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Give the equation of a rational function which has all of the properties above.
Answers
To find the equation of a rational function that satisfies the given properties, we can use the fact that a rational function can be written as the quotient of two polynomials. We know that the function has x-intercepts at (2,0) and (6,0), which means that the denominator must contain factors of (x-2) and (x-6):
r(x) = A(x-2)(x-6)/...
where A is a constant coefficient that we need to determine and the ellipsis (...) represents the remaining factors of the denominator.
We also know that there is a hole at x=1, which means that both the numerator and denominator have a factor of (x-1). To ensure that the hole cancels out, we can simplify the expression by dividing both the numerator and denominator by (x-1):
r(x) = A(x-2)(x-6)/(x-1)(x-1)... (1)
Now, we need to include the vertical asymptotes at x=5 and x=3. This means that the denominator must contain factors of (x-5) and (x-3). However, we also need to ensure that the end-behavior of the function is given by f(x) -> 1 as x -> ± infinity. This can be achieved by making the degree of the numerator equal to the degree of the denominator, and ensuring that the leading coefficient of the numerator is equal to the leading coefficient of the denominator.
To achieve this, we can set the denominator to be (x-5)(x-3)^2, and choose the numerator to be a constant multiple of (x-1)(x-1):
r(x) = A(x-2)(x-6)/(x-1)(x-1)(x-5)(x-3)^2
Now, we need to determine the value of the constant A. Since we know that r(x) has a hole at x=1, we can use the fact that the limit of r(x) as x approaches 1 must exist and be finite. This means that the factors of (x-1) must cancel out in the numerator and denominator, and we can use this to solve for A:
lim(x->1) r(x) = lim(x->1) A(x-2)(x-6)/(x-1)(x-1)(x-5)(x-3)^2
= A(-1)(-5)/(-4)^3
= 5A/64
We know that the limit must exist and be finite, and we also know that it must equal some number k, since r(x) has a hole at x=1. Therefore, we can set 5A/64 = k, and solve for A:
A = 64k/5
Substituting this value of A into equation (1), we get the final equation for r(x):
r(x) = (64k/5)(x-2)(x-6)/(x-1)(x-1)(x-5)(x-3)^2
where k is any non-zero constant.
Answer:
One possible rational function that satisfies the given properties is:
r(x) = (x-2)(x-6) / [(x-3)(x-5)]
Step-by-step explanation:
x-intercepts at (2,0) and (6,0)
The x-intercepts are the points where the function crosses the x-axis, i.e., where the function value is zero. Since we are given that the function has x-intercepts at (2,0) and (6,0), we know that the function can be factored as:
r(x) = A(x-2)(x-6)
where A is a constant that we need to determine.
A hole at x=1 and vertical asymptotes at x=5 and x=3
A hole in a rational function occurs when there are factors in the numerator and denominator that cancel out, leaving a "hole" in the graph. In this case, we are given that there is a hole at x=1, which means that there must be a common factor of (x-1) in both the numerator and denominator. So, we can write:
r(x) = A(x-2)(x-6) / (B(x-1)(x-3)(x-5))
where B is another constant that we need to determine.
We are also given that there are vertical asymptotes at x=5 and x=3, which means that the denominator must have factors of (x-5) and (x-3) that do not cancel out with any factors in the numerator. So, we can write:
r(x) = A(x-2)(x-6) / (B(x-1)(x-3)(x-5)) = [A(x-2)(x-6)] / [(B(x-1))(x-3)(x-5)]
End behavior given by x → ∞ ,f(x) → 1 and x → -∞ ,f(x) → 1
The end behavior of a rational function is determined by the degree of the numerator and denominator. Since the numerator and denominator in this case have the same degree (3), we know that the end behavior is given by the ratio of the leading coefficients, which is A/B. We are told that the end behavior approaches 1 as x approaches infinity or negative infinity, so we can set A/B = 1 and solve for A in terms of B:
A/B = 1, so A = B
Putting it all together
We now have enough information to write the equation for the rational function with the given properties:
r(x) = A(x-2)(x-6) / (B(x-1)(x-3)(x-5))
Using A = B, we get:
r(x) = A(x-2)(x-6) / [A(x-1)(x-3)(x-5)]
Canceling out the common factor of (x-1) in the numerator and denominator, we get:
r(x) = (x-2)(x-6) / [(x-3)(x-5)]
which is the equation for the desired rational function.
