Answer:
[tex]3(5w + 4) < 12w + 15[/tex]
[tex]15w + 12 < 12w + 15[/tex]
[tex]3w < 3[/tex]
[tex]w < 1[/tex]
A is the correct answer.
How much did she pay?
Megan earned £158,900 before tax last year. Her total tax bill was £53,205, which was calculated by finding the tax on each part of her earnings within different tax brackets and adding them up.
What is rate?A rate is a measure of how one quantity changes in relation to another quantity. It is usually expressed as a ratio of two different units, such as miles per hour or dollars per pound. Rates can be used to compare different values and to make predictions about future outcomes. Common examples of rates include interest rates, exchange rates, and tax rates.
According to the given information:To calculate Megan's tax bill, we need to find the tax on each part of her earnings that falls within the different tax brackets, and then add them up.
For the first £11,000 of Megan's earnings, there is no tax.
For the next £32,000 of Megan's earnings (from £11,001 to £43,000), the tax rate is 20%, so the tax on this portion of her earnings is:
0.2 x 32,000 = £6,400
For the next £107,000 of Megan's earnings (from £43,001 to £150,000), the tax rate is 40%, so the tax on this portion of her earnings is:
0.4 x 107,000 = £42,800
For the portion of Megan's earnings over £150,000, the tax rate is 45%, so the tax on this portion of her earnings is:
0.45 x 8,900 = £4,005
Therefore, Megan's total tax bill is:
£6,400 + £42,800 + £4,005 = £53,205
So, Megan paid £53,205 in total tax for the last year.
Therefore, Megan earned £158,900 before tax last year. Her total tax bill was £53,205, which was calculated by finding the tax on each part of her earnings within different tax brackets and adding them up.
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Solve the systems by graphing.
y = -2x - 8
X-Y = 5
Answer: x=1, y=-6
Step-by-step explanation:
YESSIR
8) A kite descends 57 feet in 3 seconds. Which integer represents the change, in fee
position of the kite after one second?
54
-29
-19
O-3
In the proportion , in 1 second the position of kite is C)19 feet.
What is proportion?
A percentage is created when two ratios are equal to one another. We write proportions to construct equivalent ratios and to resolve unclear values. a comparison of two integers and their proportions. According to the law of proportion, two sets of given numbers are said to be directly proportional to one another if they grow or shrink in the same ratio.
Here in 3 seconds a kite's position = 57 feet.
Then , in 1 second a kite's position = x feet.
Now using proportion then,
=> 3 seconds = 57
1 second = x
=> x = 57/3 = 19 feet.
Hence In 1 second the position of kite is C)19 feet.
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URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Match the description table or graph or equation with the correct type of function
A. Linear function table
B. Non linear function graph
C. Linear function equation
D. Non linear function equation
E. Linear function graph
F. Non linear function table
The description table or graph or equation with the correct type of function:
Linear function table: A
Non-linear function graph: B
Linear function equation: C
Non-linear function equation: D
Linear function graph: E
Non-linear function table: F
What is function?A function is a mathematical concept that describes a relationship between two sets of values, where each input value (independent variable) corresponds to exactly one output value (dependent variable). In other words, a function is a rule or a formula that takes an input value and produces a unique output value.
Here,
The table of values shows a constant rate of change, which indicates a linear function. We can verify this by calculating the slope between any two points, which should be the same. For example, the slope between (-2, -8) and (2, 0) is (0 - (-8))/(2 - (-2)) = 8/4 = 2. So the function is of the form y = mx + b, where m = 2 and b = -4. This gives us the equation y = 2x - 4, which is a linear function. Therefore, the correct match is A.
The graph shows a curve, which indicates a non-linear function. It does not resemble a straight line, so we can eliminate linear functions. The shape of the graph suggests that it might be a quadratic function, which has a U-shaped curve. Quadratic functions can be represented by an equation of the form y = ax² + bx + c. We can't determine the equation of the function from the graph alone, but we can conclude that it is a non-linear function. Therefore, the correct match is B.
The equation 3x - 4y = 27 can be rearranged to solve for y in terms of x: y = (3/4)x - (27/4). This equation is of the form y = mx + b, where m = 3/4 and b = -27/4, which indicates a linear function. Therefore, the correct match is C.
The equation 3x² - 2x = 4y is a quadratic equation, which has a variable raised to the second power. Quadratic functions are non-linear, so the correct match is D.
