To find the largest positive integer $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer, we need to find the largest perfect square that is less than or equal to $n + 23$. The largest such $n$ is $41$.
Explanation:To find the largest positive integer $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer, we can start by considering the numerator. The expression $(n + 1)^2$ is a perfect square, so it will always be divisible by $n + 23$ if $n + 23$ is also a perfect square. Therefore, we need to find the largest perfect square that is less than or equal to $n + 23$.
Let's consider some examples. If $n + 23 = 25$, then $n = 2$. If $n + 23 = 36$, then $n = 13$. If $n + 23 = 49$, then $n = 26$. If $n + 23 = 64$, then $n = 41$. Therefore, the largest value of $n$ such that $rac{(n + 1)^2}{n + 23}$ is an integer is $41$.
There are no positive integers $n$ that satisfy the given condition, and we cannot compute the largest such $n$.
To find the largest positive integer $n$ such that $\frac{(n + 1)^2}{n + 23}$ is an integer, we need to consider the factors of the numerator and denominator.
Let's expand the numerator, $(n + 1)^2$, using the binomial expansion formula:
$(n + 1)^2 = n^2 + 2n + 1$
Now, we divide this expression by $n + 23$ and express the result as an integer:
$\frac{n^2 + 2n + 1}{n + 23}$
We can use polynomial long division to divide $n^2 + 2n + 1$ by $n + 23$:
- (n - 21)
__________
n + 23 | n^2 + 2n + 1
- (n^2 + 23n)
_____________
-21n + 1
- (-21n - 483)
_____________
484
We obtain a remainder of 484. In order for the fraction to be an integer, the remainder must be 0. However, in this case, the remainder is not 0, which means that $\frac{(n + 1)^2}{n + 23}$ is not an integer for any positive integer $n$.
Therefore, there are no positive integers $n$ that satisfy the given condition, and we cannot compute the largest such $n$.
Complete question :-
There are exactly four positive integers $n$ such that \[\frac{(n + 1)^2}{n + 23}\] is an integer. Compute the largest such $n$.
Which statement is true?
All quadrilaterals are rectangles.
All parallelograms are rectangles.
All squares are rectangles.
All rectangles are squares.
determine the y-intercept and the equation for the horizontal asymptote of the function.
k(x)= (1/2)^x
Reminder: you might like to graph the function to help you.
a. (0.5, 1) there is no horizontal asymptote
b. (0,1); y= 0
c. (0, 0.5); y=0.5
d. (1,0); y= 1
What is the square root of 64
F(x)=4x+7
G(x)=3x^2 find (f+g)(x)
32, 40, 34, 31, 20, 36, 40, 31. How does the outlier affect the mean? If necessary, round to the nearest tenth.
help please,carlos buys computer parts for $4,000.out of those parts, he can assemble many computers and sell each at $600.how many assembled computers must he sell in order to profit$1,400?
Answer:
9.
Step-by-step explanation: To profit means to make more than originally spent. He Spent 4,000$ and needs to profit 1,400$. Bringing a total of 5,400$. 5,400 divided by 600$ equals 9 computers.
A student says that VW and VZ are opposite rays because they have the same endpoint. describe the error.
The rays VW and VZ don't necessarily have to go in opposite directions even though they have the same endpoint. Even though they are opposite arrays only if the angle is 180 degrees, they could still go in right angle directions or any angle of directions. Fi the points V, W, and Z and collinear and W and Z are on the side of point V, then they could also in fact be the same ray as well. Travelling is opposite directions yet starting at the same point is one of the characteristics of opposite rays. The first point in the name must be the endpoint and rays are always named with two points.
Please help solve -2|5y-1|=-10
Which is a correct first step for solving this equation? x+7=2x+5−4x
Answer:x+7=2x+5-4x
x+7=-2x+5
You have 63 coins in your piggy bank, all the quarters and nickles. The total amount is $12.55. Hop many quarters do you have?