Related Questions
A hamster running in a wheel of radius 10 cm spins the wheel one revolution in 5 seconds.
a) What is the angular velocity of the wheel?
b) At what linear velocity is the hamster running?
just need the answer
Answers
a) The angular velocity = 0.4π rad/s b) The linear velocity of the hamster = 4π cm/s.
What is linear velocity?Linear velocity is the rate of change of an object's position with respect to time. It measures how fast an object is moving in a straight line, and is often expressed in units of distance per time, such as meters per second or kilometers per hour.
According to given information:a) The angular velocity of the wheel can be calculated as follows:
Angular velocity = (2π * number of revolutions) / time
Here, the wheel completes one revolution in 5 seconds, so the number of revolutions is 1 and the time is 5 seconds. Therefore,
Angular velocity = (2π * 1) / 5
Angular velocity = 0.4π rad/s (or approximately 1.26 rad/s)
b) The linear velocity of the hamster can be calculated as follows:
Linear velocity = radius * angular velocity
Here, the radius of the wheel is 10 cm, and the angular velocity we found in part (a) is 0.4π rad/s. Therefore,
Linear velocity = 10 cm * 0.4π rad/s
Linear velocity = 4π cm/s (or approximately 12.57 cm/s)
So the hamster is running at a linear velocity of approximately 12.57 cm/s.
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find the factors to this polynomials i need help SHOW YOUR WORK.
Answers
Answer:
hope this will help you...
A patient takes 110 mg of a drug at the same time every day: Just before each tablet is taken, 5% of the drug present in th
preceeding time step remains in the body. What quantity of the drug remains in the body in the long run?
Hint: First find the quantity of drug present in the body after the second tablet, third tablet, etc to generalize the sequenc
formula for the nth tablet. Then use that sequence to predict the drug remains in the long run.
Answers
The requried quantity of drug that remains in the body, in the long run, is 220 mg.
Let's denote the amount of drug in the body just before taking the nth tablet by [tex]D_n[/tex]. We know that [tex]D_0 = 0[/tex](since the patient hasn't taken any tablets yet) and that 5% of the drug is present in the body just before taking a tablet remains in the body. Therefore, we can write:
[tex]D_1 = 110\\D_2 = 0.95 * D_1 + 110 = 203.5\\D_3 = 0.95 * D_2 + 110 = 292.225\\D_4 = 0.95 * D_3 + 110 = 376.81375[/tex]
We can see that the formula for [tex]D_n[/tex] is:
[tex]D_n = 110 * (1 - 0.05)^n + 110 * (1 - (1 - 0.05)^n) / 0.05[/tex]
Simplifying this formula, we get:
[tex]D_n = 220 * (1 - 0.95^n)[/tex]
Now, we want to find the amount of drug that remains in the body in the long run. This means we want to find the limit of [tex]D_n[/tex] as n goes to infinity. Using the formula above, we can see that:
[tex]lim_ {n- > oo} D_n = lim_ {n- > oo} 220 * (1 - 0.95^n) = 220[/tex]
Therefore, the quantity of drug that remains in the body, in the long run, is 220 mg.
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Divide and give your answer as a decimal. Round your answer to the nearest thousandth (3 digits behind the decimal).
2/9
Answers
Answer:
0.222
Step-by-step explanation:
2 divided by 9 is:
0.222 (rounded to the nearest thousandth)
Find the compound interest on Rs. 3,500 for 2 years at the rate of 8% per annum.
Answers
Answer: The formula for compound interest is:
A = P(1 + R/100)^t
where A is the amount after t years, P is the principal amount, R is the rate of interest per annum, and t is the time period in years.
Here, P = Rs. 3,500, R = 8%, and t = 2 years.
So, the amount after 2 years will be:
A = 3,500(1 + 8/100)^2
= 3,500(1.08)^2
= 3,892.32
Therefore, the compound interest for 2 years will be:
CI = A - P
= 3,892.32 - 3,500
= 392.32
Hence, the compound interest on Rs. 3,500 for 2 years at the rate of 8% per annum is Rs. 392.32.
Step-by-step explanation:
Help corrections due in 3 hours! giving 25 points and brainlist
Answers
The value of GH in the right angle triangle, given FH and Angle FHG is 22.46 inches.
