Answer: The unit circle is an essential tool in trigonometry because it helps in understanding and visualizing the relationships between angles and the values of the sine, cosine, and tangent functions.
The unit circle is a circle with a radius of one unit and centered at the origin of a coordinate plane. It is divided into 360 degrees or 2π radians. By placing this circle on the coordinate plane, we can easily determine the sine and cosine values of angles in standard position.
For any given angle θ, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and the cosine value is the x-coordinate of that same point. The tangent function, which is the ratio of sine to cosine, can also be determined using the unit circle.
The unit circle also helps in understanding the periodicity of the sine and cosine functions. Since the circumference of the unit circle is 2π, the sine and cosine functions repeat themselves after every 2π radians or 360 degrees. This periodicity allows for the use of trigonometric identities and formulas to simplify and solve complex trigonometric equations.
In summary, the unit circle is an essential tool in trigonometry as it provides a visual representation of angles and their corresponding sine, cosine, and tangent values, and allows for the use of trigonometric identities and formulas to solve complex problems.
I just need q24. a) i dunno how to do it
The product in diagram A is: (3x)*(2x + 3y)
And the product in diagram B is: 5a*(8a + 3)
How to write the given products?
Let's start with the diagram A, we can define the long stripe as x and the short one as y.
On the top side we can see 3x.
On the left side we can se 2x + 3y
Then the product modeled is just:
(3x)*(2x + 3y)
Now let's go to diagram B.
Here we can see that the left side has a length 5a, and then we have two top sides, so this can be written as a sum:
8a + 3
The product between these two quantities is:
p = 5a*(8a + 3)
These are the two products.
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how many ounces of cornstarch are there in a box containing 300 grams of cornstarch
Answer:
Roughly 10 oz, exact amount is 10.58219
Miguel is playing a game in which a box contains four chips with numbers written on them. Two of the chips have the number 1, one chip has the number 3, and the other chip has the number 5. Miguel must choose two chips, and if both chips have the same number, he wins $2. If the two chips he chooses have different numbers, he loses $1 (–$1).
Let X = the amount of money Miguel will receive or owe. Fill out the missing values in the table. (Hint: The total possible outcomes are six because there are four chips and you are choosing two of them.)
questions to answer:
What is Miguel’s expected value from playing the game?
Based on the expected value in the previous step, how much money should Miguel expect to win or lose each time he plays?
What value should be assigned to choosing two chips with the number 1 to make the game fair? Explain your answer using a complete sentence and/or an equation.
On average, Miguel loses half a dollar every time he plays.
How to solve
To check all the events (6), we label the chips.
Suppose one chip with 1 is labeled R1 and the other B1 (as if they were red and blue). Now, lets take all combinations; for the first chip, we have 4 choices and for the 2nd chip we have 3 remaining choices.
Thus there are 12 combinations. Since we dont care about the order, there are only 6 combinations since for example R1, 3 is the same as 3, R1 for us.
The combinations are: (R1, B1), (R1, 3), (R1, 5), (B1, 3), (B1, 5), (3,5)
We have that in 1 out of the 6 events, Miguel wins 2$ and in five out of the 6 events, he loses one.
The expected value of this bet is: 1/6*2+5/6*(-1)=-3/6=-0.5$.
In general, the expected value of the bet is the sum of taking the probabilities of the outcome multiplied by the outcome; here, there is a 1/6 probability of getting the same 2 chips and so on.
On average, Miguel loses half a dollar every time he plays.
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Select the shapes that have at least 1 apex.
Right circular cone
Right circular cylinder
Right pyramid
Right triangular prism
A and B
A and C
B and C
C and D
Answer:
A and C : cones and pyramids
Step-by-step explanation:
only cones, pyramids and isosceles triangles have apexes
The shapes that have at least 1 apex are:
Right circular cone
Right pyramid
Therefore, the answer is C and D.
if anyone could help, would be much appreciated
[tex]g( \sqrt{55} ) = ( \sqrt{55} ) {}^{2} + 11 = 55 + 11 = 66[/tex]
[tex]f(g(55)) = f(66) = \sqrt{66 - 2} = \sqrt{64} = 8[/tex]
The value of the function f(g(√55)) is 3√7.