The graph shows a straight line, which indicates a linear function. We can determine the equation of the line by finding the slope and y-intercept. The slope is (5 - (-1))/(2 - (-2)) = 6/4 = 3/2, and the y-intercept is -1. So the equation of the line is y = (3/2)x - 1, which is a linear function. Therefore, the correct match is E.
The table of values does not show a constant rate of change, which indicates a non-linear function. For example, between x = -2 and x = -1, y changes by 5, but between x = 0 and x = 1, y changes by only 1. This suggests that the function is not linear. We can't determine the exact form of the function from the table alone, but we can conclude that it is a non-linear function. Therefore, the correct match is F.
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Suppose the amount of time it takes a Capt. Rigg, a pilot for a national airline, to land a plane from announcement to touch down is uniformly distributed from 0 to 25 minutes. a) Find the probability that, for a randomly selected flight, it takes Capt. Rigg at least 5 minutes to land the plane after the announcement. Round to 1 decimal place. b) What is the longest landing time included in the 10% of flights he lands the quickest. Round to 1 decimal place. c) What is the probability that it will take Capt. Rigg exactly 15 minutes to land a plane?
On solving the provided question we cans ay that Thus the probability that Capt. Rigg will wait at least 5 minutes to land the plane following the announcement is 1 - 0.5 = 0.5.
What is probability?Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many fields, including statistics, economics, science, and engineering.
probability Capt. Rigg has at least 5 minutes after the announcement to land the plane, which is equal to the area under the probability density function of the uniform distribution ranging from 5 to 25 minutes. A uniform distribution's density function on the interval [a, b] is f(x) = 1/(b-a) for an x b and f(x) = 0 otherwise.
Because a = 5 and b = 25 in this case, the density function is f(x) = 1/20 for 5 x 25, and f(x) = 0 otherwise. From 5 to 25, the area covered by this function is:
dx = [5,25] 1/20 dx = [x/20] from 5 to 25 = (25-5)/20 = 1.
Thus the likelihood that Capt. Rigg will wait at least 5 minutes to land the plane following the announcement is 1 - 0.5 = 0.5.
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9t+6
in standard form. asap!
The standard form of the expression 9t + 6 is:
9t + 1.How to find the standard form of this expression?To write the expression 9t + 6 in standard form, we need to rearrange the terms so that the t-term comes before the constant term:
9t + 6 = 9t + 6(1)= 9t + 6(1/6)= 9t + (6/6)= 9t + 1Therefore, the standard form of an algebraic expression is a way of writing the expression by arranging the terms in a specific order. In the case of linear expressions like 9t + 6, the standard form is to write the t-term before the constant term.
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Which three questions are statistical questions?
A
B
C
D
E
How many hours of sleep per night do students usually get when they have school the next day?
What is the minimum height requirement for a rollercoaster?
What are the heart rates of the students in Mr. Peterson's physical education class?
How many letters are in the last names of the students in my 6th grade class?
How many runs did the New York Yankees score in their last game?
The three statistical questions are: A. How many hours of sleep per night do students usually get when they have school the next day? B and E.
What is a statistical question?A statistical question is one that may be resolved via the collection and examination of data. The question frequently incorporates a variable that might differ from person to person or scenario to situation, and the solution is not always the same. The words "how many," "what is the average," "what is the range," "what is the distribution," etc. are frequently used in statistical queries.
The three statistical questions are:
How many hours of sleep per night do students usually get when they have school the next day?
What are the heart rates of the students in Mr. Peterson's physical education class?
How many runs did the New York Yankees score in their last game?
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Write a mixed number giving the amount shaded. Then write this amount as as improper fraction
Answer:
improper fraction would be 3 2/3 and mixed number would be 11/3
Step-by-step explanation:
Less strict regulation results in many more companies obtaining the right to mine coal in Mpumalanga. At the same time, alternative energy solutions result in a decrease in the demand for coal-generated electricity. Explain how the market for coal will be affected by these changes. Clearly indicate how the equilibrium price and equilibrium quantity of coal will be affected by these changes. Make use of a combination of diagrams and verbal explanations to explain your answer.
Each of 17 mothers-to-be received 3D ultrasound scans, which showed that 8 of them will give birth togirls. Assuming that simultaneous births do not happen
(i.e., babies are born one after another), what is the
probability that the first two babies that are born
happen to be boys?
Since each birth is independent and the probability of giving birth to a boy or a girl is equal, the probability of giving birth to a boy is 0.5, and the probability of giving birth to a girl is also 0.5.