How to find the value of GH ?Since we have the adjacent side (FH) and want to find the hypotenuse (GH), we can use the cosine function.
cos(35°) = adjacent side (FH) / hypotenuse (GH)
We know that FH = 18.4 in. So, we can write the equation as:
cos(35°) = 18.4 / GH
GH = 18.4 / cos(35°)
GH = 18.4 / 0.81915
= 22.46 inches
So, the length of GH, the hypotenuse, is approximately 22.46 inches.
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An equipment was purchased from China and arrived at Kampala at a value of shs.5,000,000 shs with transport cost of 800,000 shs. and total taxes amounted to 1,500,000 the equipment is expected to be useful for 6 years and an estimated salvage value of 1,300,000 shs. calculate the depreciation expense for each year and accumulated depreciation upto year 6. Using the straight line method.
Answers
The depreciation expense for each year is 1,000,000 shs.
Accumulated depreciation up to year 6 is 6,000,000 shs.
How to calculate depreciation using the straight line method?The depreciation using the straight line method is given by the formula:
Depreciation expense per year = (Initial cost - Salvage value) / Useful life
Initial cost = 5,000,000 + 800,000 + 1,500,000 = 7,300,000 shs.
Salvage value = 1,300,000 shs.
Depreciation expense per year = (7,300,000 - 1,300,000) / 6
= 6,000,000 / 6
= 1,000,000 shs.
Thus, the depreciation expense for each year is 1,000,000 shs.
Accumulated depreciation up to year 6 = Depreciation expense per year * Number of years
Accumulated depreciation up to year 6 = 1,000,000 * 6 = 6,000,000 shs.
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28) The number of hours per week that high school students spend on computers is normally distributed, with a mean
of 5 hours and a standard deviation of 2 hours. 70 students are chosen at random. Let y represent the mean number
of hours spent on the computer for this group. Find the probability that y is between 5.1 and 5.7.
Answers
The probability that y is between 5.1 and 5.7 is approximately 0.4292.
What is probability?Probability is a way to gauge how likely or unlikely something is to happen. It is a number between 0 and 1, where 0 denotes an improbable event and 1 denotes an inevitable one.
According to question:Since the sample size is large (n = 70), we can use the Central Limit Theorem to approximate the sampling distribution of the sample mean as normal with mean μ = 5 and standard deviation σ/sqrt(n) = 2/sqrt(70) ≈ 0.24.
Now we want to find the probability that y, the sample mean, is between 5.1 and 5.7. We can use the standard normal distribution to do this by standardizing y:
z = (y - μ) / (σ / sqrt(n)) = (y - 5) / 0.24
The probability that y is between 5.1 and 5.7 is equivalent to the probability that z is between:
z1 = (5.1 - 5) / 0.24 ≈ 0.42
and
z2 = (5.7 - 5) / 0.24 ≈ 2.92
Using a standard normal distribution table or calculator, we can find that the probability of z being between z1 and z2 is approximately 0.4292.
Therefore, the probability that y is between 5.1 and 5.7 is approximately 0.4292.
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Help please i dont know
Answers
The histogram should be modified as follows:
The bin from 6 to 15 should be increased by 2.The bin from 16 to 25 should be increased by one.The bin from 26 to 35 should be increased by one.The bin from 46 to 55 should be increased by one.What is shown by an histogram?A histogram is a type of graphical representation that displays the distribution of a dataset. It is a bar graph-like chart where the data is divided into intervals, called "bins", which are represented by adjacent rectangular bars of varying heights.
Then the height of the histogram gives the number of observations into each data interval.
Hence the histogram should be modified as follows:
The bin from 6 to 15 should be increased by 2. -> two measures of 12.The bin from 16 to 25 should be increased by one. -> measure of 16.The bin from 26 to 35 should be increased by one. -> measure of 26.The bin from 46 to 55 should be increased by one. -> measure of 48.More can be learned about histograms at https://brainly.com/question/25983327
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A bag contains 2 gold marbles, 10 silver marbles, and 29 black marbles. The rules of the game are as follows: You randomly select one marble from the bag. If it is gold, you win $6, if it is silver, you win $4. If it costs $1 to play, what is your expected profit or loss if you play this game?