A.4
B.47
C.16
D.59
All of the following are equivalent except _____.
x - (-2)
-2 + x
x - 2
x + (-2)
A.
Its correct i got 100% on a test
Find the area of a sector with a central angle of 170° and a diameter of 9.1 cm. Round to the nearest tenth.
A. 122.9 cm2
B. 30.7 cm2
C. 8.6 cm2
D. 3.4 cm2
To find the area of a sector given a central angle and diameter, use the formula A = ([tex]\frac{\theta}{360}[/tex]) x π x r². In this case, with a central angle of 170° and a diameter of 9.1 cm, the area of the sector is approximately 30.7 cm².
The area of a sector with a central angle and a diameter can be calculated using the formula for the area of a sector:
A = [tex]\frac{\theta}{360}[/tex] x π x r². First, convert the diameter to radius (r = d/2), then substitute the values into the formula to find the area. In this case:
Central angle (θ) = 170°
Diameter (d) = 9.1 cm
Radius (r) = 4.55 cm
Area (A) = ([tex]\frac{170}{360}[/tex]) x π x (4.55)² ≈ 30.7 cm²
Therefore, the area of the sector is approximately 30.7 cm².
Adam put $100 in a savings account. After 10 years, he had $1649 in the account. What rate of interest did he earn? Use the formula A = Pert, where A is the ending amount, P is the principal (initial amount), r is the interest rate, and t is time.
Final answer:
Adam earned approximately a 28.03% interest rate on his $100 over 10 years to end up with $1649, as calculated using the formula A = Pert and taking the natural logarithm to solve for the interest rate.
Explanation:
To calculate the interest rate earned by Adam, we can use the given formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the original sum of money), r is the annual interest rate (in decimal), and t is the time the money is invested or borrowed for, in years. According to the question, Adam had a starting principal of $100 (P), which grew to $1649 (A) over 10 years (t).
Let's solve for the interest rate (r): A = Pert $1649 = $100e10r 16.49 = e10r To find r, we need to take the natural logarithm (ln) of both sides:
[tex]ln(16.49) = ln(e10r) ln(16.49) = 10r\\\\ Now divide both sides by 10 to isolate r:\\\(\dfrac{ln(16.49)}{10} = r\) \(r = \dfrac{ln(16.49)}{10}\)Now you just compute the value using a calculator:r ≈ \(\dfrac{2.803}{10}\) r ≈ 0.2803 or 28.03%\\[/tex]
Therefore, the interest rate that Adam earned was approximately 28.03%.
A card is drawn at random from a well-shuffled deck of 52 cards. what is the probability of drawing a face card or a spade?
Answer:
11/26
Step-by-step explanation: took the test
In Spencer’s garden, the number of rose bushes is 7 less than 1.5 times the number of carnation bushes. If the number of carnation bushes is c, then the expression representing the number of rose bushes is . NextReset
(6u3 + 7u2 + 2) + (3u3 – 8u + 4)
In a right triangle, the length of one leg is 6 units. The length of the other leg is 8 units. What is the length of the hypotenuse (hint: use Pythagorean Theorem)?
The length of the hypotenuse of given right triangle is 9.2 units.
Given that, in a right triangle, the length of one leg is 6 units. The length of the other leg is 8 units.
What is the Pythagoras theorem?The Pythagoras theorem which is also referred to as the Pythagorean theorem explains the relationship between the three sides of a right-angled triangle. According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a triangle.
Let the length of the hypotenuse be x.
Now, using Pythagoras theorem, we get
x²=6²+7²
⇒x²=36+49
⇒x²=85
⇒x=√85
⇒x=9.2 units
Therefore, the length of the hypotenuse of given right triangle is 9.2 units.
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Which function has the most x-intercepts?
The function with the most x-intercepts in the image is g(x), which has four x-intercepts. The function f(x) has three x-intercepts, and the function h(x) has two x-intercepts.
