Given the following geometric sequence, find the common ratio: {225, 45, 9, ...}.
Answer: The required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
Step-by-step explanation: We are given to find the common ratio for the following geometric sequence :
225, 45, 9, . . .
We know that
in a geometric sequence, the ratio of any term with the preceding term is the common ratio of the sequence.
For the given geometric sequence, we have
a(1) = 225, a(2) = 45, a(3) = 9, etc.
So, the common ratio (r) is given by
[tex]r=\dfrac{a(2)}{a(1)}=\dfrac{a(3)}{a(2)}=~~.~~.~~.~~.[/tex]
We have
[tex]\dfrac{a(2)}{a(1)}=\dfrac{45}{225}=\dfrac{1}{5},\\\\\\\dfrac{a(3)}{a(2)}=\dfrac{9}{45}=\dfrac{1}{5},~etc.[/tex]
Therefore, we get
[tex]r=\dfrac{1}{5}.[/tex]
Thus, the required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
Prove or disprove each of these statements about the floor and ceiling functions.
Find the point on the parabola y^2 = 4x that is closest to the point (2, 8).
Answer:
(4, 4)
Step-by-step explanation:
There are a couple of ways to go at this:
Write an expression for the distance from a point on the parabola to the given point, then differentiate that and set the derivative to zero.Find the equation of a normal line to the parabola that goes through the given point.1. The distance formula tells us for some point (x, y) on the parabola, the distance d satisfies ...
... d² = (x -2)² +(y -8)² . . . . . . . the y in this equation is a function of x
Differentiating with respect to x and setting dd/dx=0, we have ...
... 2d(dd/dx) = 0 = 2(x -2) +2(y -8)(dy/dx)
We can factor 2 from this to get
... 0 = x -2 +(y -8)(dy/dx)
Differentiating the parabola's equation, we find ...
... 2y(dy/dx) = 4
... dy/dx = 2/y
Substituting for x (=y²/4) and dy/dx into our derivative equation above, we get
... 0 = y²/4 -2 +(y -8)(2/y) = y²/4 -16/y
... 64 = y³ . . . . . . multiply by 4y, add 64
... 4 = y . . . . . . . . cube root
... y²/4 = 16/4 = x = 4
_____
2. The derivative above tells us the slope at point (x, y) on the parabola is ...
... dy/dx = 2/y
Then the slope of the normal line at that point is ...
... -1/(dy/dx) = -y/2
The normal line through the point (2, 8) will have equation (in point-slope form) ...
... y - 8 = (-y/2)(x -2)
Substituting for x using the equation of the parabola, we get
... y - 8 = (-y/2)(y²/4 -2)
Multiplying by 8 gives ...
... 8y -64 = -y³ +8y
... y³ = 64 . . . . subtract 8y, multiply by -1
... y = 4 . . . . . . cube root
... x = y²/4 = 4
The point on the parabola that is closest to the point (2, 8) is (4, 4).
How much greater was Miami's annual rainfall than Albany's?
The annual rainfall in Albany is 0.33 inch less than the annual rainfall in Nashville. How much less rainfall did Nashville get than Miami? Show your work.
Miami rainfall 61.05 inches
Albany rainfall 46.92 inches
Miami's annual rainfall was 14.13 inches greater than Albany's. It is not possible to determine from the given information how much less rainfall Nashville had than Miami.
Explanation:To find the difference in annual rainfall between Miami and Albany, we need to subtract the rainfall of Albany from Miami.
Miami rainfall = 61.05 inches
Albany rainfall = 46.92 inches
So, Miami's annual rainfall is greater by:
61.05 inches - 46.92 inches = 14.13 inches
Now regarding the second question about Nashville and Miami, we don't have the absolute rainfall measurement for Nashville, thus we can't answer specifically how much less rainfall Nashville had than Miami.
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use the fundamental theorem of algebra to determine the number of roots for 2x^2+4x+7
Suppose you obtain a $1,300 T-note with a 9% annual rate, paid monthly, with maturity in 6 years. How much interest will be paid to you each month?