Answers
Answer:
33 thats with the 1 dollar subtracted for the entry fee
Step-by-step explanation:
first you add all the number of marbles you have then you get 41 then divide it by 12 because you have 2 gold and 10 silver and you get 3.4 and multiply that by 10 because its 6 and 4 and you get 34 subtract the entry fee
A factory that has 6 X 10³ workers makes 2.4 X 105 products each day.
What is the average amount of
products made per worker?
Answers
In a a case whereby factory that has 6 X 10³ workers makes 2.4 X 105 products each day, the average amount of products made per worker is 4 *10^1 products per worker
How can the average amount of products made per worker can be calculated?The term "average" in mathematics typically refers to a measure of central tendency, which is a way to describe the typical or common value in a set of numbers. There are several types of averages.
We can calculate the required value as
( 2.4 X 10^5 )/ (6 X 10³)
=4 *10^1 products per worker
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HELP ME PLEASE!!! HELPPP!!! A backyard pool is in the shape of a rectangle. The pool has a perimeter of 90 feet and an area of 500 ft2. Which of the following could be the pool's length and width? 15 feet and 30 feet, 10 feet and 35 feet ,50 feet and 10 feet, 25 feet and 20 feet
Answers
Answer:
(d) 25 feet and 20 feet
Step-by-step explanation:
You want the possible dimensions of a pool with an area of 500 square feet and a perimeter of 90 feet.
AreaThe area is the product of the length and width. The area of the pools offered in the answer choices are ...
(a) 15·30 = 450 . . . square feet
(b) 10·35 = 350 . . . square feet
(c) 50·10 = 500 . . . square feet
(d) 25·20 = 500 . . . square feet
The area requirement eliminates answer choices A and B.
PerimeterThe perimeter is twice the sum of length and width. The perimeters of the possible pools are ...
(c) 2(50 +10) = 120 . . . feet
(d) 2(25 +20) = 90 . . . feet
The perimeter requirement eliminates answer choice C.
The pool's possible length and width are 25 feet and 20 feet, choice D.
__
Additional comment
You could write a quadratic equation for the pool dimensions, but doing that will generally involve more work than checking the given answer choices.
If x is the width, then 45-x is the length, and the area is ...
x(45 -x) = 500
x² -45x +506.25 = -500 +506.25 . . . multiply by -1, complete the square
x = 22.5 -√6.25 = 20 . . . . take the square root; width is the smaller dimension
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(23,000-789) * 19 Explain pls
Answers
The expression (23,000-789) * 19 = = 421409.
What is expression?An expression can contain variables, constants, and mathematical operations, and it can be used to represent a wide variety of mathematical concepts and relationships.
According to given information:
To solve the expression (23,000-789) * 19, we need to perform the subtraction operation inside the parentheses first and then multiply the result by 19.
(23,000-789) equals 22,211.
So, we have:
(23,000-789) * 19 = 22,211 * 19
Now, we can multiply the two numbers using the standard multiplication method. We multiply each digit of 19 with the digits of 22,211 starting from the right and then add the results.
22,211 x 19 =
444422 (19 x 1 = 19, write 9 and carry 1) + 420789 (19 x 1 = 19, write 9 and carry 1, 19 x 2 = 38, write 8 and carry 3)
= 421409
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Solve for x.
A. 7
B. 4
C. 3
D. 5
Answers
The correct option -D. 5; Thus, the value of x for the given external secant segment and the tangent on the circle is found as:x = 5.
Explain about the secant of circle:A line that precisely intersects a circle at two points is said to be a secant.
The size of the angle created when two tangents, two secants, or two tangents cross outside of a circle is equal to one-half a positive difference between the sizes of the intercepted arcs.
Using the Theorem:
The square of a length of tangent is equal the the product of such external secant segment and the overall length of the secant if one secant and one tangent both drawn to a circle from a single exterior point:
4(4 + x) = 6²
16 + 4x = 36
4x = 36 - 16
4x = 20
x = 20/4
x = 5
Thus, the value of x for the given external secant segment and the tangent on the circle is found as:x = 5.
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carpet is sold in square yards how much carpet would a person need to buy for a rectangle that has dimensions of 10 yards by 18 feet
Answers
A person would need to buy 60 square yards of carpet for a rectangular room with dimensions of 10 yards by 18 feet.
How to determine the amount of carpet needed for a rectangular room?
To determine the amount of carpet needed for a rectangular room, you need to convert the measurements to the same unit of measurement. Since the carpet is sold in square yards, we need to convert the dimensions of the room to yards.
