The nth term of the sequence 3/2, 3, 7, 16, 35, 74,... is?
Answer:
95 is the sequence
Step-by-step explanation:
The given sequence appears to follow the pattern where each term is obtained by multiplying the previous term by 2 and then adding 1. Using this pattern, we can derive a formula for the nth term of the sequence as a(n) = 3 * 2^(n-1) - 1, where a(n) represents the nth term of the sequence and n is a positive integer.
For example, to find the 6th term of the sequence, we can substitute n = 6 into the formula to get a(6) = 3 * 2^(6-1) - 1 = 3 * 2^5 - 1 = 95. So, the 6th term of the sequence is 95.
Entre que valores varía la masa corporal MC de la mayoría de estudiantes?
The majority of students have a body mass between 46.7 and 79.1 kg, and there are 36 students with a body mass of 61.7 kg or greater.
Assuming a normal distribution of body mass among the students, we can use the concept of standard deviation to answer the question. Let's assume that the mean body mass of the students is 60 kg, and the standard deviation is 5 kg.
To find the range of body mass where the majority of students fall, we can use the empirical rule, which states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
Therefore, the range of body mass where the majority of students fall is between 55 kg (60 kg - 5 kg) and 65 kg (60 kg + 5 kg).
To find the number of students with a body mass equal to or greater than 61.7 kg, we need to find the z-score of 61.7 kg, which is
(61.7 - 60) / 5 = 0.34.
We can then use a standard normal distribution table to find the proportion of the area to the right of this z-score, which is 0.3665. This means that approximately 36.65% of the students have a body mass equal to or greater than 61.7 kg.
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--The given question is incomplete, the complete question is given
" entre que valores varia la masa corporal (MC) de la mayoría de estudiantes y que cantidad tiene corporal (MC) igual o mayor que 61,7 kg"--
3-5 If we project the are of a hip roof onto a horizontal plane, it will appear as two triangles and two trapezoids. (True or False)
Given statement "If we project the area of a hip roof onto a horizontal plane, it will appear as two triangles and two trapezoids"
Given statement is True.
Because, A hip roof has four sloping sides that meet at the top ridge, forming a pyramid-like shape. When projected onto a horizontal plane, the following shapes will appear:
Two triangles:
These are formed by the two ends of the roof, where the sloping sides meet the ridge.
Two trapezoids:
These are formed by the two sides of the roof, where the eave edges meet the ridge.
So, when projecting the area of a hip roof onto a horizontal plane, it will indeed appear as two triangles and two trapezoids.
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The diameters of cherry tomatoes produced by a large farm have an approximately Normal distribution, with a
mean diameter of 22 mm and a standard deviation of 2. 5 mm. After they are picked and washed, tomatoes with a
diameter of at most 15 mm or at least 30 mm are rejected and not sold. What proportion of such tomatoes are
rejected?
Find the z-table here.
O 0. 0033
O 0. 0201
O 0. 9799
O 0. 9967
The proportion of cherry tomatoes that are rejected based on their diameter is 0.0033. (option a).
To use the CDF, we need to standardize the diameter values using the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the diameter value, μ is the mean diameter, and σ is the standard deviation.
For tomatoes with a diameter of at most 15 mm, we have:
z = (15 - 22) / 2.5 = -2.8
For tomatoes with a diameter of at least 30 mm, we have:
z = (30 - 22) / 2.5 = 3.2
Next, we use the z-table to find the area under the Normal distribution curve that corresponds to these z-scores. For a z-score of -2.8, the area is 0.0026. For a z-score of 3.2, the area is 0.9993.
Finally, we add these two areas together to get the total proportion of rejected tomatoes:
proportion = 0.0026 + (1 - 0.9993) = 0.0033
Hence the correct option is (a).
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for every 4 blacks there are 3 whites if there are 60 whites how many blacks are they
Answer: there should be 80 blacks
Step-by-step explanation:
i promise
PLEASE HELP FAST!!!!!
Lucy received a $1300 bonus. She decided to invest it in a 3-year certificate of deposit (CD) with an annual interest rate of 1.27% compounded monthly.
Answer the questions below. Do not round any intermediate computations, and round your final answers to the nearest cent. If necessary, refer to the list of financial formulas.
(a) Assuming no withdrawals are made, how much money is in Lucy's account after 3 years?
$____
(b) How much interest is earned on Lucy's investment after 3 Years?