Answer:
simple interest = $9.75
Step-by-step explanation:
given data:
Principle = $1300
annual rate = 9% [tex]= \frac{9}{`12} = 0.75 [/tex]
time = 6 year =
we knwo that simple interest is given as
Simple interest [tex]= \frac{P\times R\times T}{100}[/tex]
FOR ABOVE QUESTION
Time is 1 month
simple interest [tex]= \frac{1300\times 0.75 \times 1}{100}[/tex]
simple interest = $9.75
A 3-foot piece of wire costs $0.76. What is the unit price, rounded to the nearest cent?
divide total cost by length:
0.76 / 3ft = 0.2533 cents per foot
rounded to nearest cent = 0.25 cents per foot
An experiment results in one of the sample points upper e 1e1, upper e 2e2, upper e 3e3, upper e 4e4, or upper e 5e5. complete parts a through
c.
a. find p(upper e 3e3) if p(upper e 1e1)equals=0.10.1, p(upper e 2e2)equals=0.10.1, p(upper e 4e4)equals=0.20.2, and p(upper e 5e5)equals=0.30.3.
the graph of g(x) is the graph of f(x)=8x+20 stretched horizontally by a factor of 4.
Which equation describes the function g?
1. g(x)= 8x+5
2. g(x)= 2x+5
3. g(x)= 32x+20
4. g(x)= 2x+20
Determine the common ratio and find the next three terms of the geometric sequence.
10, 2, 0.4, ...
a.
0.2; -0.4, -2, -10
c.
0.02; 0.08, 0.016, 0.0032
b.
0.02; -0.4, -2, -10
d.
0.2; 0.08, 0.016, 0.0032
Answer:
d. 0.2; 0.08, 0.016, 0.0032
Step-by-step explanation:
The common ratio is the ratio of adjacent terms:
r = 2/10 = 0.4/2 = 0.2
__
Multiplying the last term by this ratio gives the next term:
0.4×0.2 = 0.08
0.08×0.2 = 0.016
0.016×0.2 = 0.0032
The next 3 terms are 0.08, 0.016, 0.0032.
Answer:
Option D)
Common ration = [tex] \frac{1}{5}[/tex] = 0.2
The next three terms of the given series are: 0.08, 0.016, 0.0032
Step-by-step explanation:
We are given the following information in the question:
We are given a geometric sequence:
[tex]10, 2, 0.4, ...[/tex]
Geometric Series
A geometric series is a series with a constant ratio between successive termsWe have to find the common ration of the given geometric series:
[tex]\text{Common ration} = \displaystyle\frac{\text{Second term}}{\text{First term} }=\frac{2}{10} = \frac{1}{5}[/tex]
The [tex]n^{th}[/tex] term of a geometric sequence is given by:
Formula:
[tex]a_n = a_1\timesr^{n-1},\\\text{where }a_1 \text{ is the first term of the geometric series and r is the common ratio}[/tex]
[tex]a_4 = a_1\times r^{4-1} = 10\times \bigg(\displaystyle\frac{1}{5}\bigg)^3 = 0.08\\\\a_5 = a_1\times r^{5-1} = 10\times \bigg(\displaystyle\frac{1}{5}\bigg)^4 = 0.016\\\\a_6 = a_1\times r^{6-1} = 10\times \bigg(\displaystyle\frac{1}{5}\bigg)^5 = 0.0032[/tex]
how do I answer this?
An ounce is 1/16 of a pound. How many ounces are in 8 3/4 pounds?
There are 140 ounces are in 8 3/4 pounds.
Given that,
1 ounce = 1/16 pound
We can also write it as
1/16 pound = 1 ounce
Multiply 16 both sides we get,
1 pound = 16 ounces ......(i)
Therefore,
There are 16 ounces in 1 pound.
Now we can write [tex]8\frac{3}{4}[/tex] in simple fraction as,
= (8x4+3)/4
= 35/4
Now we can say that [tex]8\frac{3}{4}[/tex] pounds as 35/4 pound as both are same quantity
Now, multiply 35/4 both sides in equation (i) we get,
35/4 pound = (35/4)x16 ounces
= 140 ounces.
Hence,
[tex]8\frac{3}{4}[/tex] pounds = 140 ounces.
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How to convert pounds to ounces
Step-by-step explanation: To convert pounds into ounces, we need to start with a conversion factor for pounds and ounces which is 16 ounces = 1 pound.