What is Function?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable).
We have the function:
f(x) = √ (x - 2)
and, g(x) = x² + 11.
As, g(√55)
= (√55)² + 11
= 55 + 11
= 66
So, f(g(√55))
= √ (65 - 2)
= √63
= 3√7
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Ms. Brown's class has 7 boys and 8 girls. Mrs. Ball's class has 4 boys and 10 girls. If one student is randomly selected from each class, what is the probability they are both girls? Answer as simplified fraction.
Considering the probability of independent events, if one student is randomly selected from each class, the probability they are both girls is 8/21.
Definition of probabilityProbability establishes a relationship between the number of favorable events and the total number of possible events.
Then, the probability of any event A is defined as the ratio between the number of favorable cases (number of cases in which event A may or may not occur) and the total number of possible cases:
Probability= number of favorable cases÷ number of possible cases
Probability of Independent EventsTwo events A and B are said to be independent if and only if the probability of event B is not influenced by the occurrence of event A or vice versa.
For any number of independent events, the probability that all the events will occur is the product of the probabilities that the individual events will occur. That is, if A and B are independent events, P(A and B) = P(A)×P(B).
Probability in this caseYou know:
Ms. Brown's class= 8 girls and 7 boys= 15 studentsMrs. Ball's class= 10 girls and 4 boys= 14 studentsIf one student is randomly selected from each class, the probability that a girl is selected in each class is calculated as:
Probability in Ms. Brown's class= 8÷ 15= 8/15Probability in Mrs. Ball's class= 10÷ 14= 10/14Now, the probability that both students selected for the classes are girls, considering that these are independent events, is calculated as:
Probability of both students selected for the classes are girls = Probability in Ms. Brown's class× Probability in Mrs. Ball's class
Probability of both students selected for the classes are girls = 8/15× 10/14
Probability of both students selected for the classes are girls = 8/21
Finally, the probability they are both girls is 8/21.
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Please helppp!!!!!!!!!!!!
The length of the arc of the sector is approximately 34.6 millimeters when rounded to one decimal place.
What is angle ?
An angle is a geometric figure formed by two rays that share a common endpoint, called the vertex of the angle. The rays are often referred to as the sides of the angle. Angles are typically measured in degrees or radians and are used to describe the amount of turn between two lines or surfaces.
In a circle, the measure of the arc of a sector is proportional to the measure of the central angle that it subtends. Therefore, to find the length of the arc of the sector in this problem, we need to first find the measure of the central angle that corresponds to the inscribed angle of 312 degrees.
The measure of the central angle is equal to twice the measure of the inscribed angle, which is 312/2 = 156 degrees. This is because the inscribed angle subtends an arc that is half as long as the arc subtended by the central angle.
The length of the arc of the sector can be calculated using the formula:
arc length = (central angle / 360 degrees) x (2 x pi x radius)
Plugging in the values we have:
arc length = (156/360) x (2 x pi x 4mm)
arc length = (13/30) x (8pi)
arc length = (104/30) x pi
arc length ≈ 34.557mm
Therefore, the length of the arc of the sector is approximately 34.6 millimeters when rounded to one decimal place.
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A survey asked a group of adults and youths if they prefer reading books printed on paper or electronic books.
Book Preference
------------Print Electronic
Youths 28 40
Adults 37 38
What percent of the youths surveyed prefer reading electronic books? Round your answer to the nearest whole number percent.
Answer: 59%
Step-by-step explanation:
To find the percentage of youths who prefer reading electronic books, we need to divide the number of youths who prefer electronic books by the total number of youths surveyed and then multiply by 100.
The total number of youths surveyed is:
28 + 40 = 68
The number of youths who prefer electronic books is:
40
Dividing the number of youths who prefer electronic books by the total number of youths surveyed and multiplying by 100, we get:
(40/68) x 100% = 58.82%
Rounding to the nearest whole number percent, we get:
59%
Therefore, approximately 59% of the youths surveyed prefer reading electronic books.