$____
Answer:
(a) $1350.46
(b) $50.46
Step-by-step explanation:
You want to know the future value of a $1300 CD earning 1.27% compounded monthly for 3 years, along with the amount of interest earned.
Future valueThe compound interest formula tells you the future value is ...
FV = P(1 +r/12)^(12·t)
where r is the annual interest rate and t is the number of years.
ApplicationUsing the given values, we have ...
FV = $1300(1 +0.0127/12)^(12·3) ≈ $1350.46
(a) The value after 3 years is $1350.46.
The interest earned is the difference between the final value and the amount of the original investment:
Interest = $1350.46 - 1300.00 = $50.46
(b) The interest earned by the CD is $50.46.
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Tickets to a school dance are sold 10$ each. The student council spent a total of 400$ on set-up costs for the dance. The school's profit (p) , in dollars , from the dance depends on the number of tickets sold (t)
3-9 On a 1500 square foot gable roof, there will be approximately 11% waste. (True or False)
Given statement: On a 1500 square foot gable roof, there will be approximately 11% waste.
The given statement is False.
The statement provides information on the amount of waste but does not provide sufficient information to determine the total area of roofing material required for the gable roof.
Therefore, it is impossible to determine whether the waste percentage is accurate or not.
The statement "On a 1500 square foot gable roof, there will be approximately 11% waste" can be either True or False depending on the specific roofing materials and project details.
However, it's not uncommon for roofing projects to have waste percentages in that range (around 10-15%), especially for complex designs or when using materials like shingles.
Keep in mind that waste factors may vary depending on the type of roof, materials used, and installation methods. Always consult with a professional to get accurate waste estimates for your specific project.
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how do i solve this??
A quadratic function is defined by ƒ (x) = x² + 5x − 24.
Which of the following expresses ƒ (x) as the product of
two linear factors?
A
B
C
ƒ (x) = (x − 4) (x + 6)
f(x) = (x − 3)(x + 8)
f(x) = (x + 3) (x − 8)
D fl
f(x) = (x + 4) (x - 6)
in a game using five cards numbered 1,2,3,4,and 5, you take two cards and add the values together. if the sum is 8, you win.Would you rather pick a card and put it back before picking the second card, or keep the card in your hand wile you pick the second card? explain your reasoning.
The probability of winning the game is the same in both instances, so I will pick a card and put it back before picking the second card.
What is probability?The probability of an event is a number that indicates how likely the event is to occur and is expressed as a number in the range from 0 and 1, or, using percentage notation, in the range from 0% to 100%.
The card has total of 10 possible pairs:
(1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), and (4,5).
we have that only (3,5) n adds up to 8, .
In conclusion, the probability of winning the game is 1/10 or 10%.
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100.0
How long would it take the bowling ball to fall 150 feet? (Estimate your answer to one
decimal place. Hint: Use the equation d = 16t².)
It would take the bowling ball approximately 3.1 seconds to fall 150 feet.
What is distance?
Distance is the total movement of an object, regardless of direction. We can define distance as how much ground an object has covered, regardless of its starting or ending point.
Distance is defined as the size or amount of displacement between two positions. Note that the distance between two locations is not the same as the distance between them. The distance traveled is the total length of the path traveled between two positions.
Here given equation,
d = 16t², where t is the time in seconds and d is the distance in feet and distance is 150 feet.
We want to find the value of t.
Now,
[tex]d = 16 {t}^{2} \\ {t}^{2} = \frac{d}{16} \\ {t}^{2} = \frac{150}{16} \\ t = \sqrt{( \frac{150}{16}) } \\ t = \sqrt{9.375} \\ t = 3.0619[/tex]
So, t = 3.1 (rounded to one decimal place)
Therefore, it would take the bowling ball approximately 3.1 seconds to fall 150 feet.
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Correct question is "How long would it take the bowling ball to fall 150 feet where equation of distance is d = 16t² ?(Estimate your answer to one decimal place). "
a store is having a contest. the person who draws a pink tag from a box of tags will get a store coupon will increase in value each day the pink tag is not drawn
Let's say that the store starts with a coupon value of x. On the first day, the probability of drawing the pink tag is 1/n, where n is the total number of tags in the box.
Define probability.Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain. Probabilities can be expressed as fractions, decimals, or percentages.
what do you mean by coupon?A Coupon refers to a discount or a voucher that can be redeemed for a specified value or product at a store or business.
If the tag is not drawn, the probability of drawing it on the second day is 1/(n-1), since there is one less tag in the box. If the tag is not drawn on the second day, the probability of drawing it on the third day is 1/(n-2), and so on.
