / (-2 - (-8))
= -3/2
The line perpendicular to TU passing through V has a slope equal to the negative reciprocal of mTU:
mVQ = 2/3
The equation of the line passing through V and perpendicular to TU is:
y - 9 = (2/3)(x + 9)
Solving for x and y at the point where this line intersects TU, we get:
y = (2/3)x + 19
(2/3)x + 19 = -3x/2 + 7
x = -8/7
y = 94/21
The perpendicular distance from V to TU is the absolute value of y - 8:
|94/21 - 8| = 2/21
So, the area of triangle TUV is:
Area = 1/2 * TU * (2/21)
= (1/21)√(2)
To find the area of rectangle QRS, we need to find the length and width. We can use the distance formula to find the length QR and the width QS:
QR = √[(9 - (-8))² + (9 - 2)²]
= √[289]
= 17
QS = √[(9 - (-9))² + (2 - 2)²]
= √[324]
= 18
So, the area of rectangle QRS is:
Area = QR * QS
= 17 * 18
= 306
Area of triangle QRS = Area of rectangle QRS - Area of triangle TUV
= 306 - (1/21)√(2)
≈ 13.4 units
So, the answer is (C) 13.
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Complete question:
Triangle TUV, with vertices T(-8,2), U(-2,8), and V(-9,9), is drawn inside a rectangle. What is the area, in square units, of the triangle TUV?
A. 7
B. 10
C. 13
D. 18
caden is 208 miles away from rahquez. they are traveling towards each other. if rahquez travels 6 mph faster than caden and they meet after 8 hours, how fast was each traveling?
Caden's speed is 10 mph and Rahquez's speed is 16 mph faster, which is 16+6 = 32 mph
Let's use the formula: distance = rate x time
Since Caden and Rahquez are traveling towards each other, their combined distance will be 208 miles. Let's call Caden's speed "x" and Rahquez's speed "x+6", since Rahquez is traveling 6 mph faster than Caden.
Using the formula above, we can set up the equation:
208 = (x + x + 6) * 8
Simplifying, we get:
208 = (2x + 6) * 8
208 = 16x + 48
160 = 16x
x = 10
Therefore, Caden's speed is 10 mph and Rahquez's speed is 32 mph.
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The population of a country is initially two million people and is increasing at a rate of 4% per year. The country’s annual food supply is initially adequate for four million people and is increasing at a constant rate adequate for an additional 0.5 million people per year.If the country doubled its initial food supply and maintained a constant rate of increase in the
supply adequate for an additional 0.5 million people per year, would shortages still occur? In
approximately which year? . If the country doubled the rate at which its food supply increases, in addition to doubling its
initial food supply, would shortages still occur?
in approximately 22 years, the population would exceed the food supply, even if the country doubles its initial food supply and doubles the rate at which its food supply increases.
How to calculate the rate?
To answer these questions, we need to calculate the population and food supply at different points in time and compare them.
Let's first calculate the population after t years:
Population after t years = 2,000,000 * (1 + 4%) raise to the power t
And the food supply after t years:
Food supply after t years = 4,000,000 + 0.5 million * t
Now, let's answer the first question:
If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur?
If the country doubles its initial food supply, the new food supply would be 8,000,000, and it would still increase at a rate of 0.5 million people per year. Let's calculate the year when the population exceeds the food supply:
Population = Food supply
2,000,000 * (1 + 4%)^t = 8,000,000 + 0.5 million * t
Solving for t, we get t ≈ 17.77 years.
So, in approximately 18 years, the population would exceed the food supply, even if the country doubles its initial food supply and maintains a constant rate of increase in the supply adequate for an additional 0.5 million people per year.
Now, let's answer the second question:
If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur?
If the country doubles the rate at which its food supply increases, the new rate would be 1 million people per year. Let's calculate the year when the population exceeds the food supply:
Population = Food supply
2,000,000 * (1 + 4%) raise to the power t = 16,000,000 + 1 million * t
Solving for t, we get t ≈ 21.96 years.
So, in approximately 22 years, the population would exceed the food supply, even if the country doubles its initial food supply and doubles the rate at which its food supply increases.
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Darius wrote two integers on the chalkboard. The integers she wrote were 256 and -17. How much larger is 256 than -17?