Answer:
59%
Step-by-step explanation:
Please help
A passenger train and a freight train traveling in the same direction leave San Jose at 3 pm. The passenger train travels 3 times at fast. At 6 pm, the trains are 240 miles apart. How fast is each traveling?
Therefore , the solution of the given problem of speed comes out to be train is moving at 60 miles per hour while the freight train is moving at 20 miles per hour.
What is speed, exactly?By keeping an eye on something from a distance, we can gauge its speed. An object's speed determines how far it can travel in a given period of time. Utilizing the equation velocity = distance/time, velocity is determined. Kilometers per hour (mileage), kilometres per hour (km/h), and metres per second (m/s) are the three most widely used measures of speed (mph).
Here,
The freight train would cover the following distance in three hours at the speed of "x" miles per hour:
=> Distance = speed x duration = x miles/hour x 3 hours = 3x miles.
The passenger train would cover the following distance in three hours at "3x" kilometres per hour:
Distance is calculated as follows: 3 kilometres per hour times 3 hours equals 9 miles.
We can now determine the 240-mile overall distance between the two trains at 6 o'clock:
Overall distance equals the sum of the routes taken by the passenger and freight trains.
=> 240 miles = 3x miles + 9x miles
=> 240 miles = 12x miles
Therefore,
=> x=20 kilometres per hour
=> 3x = 60 mph
As a result, the passenger train is moving at 60 miles per hour while the freight train is moving at 20 miles per hour.
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Construct a cumulative frequency distribution table from the frequency table given below. Class Interval 0 − 8 8 − 16 16 − 24 24 − 32 32 − 40 Frequency 9 13 12 7 15
The frequency of all the five intervals is 9 + 13 + 12 + 7 + 15 = 56.
The cumulative frequency distribution table from the given frequency table:Class IntervalFrequencyCumulative Frequency0 − 89 89 + 1312 25 + 12 + 7 32 + 1515 47 + 15 62It is clear that the frequency of the first interval is 9. Therefore, the frequency of the first interval is 9.Next, the frequency of the second interval is 13. Therefore, the frequency of the first and second intervals is 9 + 13 = 22.Next, the frequency of the third interval is 12. Therefore, the frequency of the first, second and third intervals is 9 + 13 + 12 = 34.Then, the frequency of the fourth interval is 7. Therefore, the frequency of the first, second, third, and fourth intervals is 9 + 13 + 12 + 7 = 41.Finally, the frequency of the fifth interval is 15. Therefore, the frequency of all the five intervals is 9 + 13 + 12 + 7 + 15 = 56.
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One angle of a triangle measures 80°. The other two angles are in a ratio of 3:7. What are the measures of those two angles?
Answer:
30 degrees and 70 degrees
Step-by-step explanation:
A triangle has 180 degrees, and if you take away the known 80, you are left with 100 degrees to be split in a 3:7 ratio.
On a scale drawing of a dock for a marina, 1 inch equals 18 feet. The dock is 250 feet long. What is the length of the dock on the scale drawing?
The length of the dock on the scale drawing would be 13.89 inches (250/18 = 13.89).
On a scale drawing of a dock for a marina, 1 inch equals 18 feet. This means that for every inch on the scale drawing, it represents 18 feet in reality. The dock is 250 feet long. To calculate the length of the dock on the scale drawing, divide 250 by 18. This gives the result of 13.89. This means that the length of the dock on the scale drawing is 13.89 inches. This is because for every inch on the scale drawing, it represents 18 feet in reality. Therefore, the length of the dock on the scale drawing is 13.89 inches (250/18 = 13.89). To ensure accuracy, the calculation should be double checked. This can be done by multiplying 13.89 by 18 which should give the result of 250. This confirms that the length of the dock on the scale drawing is 13.89 inches.