Next, since we are going from a larger unit "pounds" to a smaller unit "ounces" we will multiply the pounds by our conversion factor.
Finally, our product will be our answer.
Which digit represents "hundreds" in the number 8732?
When "P" Dollars is invested at interest rate "i", compounded annually, for "t" tears, the investment grows to "A" dollars, where A=P(1+i)^t. When Sara enters 11th grade, her grandparents deposit $10,000 in a college savings account. Find the interest rate "i", if the $10,000 grows to $11,193.64 in two years.
Final answer:
The annual interest rate compounded annually that grows $10,000 to $11,193.64 in two years is approximately 5.882%.
Explanation:
To find the interest rate i if the $10,000 grows to $11,193.64 in two years, we can rearrange the compound interest formula to solve for i:
[tex]A = P(1 + i)^t[/tex]
Given:
A = $11,193.64
P = $10,000
t = 2
Rearranging the formula to solve for i, we get:
[tex](A / P) = (1 + i)^t[/tex] → (11,193.64 / 10,000) = (1 + i)²
Now we must solve for i:
1.119364 = (1 + i)² → i = [tex]\sqrt{(1.119364)}[/tex] - 1
Calculating the square root of 1.119364 and subtracting 1 gives us:
i ≈ 0.05882 or 5.882%
Therefore, the annual interest rate compounded is approximately 5.882%.
12^10·75^15/15^25·80^5
To solve this expression, apply the rules of exponents and convert the fractions to decimal values. Simplify the expression and use a calculator to find the decimal values of the powers. Divide the values and express the final result in scientific notation.
Explanation:To solve this expression, we can first look at the different components. 12^10 means 12 raised to the power of 10. 75^15 means 75 raised to the power of 15. 15^25 means 15 raised to the power of 25. And finally, 80^5 means 80 raised to the power of 5.
Now, we can substitute these values back into the original expression: (12^10 · 75^15)/(15^25 · 80^5).
By using the rules of exponents, we can simplify this expression. For example, when you multiply two powers with the same base, you add the exponents. When you divide two powers with the same base, you subtract the exponents. Applying these rules, we get:
12^10 · 75^15/15^25 · 80^5 = (12/15)^10 · (75/80)^15/15^25 · 80^5 = (4/5)^10 · (3/4)^15/15^25 · 80^5.
To further simplify, we can convert the fractions into decimal values: 4/5 is equal to 0.8 and 3/4 is equal to 0.75. Substituting these values, we get:
(0.8)^10 · (0.75)^15/15^25 · 80^5 = 0.8^10 · 0.75^15/15^25 · 80^5.
We can use a calculator to find the decimal values of 0.8^10 and 0.75^15. After calculating the values and substituting them back into the expression, we get:
0.4 × 10^2 · 1.99 × 10^4/3.12 × 10^4 · 2.32 × 10^6 = 0.4 × 1.99/3.12 × 2.32 × 10^2 × 10^4 × 10^6 = 0.796/7.244 × 10^2 × 10^4 × 10^6.
Simplifying further, we get:
0.796/7.244 × 10^(2+4+6) = 0.796/7.244 × 10^12.
Dividing 0.796 by 7.244, we get approximately 0.1099373. So, the simplified expression is approximately 0.1099373 × 10^12.
In 2018, a nation’s population was 10 million. Its nominal GDP was $40 billion, and its price index was 100. In 2019, its population had increased to 12 million, its nominal GDP had risen to $57.6 billion, and its price index had increased to 120. What was this nation’s economic growth rate during the year?
The nominal GDP in base year 2014 was $40 billion. The nominal GDP in year 2015 with price index 120 was $57.6 billion. The real GDP in 2015 can be calculated as follows :
GDP (real) = GDP (nominal) / price index * 100
GDP (real) = 57.6 / 120 * 100
GDP (real) = $48 billion
The growth rate in real GDP from 2014 to 2015 is 1.2%.
Growth rate = 48 * (100/40) = 1.2%
Therefore the growth rate is 12%
The nation's real GDP increased from $40 billion in 2018 to $48 billion in 2019, resulting in an economic growth rate of 20% for that year.