The probability of the first baby being a boy is 0.5. The probability of the second baby also being a boy is also 0.5, since the gender of one baby does not affect the gender of the other.
The probability of both events occurring together (i.e., the probability of having two boys in a row) is equal to the product of their probabilities. Thus, the probability of having two boys in a row is 0.5 x 0.5 = 0.25 or 25%.
Since simultaneous births do not happen (i.e., babies are born one after another), the probability of the first two babies that are born being boys is the same as the probability of any two specific babies being boys. Therefore, the probability of the first two babies being boys is 0.25 or 25%.
Note that the fact that 8 out of the 17 mothers will give birth to girls is irrelevant to this particular probability problem, since we are only interested in the probability of two specific babies being boys.
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What’s 890 / 5621 /56 +8987 x 67 - square root
The simplified expression is: √602755 = 776.372977
What do you mean by term Square root ?
The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol √, which is called the radical symbol. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. The square root of a non-negative number is always a non-negative number.
Assuming
(890 / 5621) / 56 + 8987 x 67 - sqrt(x)
where x is some unknown value, we can simplify the expression as follows:
First, we can simplify the division:
(890 / 5621) / 56 = 0.000281
Then we can substitute this value into the expression and simplify the multiplication:
0.000281 + 8987 x 67 - sqrt(x) = 602755 - sqrt(x)
At this point, we cannot simplify the square root further without knowing the value of x.
Therefore, the simplified expression is:
√602755 - 776.372977
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Four more than the quotient of a number and 6 equals 9 Turn into a equation
Answer:
We can get creative and use many equations using these numbers.
Step-by-step explanation:
4 more than (addition) the quotiont (division) Of a number (a variable) 6 is equal to 9.
let the variable for number be n.
we can use 4n/6=9
x=13.5
or 13 5/10
Answer:
Step-by-step explanation:
(x ÷ 6) + 4 = 9
(x ÷ 6) +4 - 4= 9 -4
x ÷ 6 = 5
x/6 = 5
x/6 · 6/1 = 5 · 6
x = 30
A couple of two way radios were purchased from different stores. Two way radio A can reach 8 miles in any direction. Two way radio B can reach 11.27 kilometers in any direction.
Part A: How many square miles does two way radio A cover? Use 3.14 for TT and round to the nearest whole number.
B. Part A: How many square miles does two way radio B cover? Use 3.14 for TT and round to the nearest whole number.
C. If 1 mile= 1.61 kilometers, which two way radio covers the larger area?
D. Using the radius of each circle, determine the scale factor relationship between the radio coverages.
Answer:
Step-by-step explanation:
A) To find the area covered by two-way radio A, we need to calculate the area of a circle with a radius of 8 miles:
Area = πr² = 3.14 x 8² = 200.96 square miles
Rounding to the nearest whole number, two-way radio A covers 201 square miles.
B) To find the area covered by two-way radio B, we need to convert the radius from kilometers to miles and then calculate the area of a circle with that radius:
Radius in miles = 11.27 / 1.61 = 6.9988 miles (rounded to 4 decimal places)
Area = πr² = 3.14 x (6.9988)² = 153.94 square miles
Rounding to the nearest whole number, two-way radio B covers 154 square miles.
C) Two-way radio A covers a larger area than two-way radio B (201 square miles vs 154 square miles).
D) The scale factor relationship between the radio coverages can be found by dividing the radius of radio A by the radius of radio B:
Scale factor = radius of A / radius of B = 8 miles / (11.27 km / 1.61 km/mile) = 4.97
This means that the coverage of two-way radio A is almost 5 times larger than that of two-way radio B.
State what each variable may be so that the equation is true.
You must have at least one negative number.
Explain how you chose the values for a and b.
2ª 2b = 20
Edis
The answer is 2a + 2b + 3 = 20
Trigonometry review applications
I need help with 15-18
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It has many real-world applications such as in (engineering, architecture, and navigation.)
15. One application of trigonometry is in determining the height of a building. This can be done by measuring the angle of elevation from a known distance away from the building and using trigonometric functions to find the height.
16. Another application of trigonometry is in surveying land. Surveyors use trigonometry to measure distances and angles between different points on a piece of land, which helps them create accurate maps and boundary lines.
17. Trigonometry is also used in physics to calculate the motion of objects that move in a circular or oscillatory motion, such as a pendulum or a rotating wheel.