Answer: 239
Step-by-step explanation:
Darius wrote 2 integers 256 and -17. The question is asking you to solve how much larger is 256 than -17. This means you will subtract 256 by -17 which will give you 239 (256-17). The answer remains positive because 256 is larger
Hope this helps
Answer: They are exactly 273 places away.
Step-by-step explanation:
If a number is larger than another number then you need to subtract it to get the distance. In this case, you look at the subtraction to see how far away they are.
Hope this helps!
Have a blessed day!
a 336-m long fence is to be cut into pieces to make three enclosures, each of which is square. how should the fence be cut up in order to minimize the total area enclosed by the fence?
The fence ought to be cut into 12 pieces, every one of length 28 m, to make three squares, each with a side length of 28 m. This will limit the total area encased by the fence.
To limit the total area encased by the fence, the three squares ought to have equivalent areas. Let x be the length of each side of the squares. Then the perimeter of each square is 4x, and the total length of the fence is 3(4x) = 12x. Since the total length of the fence is given to be 336 m, we have:
12x = 336
Addressing for x, we get:
x = 28
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Which of these is a pythagorean triple?
Responses
9, 40, 41
7, 26, 89
1, 2, 3
36, 48, 62
Answer:
The first one
Step-by-step explanation:
Use the pythagoras theorem=a ^2+b^2 =c^2
Use a is 9 b is 40 because the largest value is always the hypotenuse and the hypotenuse is always c.
so you do 9 squared add 40 squared to find c squared.
Square root the answer and you get 41 so it is a pythagoras triple
if the path around the edge of a circular lawn is 132 feet, what is the approximate area inside the circular path? complete the answer sentence. solve on paper. then, enter your answer on zearn. you may consider writing your answer to the hundredths place and using the calculator to help you solve.
As a result, the circumference of the circular walkway is approximately 367.06 square feet.
what is circle ?A circle is a geometric shape made up of all points that are equally spaced apart from the center. It is a flat, two-dimensional object whose radius or diameter serves as its sole defining characteristic. A circle's circumference is the distance around it, while the area of the circle is the amount of space inside it.
given
We must deduct the area of the smaller circle from the size of the larger circle in order to determine the area inside the circular path. Let's use the abbreviations "r" for the smaller circle's radius and "R" for the larger circle's radius. We are aware that the circumference of the smaller circle, which is 132 feet, is equal to the distance around the circular path.
The wider circle's circumference is 132 feet.
2πR = 132
R = 21
The breadth of the circular path, which is equal to the difference between the radii of the two circles, can now be used to calculate the radius of the smaller circle:
R - r = diameter of the circle
21 - r = diameter of the circle
r = 18
The smaller circle therefore has a radius of 18 feet.
Inside the arc of the path, there is:
R2 - r2 equals (21 - 18) = (44 - 324) = (117) 367.06 square feet.
As a result, the circumference of the circular walkway is approximately 367.06 square feet.
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The mirror has a frame. The diameter of the mirror with
the frame is 17 inches. To the nearest hundredth, what
is the area of the mirror with the frame?
Answer:
226.865 square inches.
Step-by-step explanation:
The equation for the area of a circle is [tex]\pi r^2[/tex]. Let's use 3.14 for pi. The radius can be found by dividing 17/2 = 8.5. 8.5 squared is 72.25. Finally, multiply by pi to get 226.865.
a square has side length of 7.3 in . if the area is multiplied by 19 , what happens to the perimeter?
The new perimeter is 127.2 inches if it is multiplied by 19.
What do you mean by the are and perimeter of the square ?
The area of a square is calculated by squaring the length of one of its sides. If the area of a square is multiplied by a factor, the length of the sides will be multiplied by the square root of that factor. One way to express the concept of perimeter for a square is to state that it is the total distance around the outer boundary of the shape, which is equal to the sum of the lengths of all four sides.
Determining the the value of perimeter :
The area of the square is [tex]7.3 \times 7.3 = 53.29[/tex] square inches.
If this area is multiplied by 19, the new area will be 1012.51 square inches.
To find the new side length of the square, we take the square root of the new area:
[tex]\sqrt{1012.51} = 31.8[/tex] (rounded to one decimal place)
Therefore, the new side length of the square is 31.8 inches.
To find the new perimeter, we multiply the new side length by 4:
[tex]31.8 \times 4 = 127.2[/tex]inches
So the new perimeter is 127.2 inches.