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GIVEN: Circle with center C and point D.
CONSTRUCT: Equilateral triangle DEF so that points E and F are on circle C.
Circle with center C and point D. Triangle CDE and triangle CDF are both equilateral, and we have DE = EF = FD.
What is construction?
In the context of geometry, construction refers to the process of creating geometric figures or shapes using only a compass and straightedge (also called a ruler). The goal of construction is to create accurate and precise geometric figures using only these two tools and a given set of instructions.
To construct an equilateral triangle DEF with points E and F on circle C, we will follow the following steps:
Step 1: Draw a circle with center C and point D on it.
Step 2: Draw a line through points C and D. This line will be the base of our equilateral triangle DEF.
Step 3: Construct the perpendicular bisector of line CD. This can be done by drawing two circles with centers at C and D, both with radius greater than half the length of CD. The two circles will intersect at two points, and the line passing through these two points will be the perpendicular bisector of CD.
Step 4: Let G be the intersection point of the perpendicular bisector and circle C that is closest to point D. Draw lines from G to points C and D. These lines will intersect circle C at points E and F, respectively.
Step 5: Check that the length of CE is equal to the length of CF. This can be done using a compass and ruler.
Step 6: Finally, draw lines from points E and F to point D. The resulting triangle DEF will be equilateral.
To prove that triangle DEF is equilateral, we can show that DE = EF = FD. We already know that CE = CF from step 5. Since C is the center of the circle, CE = CF = CD. By construction, angle ECD and angle FCD are both 60 degrees, since they are inscribed angles that intercept the same arc.
Therefore, triangle CDE and triangle CDF are both equilateral, and we have DE = EF = FD.
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2. A carnival game uses a fair spinner with 16 sections. There are 2 white sections, 4 red sections, and 10 blue sections on the spinner. Part A Find the probability of each event. Express your answer as a fraction in simplest form. P(blue) = = P(green) = P(red) =
The probability of landing on a blue section on the spinner is 10/16, which simplifies to 5/8.
The probability of landing on a green section is 0/16, which simplifies to 0.
The probability of landing on a red section is 4/16, which simplifies to 1/4.
What is probability?The probability of any event occurring is calculated by finding the total number of possible outcomes that would result in that event occurring, and then dividing that number by the total number of possible outcomes.
In this case, the total number of possible outcomes is 16, since there are 16 sections on the spinner.
The total number of outcomes that would result in a blue section being chosen is 10, since there are 10 blue sections on the spinner.
Dividing 10 by 16 gives us the probability of landing on a blue section, which is 5/8.
The same process is used to calculate the probabilities for green and red sections, with the only difference being that the number of sections for each color is different.
Since there are no green sections, the probability of landing on a green section is 0.
There are 4 red sections, so the probability of landing on a red section is 4/16, which simplifies to 1/4.
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The area of this trapezoid is 25.5 ft2.
What is the height of the trapezoid? Show your work.
The calculated height of the trapezoid is 5 ft.
What is the height of the trapezoid?The formula for the area of a trapezoid is:
A = (b1 + b2)h/2
where A is the area, b1 and b2 are the lengths of the parallel sides, and h is the height.
We are given that the area of the trapezoid is 25.5 ft^2, and the lengths of the parallel sides are 3.2 ft and 7 ft.
Substituting the given values into the formula, we get:
25.5 = (3.2 + 7)h/2
Simplifying the expression on the right side, we get:
25.5 = 5.1h
Dividing both sides by 5.1, we get:
h = 25.5/5.1
h = 5
Therefore, the height of the trapezoid is 5 ft.
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Which number sentence shows the area of this figure
Answer:
?
Step-by-step explanation:
It would be a bit helpful if you uploaded a picture, since I don't see one.
Add or Subtract then complete the chart
The complete chart here is
|---Simplify the Polynomial---|--Degree--|--Leading Coefficient--|--Constant--|
|------7a^4 - 3a^3 - 2a + 2------|------4-------|---------------7----------------|--------2-------|
Explain polynomial
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using mathematical operations such as addition, subtraction, multiplication, and exponentiation. Polynomials can have one or more terms and are commonly used to model relationships in science and engineering.
What is meant by coefficient?
The coefficient is a numerical factor that is multiplied by a variable or a product of variables in an algebraic expression or equation.
According to the given information
To add the two polynomials, we combine like terms. That is, we add the coefficients of terms that have the same degree.