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Helen is about to ride a straight water slide. The launching platform is at the top of a tower that is 21 feet tall. The splash pool at the end of the slide is 72 feet from the base of the tower. How long is the water slide itself?
feet=
The length of the slide itself in feet would be = 75 ft.
How to calculate the length of the water slide?To calculate the length of the water slide the Pythagorean formula should be used which is written as follows;
C² = a² + b²
a = length of the tower = 21 ft
b = distance from the base of the tower = 72ft
C = length of the water slide = ?
c² = 21²+72²
C² = 441 + 5184
c² = 5625
C = √5625
C = 75 ft
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face A
2cm
20%
face B
2cm
Both watch faces are circular with radius 2cm.
The materials used to make both watch faces have the same thickness.
A is made entirely of plastic.
B has a 20° sector of metal and à 340° sector of plastic.
The ratio of the cost per cm² of the metal to the cost per cm² of the plastic is 3:2
Work out the ratio of the cost of the materials for A to the cost of the materials for B.
Give your answer in its simplest form.
You must show all your working.
Therefore , the solution of the given problem of ratio comes out to be 8:17 .
Describe ratio.The simple formula "a / b," where "b" may be greater than zero, can be used to determine a group of integers "a" and "b" that appear to be equal. Two ratios are combined to form a ratio. If there were only one man and three ladies, the fraction would be 1:1. In the group, there are 3/4 women and 1/4 males. Parts can be a division between things, a figure, or a particular percentage of a component's volume.
Here,
Let x be the price of plastic per square centimetre used to create visage A.
Consequently, the price of the components for aspect A is:
=> 4πx
The metal sector's size on face B is:
=> (20/360)π(2cm)² = (1/9)π cm²
The plastic sector's size on face B is:
=> (340/360)π(2cm)² = (17/9)π cm²
Face B's entire surface size is:
=> (1/9)π + (17/9)π = 2π cm²
Let y be the plastic's expense per square centimetre used to create face B.
Consequently, the price per cm2 of the metal used to create face B is three and a half times the price per cm2 of the plastic, leading to:
=> (3/2)y = (3/2)(2x) = 3x
The material cost for the metal section on face B is inversely correlated with its area, so:
=> (1/9)π(3x)
The price of the plastic used in the section on face B is inversely proportional to its area, so:
=> (17/9)πy
=> 8πx : 17πy = 8 : 17
The response is 8:17,
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Ava spent 3 hours yesterday practicing the piano. She is going to spend at least 3 hours practicing today which number line shows
this
The number line that models the total amount of time Ava will practice piano in all is shown in the figure and Ava will practice piano for a total of 6 hours.
We can model the total amount of time Ava will practice piano in all using a number line. Since she practiced for 3 hours yesterday and is going to practice for another 3 hours today, we can represent this on a number line as follows:
The number line above represents the total amount of time Ava will practice piano in all. The first mark (0) represents the starting point, which is when she has not started practicing yet. The second mark (3) represents the time she spent practicing yesterday, and the third mark (6) represents the time she will spend practicing today. The distance between the first mark and the third mark (0 to 6) is the total amount of time Ava will practice piano in all, which is 6 hours.
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Crystal is buying new carpet for her room. Carpeting is on sale for 20% off. Find the sale price of the carpeting if the original price was $480.
Answer: $384
Step-by-step explanation: 480 - 20% = 384
Select all equations that are equivalent to x2+6x=16.
The equivalent equation of x^2 + 6x = 16 are (x + 3)^2 = 25 and x^2 + 6x + 9 = 25
How to determine the equivalent equationFrom the question, we have the following parameters that can be used in our computation:
x2+6x=16
Express properly
So, we have the following representation
x^2 + 6x = 16
Add 9 to both sides of the equation
x^2 + 6x + 9 = 25 --- this represents option (c)
Expand
x^2 + 3x + 3x + 9 = 25
When the expression is factorized, we have
(x + 3)^2 = 25
Hence, the equivalent equation is (x + 3)^2 = 25
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Complete question
Select all equations that are equivalent to x2+6x=16.
The options are added as an attachment
16) Choose the picture that correctly shows the first image being reflected across *
line s, then across line t.