18. Finally, trigonometry is used in astronomy to calculate the position and movement of celestial bodies. Astronomers use trigonometric functions to measure the distance between stars and galaxies
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Alexander built a toy box in the shape of a rectangular prism with an open top. The diagram below shows the toy box and the net of the toy box. NET OF TOY BOX is 14 cm 10 cm 9 cm What is the surface area, in square centimeters, of the toy box?
The surface area of the toy box is 712 square centimeters.
Describe Surface Area?Surface area is the measure of the total area that the surface of an object occupies. In other words, it is the sum of the areas of all the faces or surfaces of the object.
For example, consider a rectangular prism with dimensions length (l), width (w), and height (h). The surface area of this object would be the sum of the areas of all six faces, which are:
Two rectangular faces with area lw
Two rectangular faces with area wh
Two rectangular faces with area lh
So the total surface area of the rectangular prism would be:
Surface area = 2lw + 2wh + 2lh
The concept of surface area is useful in various fields, such as architecture, engineering, and physics. For example, in construction, knowing the surface area of a structure can help estimate the amount of materials needed for construction, such as paint or roofing tiles. In physics, surface area is important in determining the heat transfer and rate of reaction of a substance.
The surface area of the toy box can be found by summing the areas of its six faces. From the given net, we can see that the toy box has dimensions of 14 cm by 10 cm by 9 cm.
The top and bottom faces have areas of 14 cm × 10 cm = 140 cm² each.
The front and back faces have areas of 14 cm × 9 cm = 126 cm² each.
The left and right faces have areas of 10 cm × 9 cm = 90 cm² each.
Therefore, the total surface area is:
2(140 cm²) + 2(126 cm²) + 2(90 cm²) = 280 cm² + 252 cm² + 180 cm² = 712 cm²
Thus, the surface area of the toy box is 712 square centimeters.
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Using your points from Part A, What is the total amount of water, in gallons, in the pool after 47 minutes? Show your work or explain how you determined your answer.
The total amount of water in the pool after 47 minutes is 1,470 gallons.
Determine the initial volume of water in the pool (if any) from Part A.
Calculate the rate at which water is being added to or removed from the pool (in gallons per minute).
This information should also be provided in Part A.
Multiply the rate of water flow by 47 minutes to find the total amount of water added or removed during that time.
For example, if the rate is 10 gallons per minute, you would calculate 10 * 47 = 470 gallons.
Add the initial volume of water to the total amount of water added or removed during the 47 minutes.
If the initial volume was, for example, 1,000 gallons and you added 470 gallons, the total volume would be 1,000 + 470 = 1,470 gallons.
The final result is the total amount of water in the pool after 47 minutes.
In the example above, the total volume would be 1,470 gallons.
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A jar contains 8 red marbles numbered 1 to 8, 12 blue marbles numbered 1 to 12, and 4 white marbles numbered 1 to 4. A marble is drawn at random from the jar. Find the probability of the given event. Write your answers as integers or reduced fractions. (a) The marble is red. (b) The marble is not red. (c) The marble has the number 2 written on it. (d) The marble is blue with the number 1 written on it. (d) The marble has the number 15 written on it.
(e) There is no marble with the number 15 on it, so the probability of drawing such a marble is 0.
What is probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.
(a) The probability of drawing a red marble is the number of red marbles divided by the total number of marbles in the jar. So, the probability is 8/(8+12+4) = 8/24 = 1/3.
(b) The probability of not drawing a red marble is the number of non-red marbles divided by the total number of marbles in the jar. So, the probability is (12+4)/(8+12+4) = 16/24 = 2/3.
(c) The probability of drawing a marble with the number 2 on it is the number of marbles with the number 2 divided by the total number of marbles in the jar. There is only one red marble with the number 2, one blue marble with the number 2, and no white marbles with the number 2. So, the probability is 2/24 = 1/12.
(d) The probability of drawing a blue marble with the number 1 on it is the number of blue marbles with the number 1 divided by the total number of marbles in the jar. There is only one blue marble with the number 1. So, the probability is 1/24.
(e) There is no marble with the number 15 on it, so the probability of drawing such a marble is 0.
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can someone help me pleasee
Find the x- and y- intercepts, if they exist.
f(x)= log2(x+2)-5
Accοrding the given questiοn the x- and y- intercepts, are as fοllοws,
x-intercept =(28, 0), y-intercept = (0, -4)
What are intercepts?An intercept is the pοint οf intersectiοn οf the line with the cοοrdinate axes. If the line intersects with the X axis, then the y cοοrdinate οf the pοint οf intersectiοn is zerο and the intercept is called the x intercept.