(7a⁴ - 6a³ + 1) + (3a³ - 2a + 1)
= 7a⁴ + (-6a³ + 3a³) + (-2a) + (1 + 1)
= 7a⁴ - 3a³ - 2a + 2
So, the simplified polynomial is `7a⁴ - 3a³ - 2a + 2`.
The degree of this polynomial is `4` since the term with the highest degree is `7a⁴`.
The leading coefficient of this polynomial is `7` since it is the coefficient of the term with the highest degree.
The constant term of this polynomial is `2` since it is the term with a degree of `0`.
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you have a cake that is 16 inches by 18 inches.you want each piece to be exactly the same size.how many people can u serve equal sized piece of cake
The number of people are 72.
What is HCF?Highest Common Factor is the full name for HCF in mathematics.
According to the laws of mathematics, the highest positive integer that divides two or more positive integers without leaving a residual is known as the greatest common divisor, or gcd.
What is LCM?Least Common Multiple is the full name for LCM in mathematics.
LCM (a,b) in mathematics stands for the least common multiple, or LCM, of two numbers, such as a and b. The smallest or least positive integer that is divisible by both a and b is known as the LCM.
We need to know the size of the pieces in order to calculate how many people can be fed equal-sized pieces of cake from a 16 by 18-inch cake.
Assume that each piece is a square with sides measuring "x" in length. Each piece's area would be x², and the cake's overall area would be 16 x 18 inches, or 288 square inches.
We must divide the entire area of the cake by the area of each piece to determine the maximum number of pieces that can be cut from the cake:
288 / x² = (16*18) / x² = 288 / x²
Choosing a "x" value that evenly splits into 16 and 18 is necessary if we want every piece to be the exact same size. Since 2 is the greatest possible value, we can divide the cake into 8 rows of 9 squares, yielding a total of 72 pieces.
As a result, a cake measuring 16 inches by 18 inches can be divided into 72 equal-sized pieces, each measuring 2 inches by 2 inches.
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We can serve 18 people equal-sized pieces of cake from a cake that is 16 inches by 18 inches, assuming each piece is a square and we want the pieces to be exactly the same size.
What is area?A two-dimensional figure, form, or planar lamina's area is a measurement of how much space it takes up in the plane.
To find out how many people can be served equal-sized pieces of cake from a cake that is 16 inches by 18 inches, we need to first determine the size of each piece of cake.
The total area of the cake is:
16 inches x 18 inches = 288 square inches
To divide the cake into equal-sized pieces, we need to determine the size of each piece. Let's assume we want each piece to be a square. To find the size of each square, we need to find the square root of the total area of the cake:
√(288 square inches) ≈ 16.97 inches
Since we want the pieces to be exactly the same size, we'll round down to the nearest inch, which gives us:
Each piece of cake will be approximately 16 inches by 16 inches.
To find out how many people can be served equal-sized pieces of cake, we need to divide the total area of the cake by the area of each piece:
288 square inches ÷ (16 inches x 16 inches) = 18
Therefore, we can serve 18 people equal-sized pieces of cake from a cake that is 16 inches by 18 inches, assuming each piece is a square and we want the pieces to be exactly the same size.
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THE RIGHT ANSWER GETS BRAINLIEST AND 30 POINTS ‼️‼️‼️‼️‼️‼️‼️‼️
The area of the figure is 160.6 unit²
How to find the area of the figure?
Since the figure is composed of a parallelogram and one semicircle, the area will be:
Area of figure = area of parallelogram + area of semicircle
Area of figure = bh + 1/2πr²
where b = 10, h = 6 and r = 8
Area of figure = (10*6) + (1/2 * 22/7 * 8²)
Area of figure = 60 + 100.6
Area of figure = 160.6 unit²
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Find the missing value in the equivalent ratio 12:18 = 16:___
To find the missing value in the equivalent ratio 12:18 = 16:___, you can use cross-multiplication.
First, you multiply the first term of the first ratio by the second term of the second ratio, and then you set it equal to the product of the second term of the first ratio and the missing value in the second ratio.
So, you get:
12 x ___ = 18 x 16
To solve for the missing value, you can divide both sides of the equation by 12:
___ = (18 x 16)/12
Simplifying the right-hand side, you get:
___ = 24
Therefore, the missing value in the equivalent ratio 12:18 = 16:___ is 24
Answer:24
Step-by-step explanation:
12:18
= 6:9
= 2:3
=16:24
First, you simplify it as much as you can and then scale it up to the number necessary.