To summarize, when the area of the square with side length 7.3 inches is multiplied by 19, the perimeter is multiplied by the square root of 19, which is approximately 4.36. The new perimeter is 127.2 inches.
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Which one of the following decisions most clearly reflects a lack of understanding of the concept of sunk costs? You pay to have your car towed back to the repair shop because it was not fixed properly the first time. You decide to get a master's degree because you cannot find a job in the field in which you majored. You decide to purchase a piece of machinery for your business that will eliminate three employees' positions. You study eight hours for a final exam even though there is no way now that you can pass the course.
The decision that most clearly reflects a lack of understanding of the concept of sunk costs is "You study eight hours for a final exam even though there is no way now that you can pass the course".
What are sunk costs?Sunk costs refer to any costs that have already been incurred and cannot be recovered. Sunk costs are irrelevant in decision-making because they are already gone and cannot be recovered. Decision-makers must only consider future costs and benefits when making decisions.
What does it mean to lack an understanding of sunk costs?A lack of understanding of sunk costs may result in individuals making decisions based on past costs rather than future costs and benefits. Decision-making that is influenced by sunk costs may result in inefficient resource allocation and lost opportunities.
Individuals must ignore sunk costs and instead focus on future costs and benefits to make effective decisions.
In the given options, "You study eight hours for a final exam even though there is no way now that you can pass the course" is the decision that most clearly reflects a lack of understanding of the concept of sunk costs.
Even though the student has already spent 8 hours studying, the sunk costs are irrelevant because the student cannot pass the course now. The student should focus on future costs and benefits and use the remaining time to study for other courses.
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which events are independent? check all that apply.each number 1 through 10 is written on a slip of paper, placed in a hat, and randomly picked. sarah picks a number less than 5, keeps it, and then picks an odd number.
The events that are independent are,
1) Henry rolling a multiple of 2 and landing on a red portion of the spinner.
2) Two pairs of socks being randomly chosen from a drawer, Hayden choosing a black pair of socks, putting them on, and then choosing another black pair.
3) Olivia choosing an ace and the dart hitting the bull's-eye
The events that are independent are:1) Henry rolling a multiple of 2 and landing on a red portion of the spinner.
2) Two pairs of socks being randomly chosen from a drawer, Hayden choosing a black pair of socks, putting them on, and then choosing another black pair.
3) Olivia choosing an ace and the dart hitting the bull's-eye.
The other events are dependent because the outcome of the first event affects the outcome of the second event. Specifically
1) Sarah picking a number less than 5 and then picking an odd number. The second event depends on the outcome of the first event.
2) A number cube being rolled and a spinner being spun. The two events are independent of each other, but the outcomes of each event are not independent.
3) Two cards being randomly chosen from a standard deck, and Eliza choosing a jack, replacing it, and then choosing a black card. The outcome of the second event depends on the outcome of the first event, and the outcome of the first event affects the composition of the deck for the second event.
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The given question is incomplete, the complete question is:
Which events are independent? Check all that apply. Each number 1 through 10 is written on a slip of paper, placed in a hat, and randomly picked. Sarah picks a number less than 5, keeps it, and then picks an odd number. A number cube is rolled and a spinner is spun. Henry rolls a multiple of 2 and lands on a red portion of the spinner. Two cards are randomly chosen from a standard deck. Eliza chooses a jack, replaces it, and then chooses a black card. Two pairs of socks are randomly chosen from a drawer. Hayden chooses a black pair of socks, puts them on, and then chooses another black pair. A card is randomly chosen from a standard deck and a dart is randomly thrown. Olivia chooses an ace and the dart hits the bull’s-eye.
How can the area of this triangle be determined by forming a rectangle?
Select from the drop-down menus to correctly complete the statements.
The length of the rectangle is what units.
The width of the rectangle is what units.
The area of the rectangle is what square units.
The area of the triangle is half the area of the rectangle, so the area of the triangle is
what square units.
Answer:
the area ot the triangle is half the area of the rectangle, so the area of the triangle is what sqauare units
Answer: It's much easier to form a rectangle and multiply the width and the length and then divide it by 2
Subtract the following polynomials.
The subtraction of the polynomials (3.1x + 2.8z) - (4.3x - 1.2z) is -1.2x + 4x
How to subtract polynomials?A polynomial is an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power.