Tο find the x-intercept, we need tο sοlve fοr x when y = 0:
0 = lοg2(x + 2) - 5
5 = lοg2(x + 2)
[tex]2^5 = x + 2[/tex]
x = 30 - 2 = 28
Therefοre, the x-intercept is (28, 0).
Tο find the y-intercept, we need tο evaluate the functiοn when x = 0:
f(0) = lοg2(0 + 2) - 5
f(0) = lοg2(2) - 5
f(0) = 1 - 5
f(0) = -4
Therefοre, the y-intercept is (0, -4)
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) 100 जना मानिसमा गरिएको सर्वेक्षणमा 65 जनाले दुध र 55 जनाले दही मन पराउँछन् । यदि 15 जनाले दुवै मन पराउँदैनन् भने भेनचित्र प्रयोग गरी दुध र दही दुवै मन पराउने मानिसको सङ्ख्या पत्ता लगाउनुहोस् । In a survey of 100 people, 65 like milk and 55 like curd. If 15 like none of these, by using Venn-diagram, find how many people like both milk and curd. 12- मूल्य अभिवृद्धि कर सहित बिक्री गरियो । यदि
Answer:
X=35
Step-by-step explanation:
if the translated part is the full question; the answer is attached ⬆️.
Hope it helps
Write te equation of a circle with (-17;-9) and (-19;-9) being the ends of the diameter
Answer:
[tex](x + 18)^{2} + (y + 9)^{2} = 1[/tex]
Step-by-step explanation:
Suppose you want to have $300,000 for retirement in 30 years. Your account earns 9% interest
How much would you need to deposit in the account each month?
How much interest will you earn?
Using percentage, we can find that:
You need to deposit $833.33 every month and the interest earned will be = $27,000.
Define percentage?The Latin term "per centum," which means "by the hundred," is the source of the English word "percentage." Fractions with a denominator of 100 are percentages. In other words, it is a link where the value of "whole" is always taken to be 100.
Total amount you want to have = $300,000
Time is = 30 years
= 30 × 12
= 360 months.
Interest earned is 9%.
Now, the amount you need to deposit every month =
Total amount/Total time
= 300000/360
= $833.33
So, you need to deposit $833.33 every month.
Now interest earned in 30 years is given = 9%
So, amount earned = 9% of 300,000
= 9/100 × 300,000
= $27,000.
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What’s the answer to this math problem that’s in the picture?
The value of the account at the end of 50 years is $432,125.19.
1. For the first 20 years:
The monthly contribution is $110, and the interest rate is 5.5% compounded monthly.
Using the formula:
FV = P [[tex](1 + r)^n[/tex] - 1] / r
where:
FV = Future value
P = Monthly payment
r = Monthly interest rate
n = Number of months
Plugging in the values:
FV1 = $110 x [tex][(1 + 0.055/12)^{(20)(12)[/tex] - 1] / (0.055/12)
= $67,100.54
2. For the next 25 years:
The monthly contribution increases to $225, and the interest rate remains the same at 5.5% compounded monthly.
Plugging in the values:
FV2 = $225 [[tex][(1 + 0.055/12)^{(25)(12)[/tex] - 1] / (0.055/12)
= $307,527.10
3. For the last 5 years:
The monthly contribution further increases to $350, and the interest rate is still 5.5% compounded monthly. Again, we will use the same formula to calculate the future value of the contributions made during this period.
Plugging in the values:
FV3 = $350 [[tex][(1 + 0.055/12)^{(5)(12)[/tex] - 1] / (0.055/12)
= $57,497.55
Finally, the total value of the account at the end of 50 years
Total Value = FV1 + FV2 + FV3
= $67,100.54 + $307,527.10 + $57,497.55
= $432,125.19
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decompose the mixed number in two different ways 1 4/10
The mixed number 1 4/10 can be decomposed as 14/10 or 1 2/5.