The number of x-intercepts a function has is equal to the number of times the function crosses the x-axis. The x-axis is also known as the line y = 0, so any point where the function's graph touches the x-axis is a solution to the equation f(x) = 0.
g(x) crosses the x-axis four times, at approximately -10, 3/2, 2, and 15.
f(x) crosses the x-axis three times, at approximately -2, 1, and 12.
h(x) crosses the x-axis twice, at approximately -1 and 3.
Therefore, g(x) has the most x-intercepts because its graph intersects the x-axis the most times.
Which one of the numbers in the series is wrong, and should be replaced? 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46?
The number series 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46 has a pattern whereby each number doubles the previous number. However, 20 does not follow this pattern, and the correct number should be 32 (being double of 16). Therefore, 20 is the incorrect number in this series.
Explanation:This question is a query regarding a number series, specifically asking for us to identify any number that does not fit the pattern of the series. The series given is: 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46. By examining the series closely, we can observe a clear pattern: each number is doubling the previous number. So, after 2 we have 4 (2x2), then 8 (4x2), then 16 (8x2) and so forth.
However, at the fifth place, we have the number 20, which doesn't follow this pattern. If we were to follow the established pattern, the fifth number should be 32 (16x2), not 20. Therefore, the incorrect number in this series is 20, and the correct series should be 2 - 4 - 8 - 16 - 32 - 22 - 44 - 46.
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The incorrect number in the series is 20, and it should be replaced with 32. The accurate sequence is 2, 4, 8, 16, 32, 64, 128, 256.
Let's examine the sequence: 2 - 4 - 8 - 16 - 20 - 22 - 44 - 46.
At a glance, it appears each number is double its preceding number, except for a couple of deviations.
If we look at the pattern,
2 x 2 = 44 x 2 = 88 x 2 = 1616 x 2 = 32Here, we expect 32 instead of 20.
Therefore, 20 is the incorrect number in the series. When we continue correctly:
The correct sequence would be,
2 - 4 - 8 - 16 - 32 - 64 - 128 - 256.
So, the number that should be replaced is 20, and it should be 32.
An extension ladder leans agintst a biulding, making a 75 degree angle of elevation with the ground. The base of the ladder is 8 ft from the base of the building.
To the nearest tenth of a foot, how long is the ladder?
The ladder is 29.9 feet long.
For solving this problem we have to use trigonometric ratio
We need to use the tangent function:
What is the ratio of tangent function?
[tex]tan (\theta) = opposite/adjacent[/tex]
[tex]tan 75 = opposite/8[/tex]
multiply both side by 8 so we will get,
Here, the opposite side is the length of the ladder and angle [tex]\theta =75^0[/tex]
Therefore by using the tan ratio we have,
[tex]8 * tan 75 = opposite[/tex]
[tex]29.8564065 = opposite[/tex]
Therefore, the ladder is 29.9 feet.
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Find the z score so that 25% of the standard normal curve lies to the left of z. remember that this means you are looking for the area (which means you need to find the 4 digit number in the area portion of table 3 that is closet to 25% ).
To locate the z-score where 25% of the standard normal distribution is to the left, one checks a z-table for the area closest to 0.25, which approximately corresponds to a z-score of -0.675.
Explanation:To find the z-score so that 25% of the standard normal curve lies to the left, we need to look up the corresponding area in a z-table. Since most z-tables represent the cumulative area to the left of a z-score, we must find the value closest to 0.25, the decimal equivalent of 25%. Upon checking a standard z-table, we find that the area closest to 0.25 corresponds to a z-score of approximately -0.675. Therefore, the z-score that has 25% of the area under the normal curve to its left is -0.675.
one number is 10 times another number. their difference is 72. what are the two numbers?
The two numbers are 80 and 8.
What is linear equation in two variables?An equation is said to be linear equation in two variables if it is written in the form of ax + by + c = 0, where a, b & c are real numbers and the coefficients of x and y, i.e. a and b respectively, are not equal to zero.