(I NEED THIS ASAP PLEASE PLEASE!)
6. Reflect triangle ABC over the line x = -2. Call this new triangle A'B'C'. Then reflect triangle A'B'C' over the line x = 0. Call the resulting triangle A"B"C".
Describe a single transformation that takes ABC to A"B"C".
A Reflect triangle over the line x = -1 will take ABC to A"B"C".
What is triangle reflection theorem?
The outcome is the same as rotating the initial triangle 180 degrees around the origin, just as the theorem predicts, when we reflect the triangle over the y-axis and then cross the x and y axes while reflecting the outcome across the x axis.
The single transformation that takes ABC to A"B"C" is a reflection over the line x = -1.
The new triangle A'B'C' is formed by reflecting ABC across the line x = -2, where each point in A', B', and C' is the reflection of the corresponding point in ABC. As a result, A', B', and C''s x-coordinates are all 4 units to the left of A', B', and C''s x-coordinates.
The final triangle A"B"C is formed by reflecting A'B'C' over the line x = 0, where each point in A", B", and C" is a reflection of the corresponding point in A'B'C' over the line x = 0. Accordingly, the x-coordinates of A, B, and C" are all located two units to the right of the x-coordinates of A', B', and C'.
If we combine these two reflections into a single transformation, we can think of it as reflecting ABC over a line that is 2 units to the right of x = -2, which means it has an equation of x = -2 + 2 = -1. This line is the perpendicular bisector of the segment connecting the points (-2, 0) and (0, 0), which is the line that connects the two reflection lines x = -2 and x = 0.
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Whats a product with 2 consecutive digits
The two consecutive digits whose product is 132 are equals to 11 and 12.
Product of two consecutive digits = 132
Let us consider that the two consecutive digits are y and y+1.
Then we can write the expression as,
⇒y × ( y + 1 ) = 132
Expanding the left side of the expression, we get,
⇒ y² + y = 132
Rearranging the values and solving for y, we get,
⇒ y² + y - 132 = 0
Factorize this quadratic equation to get the value of y we have,
⇒ y² + 12y - 11y - 132 = 0
⇒ y( y + 12 ) -11 ( y + 12 ) = 0
⇒ ( y + 12)(y - 11) = 0
This implies,
The two possible values of y are -12 and 11.
However, choose the positive value of y .
Since we are dealing with digits.
Thus, y = 11, and the consecutive digits are 11 and 12.
Therefore, the two consecutive digits are 11 and 12, and their product is indeed 132.
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The given question is incomplete, I answer the question in general according to my knowledge:
The product of two consecutive digit is equal to 132 . What are the numbers?
The following data set shows the number of books checked out from a library during the first two weeks of the month: 19, 10, 15, 99, 12, 18, 15, 16, 12, 13, 18, 17, 19, 13 Which of the following statements is true based on the data set? (5 points)
Pls ASAP
PLS
Please Help With These
3-6 You can determine the slope of a roof using a flat board, a bubble level and a ruler,or a sight card can be used. (True or False)
Given statement "3-6 You can determine the slope of a roof using a flat board, a bubble level and a ruler,or a sight card can be used." is true. Because flat board, a bubble level and a ruler,or a sight card methods are commonly used to determine the slope of a roof, and both are effective.
Determining the slope of a roof is an important task for many construction and roofing projects.
One method to determine the slope of a roof is to use a flat board, a bubble level, and a ruler.
First, place the flat board horizontally on the roof surface, and use the bubble level to make sure it is level.
Then, measure the distance between the bottom of the board and the roof surface at the point where the board intersects with the roof using the ruler. This distance is the rise.
Next, measure the distance from the point where the board intersects with the roof to the end of the board using the ruler. This distance is the run. Finally, divide the rise by the run to get the slope.
Another method to determine the slope of a roof is to use a sight card. A sight card is a specially designed tool that can help to measure the angle of a slope or pitch. The sight card is held at the bottom of the roof and lined up with a known vertical or horizontal line, such as the edge of a building.
Statement is true.
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Marcellus and Anika are playing a video game where they earn 35 stars for each level they pass. Which equation represents the total number of stars earned, y, for passing x levels?
The equation represents the total number of stars is y = 35x
Which equation represents the total number of starsThe equation that represents the total number of stars earned, y, for passing x levels would be:
y = 35x
where y represents the total number of stars earned, and x represents the number of levels passed.