(3.1x + 2.8z) - (4.3x - 1.2z)
open parenthesis
3.1x + 2.8z - 4.3x + 1.2z
combine like terms
3.1x - 4.3x + 2.8z + 1.2z
-1.2x + 4x
Ultimately, -1.2x + 4x is the results of the subtraction of the polynomial.
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julio has $31.00 he earns half of that much mowing a lawn. How much money does he have in all?
Answer: $ 46.50
First divided 31 by 2
Which equals...
15.50
Then add 15.50 to 31.
46.50
The answer is $46.50
Proportional or not proportional?
Step-by-step explanation:
A is not proportional
Use the graph to record the value of y axis when x is 1, 2, 3, 4, 5, 6
And record the values of x when y is 1, 2, 3, 4, 5, 6 till the end
Using the graph Value of y
(1, 0.9) (2,1.5) (3,2.2) (4,2.5) (5, 3) (6,4) (7, 5) (10,6)
Using the graph, find the value of x when y is 1 - 10)
1.2, 2.5, 5,.....
The slope of the graph also shows that it is not proportional
B is proportional
X is 0 and y is 0
When x is 5, y is 10
Ratio x: y = 5:10 = 1:2
When x is 10, y is 20
Ratio x:y is 10: 20 = 1:2 same as above
C is not proportional
When Moyra is 32, daughter is 8
When her daughter is 22, Moyra will also be 4years older, which is 36
32: 8 = 4:1
36: 12 =3:1
the area of the figure
Answer:
432
Step-by-step explanation:
go 16x27 and that is your answer
Answer:
The area of the figureo will be 432
BECAUSE :-
You have to multiply 16 × 27 = 432
Hope Its Help You !!
Tammy said the product of 5/7
and 1 1/4
is 1 3/4
.
How can you tell that this answer is wrong?
Answer:
Step-by-step explanation:
To determine if Tammy's answer of 1 3/4 for the product of 5/7 and 1 1/4 is correct, we can perform the multiplication ourselves.
First, we need to convert 1 1/4 to an improper fraction:
1 1/4 = (4 x 1 + 1)/4 = 5/4
Then, we can multiply the fractions:
5/7 x 5/4 = 25/28
As we can see, 25/28 is not equal to 1 3/4. Therefore, Tammy's answer of 1 3/4 is incorrect. The correct answer is 25/28, which is an improper fraction. We can also express this as a mixed number: 0 25/28.
When George Washington became president in 1789, the army he had commanded in the American Revolution had
disbanded
ARMY DISBANDED MEANS SOLDIERS WERE RELEASED FROM SERVICE
When George Washington became president in 1789, the army he had
commanded in the American Revolution had disbanded. This means that
the soldiers were released from service and the military units were
dissolved after the end of the war.He pursued two intertwined interests:
military arts and western expansion. At 16 he helped survey Shenandoah
lands for Thomas, Lord Fairfax. Commissioned a lieutenant colonel in 1754,
he fought the first skirmishes of what grew into the French and Indian War.
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please help i will give brainliest along with 20 pnts. ONLY ANSWER IF YOU KNOW IT
Giovanna deposited $1600 into two accounts, half in First Oak and half in West United.
Interest earned in First Oak account after 5 years at 6% simple interest:
= 0.06 x $800 x 5
= $240
Interest earned in West United account after 5 years at 5% compounded annually:
= $800 x (1 + 0.05)^5 - $800
= $800 x 1.27628 - $800
= $221.02
Total interest earned in both accounts:
= $240 + $221.02
= $461.02
Therefore, Giovanna will have earned $461.02 in interest from both accounts at the end of 5 years.
Luca owns a food truck called The Muffin Man, and this week, he is making a specialty flavor by adding cheddar cheese and hot peppers to corn muffin mix. The mix comes in a box shaped like a rectangular prism with a volume of 60 cubic inches. The box has a length of 5 inches and a height of 8 inches
The width of the rectangular prism box is 1.5 inches if the length of the box is 5 inches and the height of the box is 8 inches.
The volume of a rectangular prism = 60 cubic inches.
The length of the box = 5 inches
The height of the box = 8 inches
To calculate the width of the rectangular prism, we can use the formula for volume:
Volume = length * width * height
Mathematically,
V = l*w*h
60 = 5w * 8
w = 60 / (5 * 8)
w = 1.5 inches
Therefore we can conclude that the width of the rectangular prism box is 1.5 inches.
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An angle measures 62° more than the measure of its complementary angle. What is the measure of each angle?