The length of the room is given in yards, so we don't need to make any conversions for that dimension. However, the width of the room is given in feet, so we need to convert it to yards by dividing by 3 (since there are 3 feet in a yard),
18 feet ÷ 3 feet/yard = 6 yards
So the dimensions of the room in yards are 10 yards by 6 yards.
To find the total area of the room, we need to multiply the length and width,
10 yards × 6 yards = 60 square yards
Therefore, a person would need to buy 60 square yards of carpet for a rectangular room with dimensions of 10 yards by 18 feet.
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Chloe claims that Point A=−6 and Point B=1 . Which of the following statements provide support for Chloe's claim? Select ALL that apply. A A<0 because A is to the left of zero B A>0 because A is to the left of zero C A 0 because B is to the left of zero F B>0 because B is to the right of zero
Answers
The statements that provide support for Chloe's claim are A<0 because A is to the left of zero, B>0 because B is to the right of zero. So correct option is A and F.
Describe Comparison Algebra?Comparison algebra is a branch of algebra that deals with inequalities and comparisons between different quantities or expressions. In comparison algebra, the goal is to determine the relationships between different expressions or quantities, such as whether one expression is greater than, less than, or equal to another expression.
Comparison algebra involves the use of comparison symbols, such as "<" (less than), ">" (greater than), and "=" (equal to), to express these relationships. For example, if we have two expressions, A and B, we can use the "<" symbol to express the relationship that A is less than B, as in A < B.
In comparison algebra, we can also manipulate inequalities and equations in similar ways as we do with regular algebraic expressions. For instance, we can add, subtract, multiply, or divide both sides of an inequality or equation by the same number or expression, while maintaining the inequality's direction.
The statements that provide support for Chloe's claim are:
A. A<0 because A is to the left of zero.
F. B>0 because B is to the right of zero.
Statement A supports Chloe's claim because Point A is to the left of zero on the number line, which means its value is negative. Statement F also supports Chloe's claim because Point B is to the right of zero on the number line, which means its value is positive.
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using truth table show that (p^q) =~pv~q please help meeee
Answers
The truth table for the given expression can be given using p= T q = T, p^q = T, ~p = F, ~q = F, ~q v ~p = T, and (p ^ q) =~p v ~q = T, thus proving the required.
What is propositional logic?The field of symbolic logic known as propositional logic, also referred to as sentential logic or statement logic, is concerned with propositions or statements, which are declarative phrases that can be true or untrue. To represent propositions and their relationships, propositional logic employs symbols and operators.
Many disciplines, including mathematics, computer science, philosophy, and others, employ propositional logic as a tool for debating the truth or falsity of propositions and analysing arguments.
The truth table for the given expression can be given as:
p | q | p ^ q | ~p | ~q | ~q v ~p | (p ^ q) =~p v ~q
-------------------------------------------------------------
T | T | T | F | F | T | T
T | F | F | F | T | T | T
F | T | F | T | F | T | T
F | F | F | T | T | T | T
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PLEASE HURRY DUE TODAY WILL MARK BRAINLESTIS RIGHT
what is √29 Place a dot on the number line at the BEST approximation
Answers
Answer:
5.4
Step-by-step explanation:
square of 29 = 5.385164807
5.385164807 estimated is 5.4
Write 80cm to 2km as rate
Answers
One per 2,500 can be used to represent the rate of 80cm to 2km.
Writing measures as a rate.To write 80cm to 2km as a rate, we need to convert the units to the same system. We can convert 80cm to kilometers by dividing by 100,000 (since there are 100,000 centimeters in a kilometer):
80 cm/100,000 = 0.0008 km
Now we can express the rate as:
0.0008 km per 2 km
Or we can simplify it by dividing both the numerator and denominator by 0.0008:
1 per 2,500
Therefore, 80cm to 2km can be expressed as a rate of 1 per 2,500.
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An ice cream stand uses the expression 1 + 0.75x to determine the cost (in dollars) of an ice cream cone that has x scoops of ice cream. How much does an ice cream cone with 3 scoops cost?
Answers
Answer:
$3.25
Step-by-step explanation:
An ice cream cone that has x scoops of ice cream costs 1 + 0.75x.
So an ice cream cone that has 3 scoops of ice cream costs
1 + 0.75×3 = 3.25 (dollars)
Help me on my math question pls!