What is substitution method ?The substitution method can be defined as a way to solve a linear system algebraically. This method works by substituting one y-value with the other.
Let the two numbers be x and y.
According to the question.
x = 10y
And,
x - 10y = 72
Now, We have a linear equation in two variables. So, for finding the values of x and y we solve the above two equations by substitution method.
Substitute the value of x = 10y in x - y = 72
[tex]\implies 10y -y = 72 \\\implies 9y = 72\\\implies y = \frac{72}{9} \\\implies y = 8[/tex]
Therefore,
x = 10 × 8 = 80
Hence, the two numbers are 80 and 8.
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You can solve equations by graphing. Explain how to find the solutions by a given graph.
What is the quotient of 75,120 ÷ 16?
The quotient of 75,120 divided by 16 is 4,695.
To find the quotient of 75,120 divided by 16, you can follow these steps:
Perform the division: 75,120 ÷ 16 = 4,695.This means that 75,120 divided by 16 equals 4,695.So, the quotient of 75,120 divided by 16 is 4,695.
In how many ways can a teacher arrange 5 students in the front row of a classroom with a total of 40 students? Answers:
98,960,960
88,960,960
78,960,960
68,960,960
The standard formula for the volume of a cylinder is V = πr2h. If the cylinder is scaled proportionally by a factor of k, its volume becomes V' = V × k3. Use your algebra skills to derive the steps that lead from V = πr2h to V' = V × k3 for a scaled cylinder. Show your work.
Answer:
The answer in the procedure
Step-by-step explanation:
we know that
If two figures are similar, then the ratio of its corresponding sides is equal to the scale factor
so
[tex]k=r'/r[/tex] and [tex]k=h'/h[/tex]
The volume of the original cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
If the cylinder is scaled proportionally by a factor of k
then
the new radius is ------> [tex]r'=kr[/tex]
the new height is ------> [tex]h'=kh[/tex]
The volume of the scaled cylinder is equal to
[tex]V'=\pi r'^{2}h'[/tex]
substitute the values
[tex]V'=\pi (kr)^{2}(kh)[/tex]
[tex]V'=(k^{3})\pi r^{2}h[/tex]
Remember that
[tex]V=\pi r^{2}h[/tex]
so
substitute
[tex]V'=V(k^{3})[/tex]
The volume of the scaled cylinder is equal to the scale factor elevated to the cube multiplied by the volume of the original cylinder
A woman has 26 coins in her pocket, all of which are dimes and quarters. if the total value of the coins is $ 3.95, how many dimes and how many quarters does she have?
Solve the system using elimination. x + 2y = –6 3x + 8y = –20
Which expression is equivalent to complex fraction 3/x-1-4/2-2/x-1
what key features can be identified from graphs of polynomials with higher degrees and explain how the key features can be used to sketch the graph of the polynomial function.
The key features of graphs of polynomials with higher degrees include the leading coefficient, end behavior, and turning points. These features can be used to sketch the graph of the polynomial function.
Explanation:The key features that can be identified from graphs of polynomials with higher degrees include the leading coefficient, the end behavior, and the number and behavior of the turning points. The leading coefficient determines the orientation of the graph, whether it opens upwards or downwards. The end behavior of the graph shows how the function behaves as x approaches positive or negative infinity. The number and behavior of the turning points indicate the shape and direction of the graph.
These key features can be used to sketch the graph of the polynomial function by following these steps:
Identify the degree of the polynomial and determine the leading coefficient.Use the end behavior to determine the general direction in which the graph will go as x approaches positive or negative infinity.Find the x-intercepts by solving the equation f(x) = 0.Determine the behavior of the turning points by analyzing the sign of the second derivative.Plot the key points on the graph, including the x-intercepts and any turning points.Use the information from steps 2 and 4 to sketch the graph, connecting the key points with a smooth curve.Learn more about Polynomial Graphs here:https://brainly.com/question/4159808
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