Answer:
Number C shows to be first image being reflected across.
A lake near Susan's house is predicted to lose water every day because of a drought. The table shows the estimated water depth in feet, y, after x days of drought. 0 100 1 99.5 2 99 3 98.5 4 98 5 97.5 Choose the function that represents the data. O y = -0,5x + 100 O y=-x05 + 100 O y=-0.5x² + 100 Oy= - +100 X 0.5
As the table shows the estimated water depth in feet, y, after x days of drought. The function that represents the data is y = -0,5x + 100. The Option A is correct.
What does a function means?A function is defined as a relationship between a set of inputs that each have one output. A function is a relationship between inputs in which each input is related to exactly one output.
Based on the given table, it appears that the water depth is decreasing by 0.5 feet every day due to the drought. Additionally, the initial depth of the lake was 100 feet.
Therefore, the function that represents the data is y = -0.5x + 100 where y represents the water depth in feet and x represents the number of days of drought.
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Please help I do not understand this .
HELP. Turn the equation into y=mx+b if not already. Figure out whether it has one solution, no solution or infinite solutions. Use graphing, substitution or elimination.
-3x+3y=4
-x+y=3
(same slope or different slop?)
(same y-intercept or different y-intercept?)
Answer:
Starting with the given system of equations:
-3x + 3y = 4
-x + y = 3
To solve for y in terms of x, we can rearrange each equation to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
For the first equation, we can add 3x to both sides and divide by 3 to obtain:
3y = 3x + 4
y = x + 4/3
So the slope of the first equation is 1 and the y-intercept is 4/3.
For the second equation, we can add x to both sides to obtain:
y = x + 3
So the slope of the second equation is also 1, but the y-intercept is different at 3.
To determine the number of solutions, we can use either substitution or elimination.
Using substitution, we can solve for x in the second equation:
x = y - 3
Then substitute this expression for x in the first equation:
-3(y - 3) + 3y = 4
Simplifying, we get:
-3y + 9 + 3y = 4
9 = 4
This is a contradiction, so the system has no solution.
Alternatively, we can use elimination to solve for y:
-3x + 3y = 4
-x + y = 3
Multiplying the second equation by 3, we get:
-3x + 3y = 4
-3x + 3y = 9
Subtracting the second equation from the first, we get:
0x + 0y = -5
This is a contradiction, so the system has no solution.
In summary, the given system of equations has no solution. The two equations have the same slope but different y-intercepts.
Hope this helps you! I'm sorry if it's wrong. If you need more help, ask me! :]
Answer:
Equation 1: y = x + 4/3
Equation 2: y = x +3
No solution, same slope, different y-intercepts.
Step-by-step explanation:
The two equations have been converted to y=mx+b or slope-intercept form. The slope is the same (1), but the intercepts are different. Equation 1 crosses the y-axis at (0, 4/3), Equation 2 crosses the y-axis at (0,3). Both equations represent parallel lines, so there is no solution because they will never intersect each other.
If P(A) = 7/13, P(B) = 5/13, and P(A and B) = 3/13, what will be P(A|B)?
If P(A) = 7/13, P(B) = 5/13, and P(A and B) = 3/13, then the probability of P(AIB) = 3/5.
Probability is a measure of the likelihood of an event occurring. Many events cannot be predicted with absolute certainty. We can only predict the probability of an event by using it, i.e. the probability of it occurring.
The probability can range from 0 to 1, where 0 means the event is unlikely to happen and 1 means it will definitely happen. All events in the sample space have a probability of 1. For example, when we flip a coin, heads or tails, there are only two possible outcomes (H, T). But when you toss two coins, there are four possible outcomes, namely {(H, H), (H, T), (T, H), (T, T)}.
According to the Question:
Given that:
P(A) = 7/13
P(B) = 5/13
P(A∩B) = 3/13
We know that :
P(A|B) = P(A∩B) / P(B).
(By formula for conditional probability)
P(AIB) = (3/13)÷ (5/13)
= 3/5
Hence, the value of P(A|B) = 3/5.