Answers
Answer:
D
Step-by-step explanation:
A rotation and translation preserves the length measures of the sides, so FD is equal to 7 inches
Answer: D
Step-by-step explanation:
complete the equatiotin 3 over 4 x 6 =
PLEASE HELP
Answers
Answer:
1/8
Step-by-step explanation:
If "over" refers to (4 * 6) as a whole,
[tex]\frac{3}{4*6}\\\\=\frac{3}{24}\\\\=\frac{1}{8}[/tex]
If "over" refers to division,
[tex]3 \div 4 \times 6\\= 0.75 \times 6\\= 4.5[/tex]
I would suppose it's the first one, so, ignore the second one if you haven't learnt BODMAS yet.
Hope this helps and be sure to mark this as brainliest! :)
In order to develop a more appealing cheeseburger, a franchise uses taste tests with 11 5 different buns, 3 different cheeses, 4 types of lettuce, and types of tomatoes. If the taste tests were done at one restaurant by one tester who takes 10 minutes to eat each cheeseburger, approximately how long would it take the tester to eat all possible cheeseburgers?
Answers
It would take approximately 1,980 minutes (33 hours) for the tester to eat all possible cheeseburgers.
) Rewrite as an exponential equation.
log8 1/64 =2
(b) Rewrite as a logarithmic equation.
3 0=1
Answers
a) the exponential equation equivalent to log8 1/64 = 2 is 1/64 = [tex]8^{(-2)}[/tex]
b) the logarithmic equation equivalent to 3^0 = 1 is log3 1 = 0.
What is exponential equation?
An exponential equation is one in which the exponent contains a variable.
(a) We can rewrite the logarithmic equation as an exponential equation by using the definition of logarithms. The logarithmic equation
log8 1/64 = 2
means that 8 raised to the power of 2 is equal to 1/64:
[tex]8^2 = 1/64[/tex]
Thus, we can write the exponential equation as:
[tex]1/64 = 8^{(-2)}[/tex]
Therefore, the exponential equation equivalent to log8 1/64 = 2 is 1/64 = [tex]8^{(-2)}.[/tex]
(b) We can rewrite the exponential equation as a logarithmic equation by using the definition of logarithms. The exponential equation
[tex]3^0 = 1[/tex]
means that the logarithm of 1 to the base 3 is equal to 0:
log3 1 = 0
Therefore, the logarithmic equation equivalent to 3^0 = 1 is log3 1 = 0.
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Equipment accusers on January 8 at cost of $212,000 has an estimated useful life of 15 years, has an estimated residual value of $14,000, and is depreciated by the straight line method
Answers
Answer: To calculate the annual depreciation expense of the equipment using the straight-line method, we need to first determine the depreciable cost of the asset, which is the cost of the asset minus its estimated residual value.
Depreciable Cost = Cost of asset - Estimated Residual Value
Depreciable Cost = $212,000 - $14,000
Depreciable Cost = $198,000
Next, we can use the following formula to calculate the annual depreciation expense:
Annual Depreciation Expense = Depreciable Cost / Useful Life
Annual Depreciation Expense = $198,000 / 15 years
Annual Depreciation Expense = $13,200 per year
Therefore, the annual depreciation expense of the equipment using the straight-line method is $13,200 per year.
Step-by-step explanation:
What is the area of the following right triangle?
Answers
Let [tex]x[/tex] be the unknown side of the triangle. Find the length of this side from the Pythagorean Theorem;
[tex]x^2+5^2=13^2[/tex][tex]x^2=13^2-5^2[/tex][tex]x^2=144[/tex][tex]x=12[/tex]Another root of the equation is also negative, but we've eliminated it because length can't be a negative value.
To find the area of a triangle, we multiply the height of the triangle by the base length and take half.
[tex]Area=(12.5)/2=30[/tex]As seen in the diagram below, Austin is building a walkway with a width of x feet to go around a swimming pool that measures 8 feet by 10 feet. If the total area of the pool and the walkway will be 360 square feet, how wide should the walkway be?
Answers
Using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.
What is a rectangle?In the Euclidean plane, a rectangle is a quadrilateral with four right angles.
A parallelogram with a right angle or an equiangular quadrilateral, where equiangular means that all of its angles are equal, are other ways to define it.
A rectangle with four equal-length sides is a square.