Last year, 89 musicians attended a jazz camp, and 15 of them were bassists. What is the experimental probability that the first musician to sign up will be a bassist?
There is a roughly 16.85% chance that a bassist will be the first musician to register for the jazz camp.
What is experimental probability?Experimental probability is a type of probability that is based on actual observations or experiments. It is also called empirical probability
According to question:The ratio of the frequency of an occurrence to the total number of trials or observations is known as the experimental probability. In this case, the event is the first musician to sign up being a bassist.
Since there were 15 bassists out of 89 musicians who attended the camp last year, the experimental probability of the first musician to sign up being a bassist is:
Experimental probability = Number of bassists / Total number of musicians
Experimental probability = 15 / 89
Experimental probability = 0.1685 or approximately 16.85%
As a result, there is a roughly 16.85% chance that a bassist will be the first musician to register for the jazz camp.
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Factor
[tex]64h^3+216k^9[/tex]
Answer:
Factor 64h^3+216k^9
Step-by-step explanation:
The given expression is a sum of two terms:
[64h^3+216k^9
Notice that each term has a common factor. For the first term, the greatest common factor (GCF) is 64h^3, and for the second term, the GCF is 216k^9. So we can factor out these GCFs to get:
64h^3+216k^9 = 64h^3(1 + 3k^6)
This expression cannot be factored any further, so the final answer is:
64h^3+216k^9 = 64h^3(1 + 3k^6)
If you can, give me brainliest please!
you have been tracking how long you spend on each course subject during your homework sessions. you want to see what percentage each subject represents. which type of chart would be best for this purpose?
a pie chart would be an effective way to quickly and easily understand the distribution of time spent on different course subjects.
For visualizing the percentage distribution of different course subjects, a pie chart would be the most suitable type of chart. Pie charts are ideal for displaying the relative proportions of different categories or groups as slices of a circular pie. Each slice of the pie represents a proportion of the whole, which in this case would be the total time spent on all course subjects during homework sessions. The size of each slice will correspond to the percentage of time spent on each course subject. Therefore, a pie chart would be an effective way to quickly and easily understand the distribution of time spent on different course subjects.
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PLEASE HELP!
The sum of the roots of a monic quadratic is -6, and the product of its roots is 7. What is the quadratic?
Answer with a quadratic expression using the variable x such as x^2 + 10x + 20
Answer:
x^2 +6x+7
Step-by-step explanation:
for roots a and b
x^2 - (a+b)x + ab = (x-a)(x-b)
John is making apple pies and apple cobblers to sell at his stand at the Farmer's Market.
A pie uses 4 cups of apples and 3 cups of flour.
A cobbler uses 2 cups of apples and 3 cups of flour.
John has 16 cups of apples and 15 cups of flour.
When John sells the pies and cobblers at the Farmer's Market, he will make $3.00 profit per pie and $2.00 profit per cobbler.
Let x = the number of pies John makes.
Let y = the number of cobblers John makes.
Enter the four constraints into the graphing calculator.
What are the vertices of the feasible region?
Hint: input your answers from questions 3, 4, and 5 into Desmos to find the vertices.
Answers from questions 3, 4, and 5
[tex]4x+2y≤16\\3x+3y≤15\\x≥0\\y≤0[/tex]
Answer:huh,
Step-by-step explanation:I don’t understand you’re saying
Tonya's income is four times as much as Nora's income. Write an Algebraic expression representing Nora's income in terms of Tonya's
An algebraic expression for representing the Nora's income in form of Tonya's income is given by y = ( x / 4 ) .
Let us consider 'x' represents the Tonya's income.
And variable 'y' represents the Nora's income.
Tonya income is equal to four times of Nora's income.
This implies,
Nora's income is equal to one fourth times of Tonya's income.
⇒ y = ( x / 4 )
Rewrite an algebraic expression to represents Tonya's income in terms of Nora's income we have,
Simplify by multiplying both the sides of the algebraic expression by 4 we get,
⇒ x = 4y
Therefore, an algebraic expression to represents the Nora's income in terms of Tonya's income is equal to y = ( x / 4 ).
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Each prism has the same volume. Which has the least surface area? Explain or show your
reasoning.
This means that the cube will have less surface area than the rectangular prism.