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Nuhome movers charges $95 per
hour for loading/unloading
Jay is moving a distance of 200 miles and needs 9 hours of loading/unloading and 7 hours of packing/ unpacking. His total moving cost including tax charges equals to the $2,053.8375.
We have to calculate Jay's moving cost if the service also charges 7.25% tax on the total. Here, Jay is moving a distance of 200 miles and needs 9 hours of loading/unloading and 7 hours of packing/ unpacking. Now, first we determine the NuHome Movers charges,
Rate of NuHome Movers charges for loading/unloading services = $95 per hour and NuHome Movers charges for 9 hours of loading/unloading services = Multiplication of number of hours by rate of service = 9× $95 = $855 --(1)
Rate of NuHome Movers charges for packing/unpacking services = $80 per hour land NuHome Movers charges for 7 hours of packing/unpacking services = Multiplication of number of hours by rate of service = 7× $80 = $560--(2)
Now, determine charge for truck rental,
Rate of NuHome Movers charges for truck rental = $2.50 per mile
and NuHome Movers charges for 200 miles truck rental = number of miles × rate of each mile = $2.50 × 200
= $500--(3)
Thus, total cost of charges which Jay will pay = charges cost for packing/unpacking services + charges cost for loading/unloading services + charges for truck rental
= $855 + $560 + $500 ( from (1), (2),(3))
= $1,915
Tax charges on total = 7.25%
Determine the Jay's total moving cost including taxes = $1,915 + $1915(7.25/100)
= $1,915 + $1915( 0.0725)
= $1,915 + $138.8375
= $2,053.8375
Hence, required cost value is $2,053.8375.
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Complete question :
NuHome Movers charges $95 per hour for loading/unloading services and $80 per hour for packing/unpacking services. Their charge is $2.50 per mile for truck rental. Jay is moving a distance of 200 miles and needs 9 hours of loading/unloading and 7 hours of packing/ unpacking. What will his moving cost be if the service also charges 7.25% tax on the total?
what is solved 5x+6=3l
Find the difference of 42 3/4−34 1/8
The difference of the fractions 42¾ - 34⅛ is equal to 8⅝.
How to evaluate the difference of the fractionsThe given fractions are expressed as mixed numbers, so we shall first rewrite them as improper fractions, find the lowest common multiple of their denominators to have a single fraction, before then subtracting the resulting numerators to have their difference.
42¾ = 171/4
34⅛ = 273/8
42¾ - 34⅛ = 171/4 - 273/8
LCM of 4 and 8 is 8, so;
42¾ - 34⅛ = (171 ×2 - 273 × 1)/8 {simplifying to a single fraction}
42¾ - 34⅛ = (342 - 273)/8
42¾ - 34⅛ = 69/8
42¾ - 34⅛ = 8⅝
Therefore, difference of the fractions 42¾ - 34⅛ is equal to 8⅝.
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Sarah invested £12 000 in a unit trust five years ago.
The value of the unit trust has increased by 7% per annum for each of the last 3 years.
Before this, the price had decreased by 3% per annum.
Calculate the current price of the unit trust.
Give your answer to the nearest whole number of pounds.
Answer:efefef
Step-by-step explanation:
efregtythth
The lateral area of the smaller cylinder is 25pi and the lateral area of the larger cylinder is 64pi. If the volume of the smaller cylinder is 150pi, what is the volume of the larger cylinder?
The volume of the larger cylinder is (25/8)πR³.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object. It is typically measured in units of cubic meters, cubic centimeters, cubic feet.
According to question:Let's denote the radius and height of the smaller cylinder as r and h, respectively, and the radius and height of the larger cylinder as R and H, respectively.