Squares are not always rectangles, but rectangles are always squares.
So, the pool and walkway together have a 360 square foot total space.
Assume that the walkway is w feet in width.
Since the pool has an additional w feet of width on both sides, the total area's measurements can be expressed as (8 + 2w) by (10 + 2w).
Thus, we can construct the following equation:
(8 + 2w) x (10 + 2w) = 360
By enlarging and condensing the left side, we obtain:
4w² + 36w - 80 = 0
Using the quadratic formula, we can find w:
w = (-b ± √(b² - 4ac)) / 2a
Here, an equals 4, b equals 36, and c equals -80. When these values are added to the formula, we obtain:
w = (-36 ± √(36² - 4(4)(-80))) / 8
w = (-36 ± √(1872)) / 8
w ≈ -5.6 or w ≈ 3.6
We can ignore the first option because the width cannot be negative. Consequently, the pathway should be around 3.6 feet wide.
Therefore, using the total area of the rectangular pool we know that the required width of the pathway is 3.6 ft respectively.
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50pts!!!!
Many investors use interest-only loans to buy shares or property. For such loans the principal stays constant and only the interest is paid back each month.
She buys an investment property for $300 000 and borrows the full amount at 7% p.a. simple interest. She rents out the property at $1500 per month and pays $3000 per year in rates and other costs to keep the property.
Find the amount of interest she needs to pay back every month.
Find her yearly income from rent.
By considering the other costs in keeping the property, calculate her overall loss in a year.
she hopes that the property’s value will increase enough to cover any loss she is making. By what percentage of the original price will the property need to increase in value per year?
Answers
Step-by-step explanation:
to find the interest she needs to pay back every month, we need to use this formula:
I = PRT/12
in this case the principle is $300,000, the rate is 7% p.a. and the amount of time is 1 year, if we substitute in our values we:
I = (300,000)(7)(1)/12 = $17,500 in interest every month
to find her yearly income from rent, we have to multiply the monthly rent by 12
1500 × 12 = $18,000
to calculate her loss percentage in a year, we have to subtract 18,000 from 3000 which is 15,000
she said that she hopes the property's value will increase to cover the loss she made.
to cover the loss of $15000 per year, the property needs to increase in value by at least $15000. the percentage increases value can be written as
Percentage increase = (difference in increase/Original price) × 100
so
percentage increase = (15000/300000) × 100 = 5%
so the property needs to increase in value by at least 5% per year to cover the loss.
The woman needs to pay $1750 monthly as interest. Her yearly income from the rent is $18,000. After considering all other costs, she is at a loss of $6000 per year, so the property needs to increase in value by 2% per year to cover this loss.
Explanation:First, let's calculate the monthly interest she needs to pay. 7% of $300,000 is $21,000 for a year. To find the monthly interest, we divide this by 12, resulting in $1750.Her yearly income from rent is $1500 multiplied by 12, giving us $18,000.To calculate her overall loss, we add the yearly interest and the costs to keep the property, then subtract the yearly rent income. So, $21,000 + $3000 - $18,000 = $6000 loss per year.Lastly, to find out by what percentage the property value needs to increase, we divide the loss by the original price and multiply by 100, giving us 2% increase per year.Learn more about Interest and Profit Calculation here:https://brainly.com/question/32651816
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please answer and explain
Answers
Answer:
Trapezoid
Step-by-step explanation:
It's a quadrilateral and it only has one pair of parallel sides.
The owner of a stereo store wants to advertise that she has many different sound systems in stock. The store carries 5 different CD players, 8 different receivers, and 10 different speakers. A sound system consists of one of each item. How many different sound systems can the store owner advertise?
Answers
The owner can advertise 400 different sound systems. The solution has been obtained by using the arithmetic operations.
What are arithmetic operations?
All real numbers can be explained using the four fundamental operations, generally known as "arithmetic operations." In mathematics, operations like quotient, product, sum, and difference occur after operations like division, multiplication, addition, and subtraction.
We are given that the owner of a stereo store carries 5 different CD players, 8 different receivers, and 10 different speakers and each sound system consists of one of each item.
So, the different sound systems that the owner can advertise is obtained by using the multiplication operation, which gives:
⇒ Different sound systems = 5 * 8 * 10
⇒ Different sound systems = 400
Hence, the owner can advertise 400 different sound systems.
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