We may use the fact that a mixed number combines a whole number and a correct fraction to deconstruct the mixed number 1 4/10 in two different ways. So, the mixed number 1 4/10 can be represented as follows:
1 4/10 = 1 + 4/10
The sum of two fractions with a common denominator can be used to decompose this mixed number. This can be accomplished by changing the whole number 1 to a fraction with the same denominator as the fraction 4/10:
1 4/10 = 1 + 4/10 = 10/10 + 4/10 = 14/10
The sum of a whole number and a fraction that is less than one can be used to further break down this mixed number. In order to achieve this, we can take one out of the mixed number and express the resulting fraction in the simplest words possible:
1 4/10 = 1 + 4/10 = 1 + (4/10)/(10/10) = 1 + 2/5 = 1 2/5
The mixed number 1 4/10 can be broken down into 14/10 or 1 2/5 as a result.
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Can someone please help with this ?!!! 30 points
A patient reports taking three teaspoons of a liquid medication every four hours at home. How should the total daily amount of this medication be documented using the metric system?
To document the total daily amount of liquid medication using the metric system, we need to convert teaspoons to milliliters, which is the standard unit of volume in the metric system.
One teaspoon is equal to approximately 5 milliliters, so three teaspoons would be equal to 15 milliliters.
If the patient is taking 15 milliliters every four hours, then they would be taking the medication 6 times a day (24 hours divided by 4 hours per dose).
Therefore, the total daily amount of the medication would be:
15 milliliters per dose x 6 doses per day = 90 milliliters per day
So, the total daily amount of the liquid medication should be documented as 90 milliliters per day in the patient's medical records using the metric system.
Vusi has been told that he should increase the overall total value of all of the eight parts by 10%, to reflect the effect of price inflation and the devaluation of the rand over the six month period when the production line was not operational.
By answering the presented question, we may conclude that As a result, equation the new total value of all eight pieces, after the 10% rise, is R132.
What is equation?In mathematics, an equation is a statement that states the equivalence of two expressions. An equation is made up of two sides separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" contends that the phrase "2x Plus 3" equals the number "9." The purpose of equation solving is to identify the value or values of the variable(s) that will make the equation true. Simple or complicated equations, regular or nonlinear, with one or more components are all possible. For example, in the equation "x2 + 2x - 3 = 0," the variable x is raised to the second power. Lines are utilized in many areas of mathematics, including algebra, calculus, and geometry.
Multiply the current total value by 1.1 to boost the overall total value of all eight components by 10%.
To begin, compute the current total value of all eight parts:
R16 + R24 + R30 + R18 + R12 + R10 + R16 + R14 = Total value
R120 is the total value.
Let us now boost the total amount by 10%:
Total new value = R120 x 1.1
R132 is the new total value.
As a result, the new total value of all eight pieces, after the 10% rise, is R132.
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Which fraction is equivalent to 0.008?
A. 2/250
B. 4/250
C. 2/?25
D. 4/25
Answer:
A. 2/250
Step-by-step explanation:
So, first use calculator on the multiple-choice question.
solve the equation for exact solution in interval [0 degree , 360 degree)
(cot theta-1)(2sin theta +square root 3)=0
Answer: We can start by using the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.
So, we can set each factor equal to zero and solve for theta:
Factor 1: cot(theta) - 1 = 0
Using the identity cot(theta) = cos(theta) / sin(theta), we can rewrite this as:
cos(theta) / sin(theta) - 1 = 0
cos(theta) - sin(theta) = 0
cos(theta) = sin(theta)
Dividing both sides by cos(theta), we get:
tan(theta) = 1
The solutions to this equation are theta = 45 degrees and theta = 225 degrees (since tangent is positive in the first and third quadrants).
Factor 2: 2sin(theta) + sqrt(3) = 0
Subtracting sqrt(3) from both sides and dividing by 2, we get:
sin(theta) = -sqrt(3)/2
The solutions to this equation are theta = 240 degrees and theta = 300 degrees (since sine is negative in the third and fourth quadrants, and sin(240) = sin(300) = -sqrt(3)/2).
Therefore, the exact solutions in the interval [0 degrees, 360 degrees) are:
theta = 45 degrees, 225 degrees, 240 degrees, 300 degrees.
Step-by-step explanation:
Name a radius
Name a chord
Name a tangent
Where is the point
of tangency?
B
E
C D
G
TH
F
The names of the terms stated on the circle are radius = DB, chord = GE, tangent = IF and point of tangency = H
Naming the terms stated on the circleA radius is a line segment that connects the center of a circle to any point on the circle's circumference.
So, the radius is DB
A chord is a line segment that connects two points on the circumference of a circle.
So, the chord is GE
A tangent is a line that intersects a circle at exactly one point, called the point of tangency.
So, the tangent is IF and the point of tangency is H
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