For example, if Marcellus and Anika passed 10 levels, they would earn:
y = 35(10) = 350 stars
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Mr. Liaw has two cylindrical travel mugs that he likes to take on his drive to work. His stainless steel mug has a height of 8.5 inches and a radius of 1.5 inches. What is the volume of the stainless steel mug?
Use ≈3.14 and round your answer to the nearest whole number
Mr. Liaw's ceramic mug has the same volume, but a radius of 2 inches. What is the height of the ceramic mug?
Use ≈3.14 and round your answer to the nearest whole number
Therefore , the solution of the given problem of volume comes out to be ceramic mug has a 5 inch height.
What does volume actually mean?A three-dimensional object's volume, which is expressed in cubic units, indicates how much space it takes up. The indications for cubic measurements are litre and in3. However, you have to be cognizant of an object's volume in order to calculate its dimensions. Converting an object's weight into metric units like iii . and kilogrammes is a standard technique.
Here,
The following formula determines a cylinder's volume:
=> V = π * r² * h
where r is the cylinder's radius and h is its height, and is roughly equivalent to 3.14.
For the mug made of stainless steel:
1.5 inches is the radius (r).
Height in inches (h) is 8.5
With these values entered into the formula, we obtain:
=> V = 3.14 * (1.5)² * 8.5
=> V ≈ 3.14 * 2.25 * 8.5
=> V ≈ 60.14
The stainless steel mug has a 60 cubic inch volume, rounded to the nearest whole number.
Regarding the mug:
Radius (r) equals two inches
Given that it has the same volume as the stainless steel mug, its volume (V) is equal to 60 cubic inches.
=> 60 = 3.14 * (2)²* h
=> h = 60 / (3.14 * 4)
=> h ≈ 4.77
The ceramic mug has a 5 inch height.
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Select the correct answer.
Exponential function f is represented by the table.
x -2 -1 0 1 2
f(x) -46 -22 -10 -4 -1
Function g is represented by the equation.
Which statement correctly compares the two functions on the interval [-1, 2]?
A. Both functions are increasing, but function g increases at a faster average rate.
B. Only function f is increasing, and only function f is negative.
C. Both functions are increasing, but function f increases at a faster average rate.
D. Only function f is increasing, but both functions are negative.
The statement that correctly compares the two functions on the interval [-1, 2] is A. Both functions are increasing, but function g increases at a faster average rate.
How to explain the functionFor function f, as x increases from -2 to 2, we can see that the corresponding values of f(x) also increase, but at a decreasing rate. This is because the function is exponential and approaching a horizontal asymptote as x goes to infinity. Therefore, f(x) is increasing, but at a decreasing rate.
For function g, we don't have enough information to determine if it's increasing or not, since only its equation is given. The correct option is A.
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What three regular polygons have special triangles formed by their radius and apothem and what are the special triangles?
The regular triangle, regular hexagon, and regular dodecagon have special triangles formed by their radius and apothem, which are congruent right triangles with legs equal to half the length of one side and hypotenuse equal to the radius or apothem, and are 30-60-90 triangles.
Regular polygons with three special triangles formed by their radius and apothem are,
Regular triangle
The radius and apothem of an equilateral triangle are equal. Therefore, the special triangles formed by the radius and apothem are congruent right triangles with legs equal to half the length of one side of the equilateral triangle and hypotenuse equal to the radius or apothem. In other words, the special triangles are 30-60-90 triangles.
Regular hexagon
The radius of a regular hexagon is equal to the length of one side, and the apothem is equal to the height of an equilateral triangle with side length equal to the length of one side of the hexagon.
Regular dodecagon
The radius of a regular dodecagon is equal to the length of one side multiplied by the square root of 2 plus the square root of 2 times the square root of 3. The apothem is equal to the height of an isosceles triangle with base equal to one side of the dodecagon and legs equal to the radius of the dodecagon.
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For the hexagon, the special triangle formed by its radius and apothem is a 30-60-90 right triangle, where the radius is the hypotenuse and the apothem is the longer leg.
The three regular polygons that have special triangles formed by their radius and apothem are the triangle, square, and hexagon.
For the triangle, the special triangle formed by its radius and apothem is a 30-60-90 right triangle, where the radius is the hypotenuse and the apothem is the shorter leg.
For the square, the special triangle formed by its radius and apothem is a 45-45-90 right triangle, where the radius is the hypotenuse and the apothem is one of the legs.
For the hexagon, the special triangle formed by its radius and apothem is a 30-60-90 right triangle, where the radius is the hypotenuse and the apothem is the longer leg.
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