What is surface area?Surface area is a measure of the total exposed area of an object, or the total area of its two-dimensional surface. It is a measurement of the total exposed area of a three-dimensional object and is used to calculate the amount of material needed to cover a given area. Surface area is commonly used in physics, chemistry, engineering, and mathematics.
The prism with the least surface area of the same volume is the cube. This is because a cube has equal length, width, and height, allowing for all of the sides to be equal in size. This means that the area of each side is the same and all six sides of the cube are congruent. This creates the least amount of surface area for a given volume, as compared to any other prism. To illustrate this, take a cube and a rectangular prism with the same volume. The cube will have 6 smaller sides, while the rectangular prism will have 5 larger sides and 1 smaller side. This means that the cube will have less surface area than the rectangular prism.
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a spinner with the colors red, yellow, blue, and green is spun. what is the theoretical probability of stopping on the color blue?
The theoretical probability of stopping on the color blue when spinning a spinner with the colors red, yellow, blue, and green is 25% or 1/4.
A spinner is a tool that spins around an arrow or a pointer that points towards the numbers, colors, or pictures around its edge. The probability of a certain event is the likelihood of that event occurring. In this case, we need to find the theoretical probability of stopping on the color blue.
We have four colors on the spinner, so the probability of stopping on blue can be found using the formula of theoretical probability.
The formula for theoretical probability is:
number of favorable outcomes / total number of outcomes.
Since we have four colors on the spinner, the total number of outcomes is 4. The favorable outcomes are the number of times the spinner will land on blue, which is 1.
So the probability of stopping on the color blue can be calculated as follows:
1 (favorable outcome) / 4 (total number of outcomes) = 1/4
So, the theoretical probability of stopping on the color blue is 1/4. This is because there is an equal chance of landing on any of the four colors, and since there are four colors, each has a 25% chance of being selected.
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PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP AND BE CORRECT
Answer:
4t - 22
Step-by-step explanation:
To find:-
Perimeter of the given figure.Answer:-
To find out the perimeter we can simply add all the side lengths of the given figure. Since the given figure here is a rectangle, we can add up all the four sides to find the perimeter.
The four sides given to us are t-6 , t-5 , t-6 and t-5 .
Hence the perimeter of the quadrilateral would be ,
Perimeter = t-5 + t-6 + t-5 + t-6
Perimeter = 4t - 10 - 12
Perimeter = 4t - 22
Hence the perimeter of the given figure us 4t - 22.
[tex]\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(5,0){2}{\line(0,1){3}}\multiput(0,0)(0,3){2}{\line(1,0){5}}\put(0.03,0.02){\framebox(0.25,0.25)}\put(0.03,2.75){\framebox(0.25,0.25)}\put(4.74,2.75){\framebox(0.25,0.25)}\put(4.74,0.02){\framebox(0.25,0.25)}\multiput(2.1,-0.7)(0,4.2){2}{$\sf\large t-5$}\multiput(-1.4,1.4)(6.8,0){2}{$\sf\large t-6$}\put(-0.5,-0.4){\bf A}\put(-0.5,3.2){\bf D}\put(5.3,-0.4){\bf B}\put(5.3,3.2){\bf C}\end{picture} [/tex]
a cable tv receiving dish is in the shape of a paraboloid of revolution. find the location of the receiver, which is placed at the focus, if the dish is 6 feet across at its opening and 2 feet deep.
the receiver is located at (0, 0, 2.25 feet) or (0, 0, 27 inches).To find the location of the receiver, we first need to determine the equation of the paraboloid.
The standard equation for a paraboloid of revolution with a vertical axis is:
z = [tex](x^2 + y^2)[/tex]/(4f)
Where:
z is the height at any point (x, y) on the paraboloid.
x and y are the horizontal coordinates of the point.
f is the focal length of the paraboloid, which is half the depth of the dish.
In this case, the dish is 6 feet across at its opening, so the diameter is 6 feet and the radius is 3 feet. Therefore, the maximum value of x and y is 3 feet. The depth of the dish is given as 2 feet.
Using these values, we can solve for the focal length:
2 = [tex](3^2 + 3^2)[/tex]/(4f)
2 = 18/(4f)
f = 18/8 = 9/4 = 2.25 feet
Now that we have the value of f, we can find the location of the receiver, which is placed at the focus of the paraboloid. The focus is located at (0, 0, f).
Therefore, the receiver is located at (0, 0, 2.25 feet) or (0, 0, 27 inches).
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