We know that the lateral area of a cylinder is given by the formula:
Lateral area = 2πrh
Using this formula for the smaller and larger cylinders, we get:
25π = 2πrh
64π = 2πRH
Dividing both sides of the first equation by 2π, we get:
rh = 25/2π
We also know that the following formula determines a cylinder's volume:
Volume = πr²h
Using this formula for the smaller cylinder, we get:
150π = πr²h
Substituting rh = 25/2π from above, we get:
150π = πr²(25/2π)
r² = 12
Taking the square root of both sides, we get:
r = √12 = 2√3
Substituting this value into rh = 25/2π, we get:
h = 25/(2πr) = 25/(4π√3)
Now, we can use the ratio of the lateral areas to find the ratio of the heights:
25π/64π = (2πrh)/(2πRH)
25/64 = rh/RH
25/64 = (2√3)(25)/(4πR)(H)
H = 2hR/r
Finally, we can use the formula for the volume of the larger cylinder to find its volume:
Volume = πR²H
Volume = πR²(2hR/r)
Volume = 2πR³h/r
Substituting the values we found for r, h, and H, we get:
Volume = 2π(R³)(25/(4π√3))/(2√3)
Volume = (25/8)πR³
Therefore, the volume of the larger cylinder is (25/8)πR³.
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A house painter charges a fee for supplies and an hourly fee for the time spent painting. A paint job
that takes 3 h costs $140. A paint job that takes 5 h costs $200.
Equation for the total cost of paint job 'y' with number of hours spent on paining 'x' is formed as:
y = 50 + 3x
y = 50 + 5x
Explain about linear equation in two variables?To determine the values of the variables, we should solve an equation in 2 variables using several techniques and making sure there are 2 equations in the supplied system of equations. If there is only one solution, the given equations are always of parallel lines, and if there isn't a solution at all, the offered equations are of intersecting lines.Let the total cost of paint job be' 'y'.
Let 'x' be the number of hours spent on paining.
Then,
Total cost = supplies + hourly fee* number of hours.
Let the supplies cost be 'z'
140 = z + 3x
200 = z + 5x
Solving both equation.
2x = 60
x = 30
z = 50
Thus, equation for the total cost of paint job 'y' with number of hours spent on paining 'x' is formed as:
y = 50 + 3x
y = 50 + 5x
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Consider a box with dimensions 3 cm × 5 cm × 11 cm. If all of its dimensions are increased by x cm, what values of x will give a box with a volume between 300 cm3 and 900 cm3?
Explain your solution and solve using the topic of inequalities.
The correct answers are x³ + 19 x² + 103 x - 135 = 0 and x³ + 19 x² + 103 x - 735 = 0.
Given:
A box with dimensions 3 cm × 5 cm × 11 cm
Current volume of given box = 3×5×11 = 165 cm^3
New volume of box = (3+x) × (5+x) × (11+x)
= (15 + 8 x + x²) × (11+x)
= 165 + 103 x + 19 x^2 + x³
For volume to be 300 cm³
=> x³+ 19 x² + 103 x + 165 = 300
=> x³ + 19 x² + 103 x - 135 = 0
For volume to be 900 cm³ -
=> x³ + 19 x² + 103 x + 165 = 900
=> x³ + 19 x²+ 103 x - 735 = 0
Therefore, x^3 + 19 x^2 + 103 x - 135 = 0 and x³ + 19 x² + 103 x - 735 = 0 are the solution.
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Work out the volume of the cone radius is 3. 5 and length is 12
To work out the volume of the cone, we can use the formula V = (1/3) * π * r² * h, where r is the radius of the base of the cone and h is the height of the cone. The volume of the cone is approximately 171.55 cubic units.
The formula for calculating the volume of a cone is given as V = (1/3) * π * r² * h. In this formula, 'r' represents the radius of the base of the cone, and 'h' represents the height of the cone. By substituting the given values, the volume of the cone is calculated to be approximately 171.55 cubic units. This formula is widely used in various fields, such as architecture, engineering, and manufacturing. Calculating the volume of a cone is useful in determining the amount of material required to construct or fill a cone-shaped object, such as a cone-shaped container or a cone-shaped structure. The formula is also used in solving real-life problems related to cones, such as calculating the volume of ice cream in a cone-shaped ice cream cone or the volume of a cone-shaped tank used for storing liquids.
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