To calculate the economic growth rate of a nation, we need to look at the increase in its real GDP. Real GDP is calculated by dividing the nominal GDP by the GDP deflator and then multiplying by 100. The GDP deflator is like a price index that reflects the level of prices of all new, domestically produced, final goods and services in an economy.
For the nation in question:
2018 Real GDP = (Nominal GDP / Price Index)To find the economic growth rate, we subtract the previous year's real GDP from the current year's real GDP, divide by the previous year's real GDP, and then multiply by 100 to get a percentage:
Economic Growth Rate = [(2019 Real GDP - 2018 Real GDP) / 2018 Real GDP]
100
Economic Growth Rate = [($48 billion - $40 billion) / $40 billion]
100 = (8 / 40)
100 = 20%
The nation's economic growth rate during the year was 20%.
Your high school freshman class consists of 760 students. In recent years only 4 out 7 students actually graduated in four years. Approximately how many of your classmates are expected to graduate in four years
To find out how many of the 760 students in the freshman class are expected to graduate in four years, we use the rate of 4 out of 7 students graduating. This calculation yields approximately 434 students expected to graduate.
To calculate the approximate number of students expected to graduate in four years from a freshman class of 760 students, given a historical graduation rate of 4 out of 7, we use a simple proportion calculation. We set up a proportion where 4 out of 7 students graduate, which can be written as 4/7 = x/760, where x represents the number of graduating students.
Calculating this, we get x = (4/7) times 760. To solve for x, multiply both sides of the equation by 760 to isolate x.
x = 760 imes (4/7) = 760 imes 0.57142857
Approximately x = 434
Therefore, we can expect that approximately 434 students in the freshman class will graduate in four years, based on the provided rate.
what is the radius of a circle with an area of 32.1 square feet
completely factor the expression 16t^3 - 50t^2 + 36t
Ken spent 1/5 of his allowance on a movie, 3/8 on snacks, and 2/7 on games. If his allowance was $20, how much did Ken have left?
Answer:
Ken is left with $2.79.
Step-by-step explanation:
We are given the following information in the question:
Ken allowance = $20
Money spent on movies =
[tex]\displaystyle\frac{1}{5}\times 20 = \$4[/tex]
Money spent n snacks =
[tex]\displaystyle\frac{3}{8}\times 20 = \$7.5[/tex]
Money spent on games =
[tex]\displaystyle\frac{2}{7}\times 20 = \$5.71[/tex]
Total money spent =
[tex]4 + 7.5 + 5.71 = \$17.21[/tex]
Money left =
[tex]=\text{Allowance}-\text{ Total money spent}\\= 20 - 17.21\\=\$2.79[/tex]
Ken is left with $2.79.
Find an integer x such that 37x $\equiv$ 1 (mod 101).}
what is the sale tax on a 17500 truck if the tax rate is 6%
Answer: the sales tax is 1050
Kirin has 28 books. This is 7 times as many books as Gail has. Kirin made a model to compare the numbers of books they have. Which equation represents how to find the value of n?
What is the slope of this line?
a. −15
b. −5
c. 5
d. 15
Answer:
C. 5
Step-by-step explanation:
I did the test
A presidential candidate plans to begin her campaign by visiting the capitals in 3 of 48 states. What is the probability that she selects the route of three specific capitals?
The probability that the candidate selects the route of three specific capitals out of 48 states is [tex]\( \frac{1}{17296} \)[/tex].
To calculate the probability of the candidate selecting the route of three specific capitals out of 48 states, we need to consider the total number of possible routes and the number of routes that include the specific capitals.
Calculate the total number of possible routes.
Since the candidate plans to visit 3 out of 48 states, the total number of possible routes is the number of ways to choose 3 states out of 48, which can be calculated using combinations:
[tex]\[ \text{Total number of routes} = \binom{48}{3} \][/tex]
Calculate the number of routes including the specific capitals.
Since the candidate plans to visit the capitals of three specific states, there is only one way to choose each of those specific states. So, the number of routes including the specific capitals is 1.
Calculate the probability.
[tex]\[ \text{Probability} = \frac{\text{Number of routes including specific capitals}}{\text{Total number of possible routes}} \][/tex]
[tex]\[ = \frac{1}{\binom{48}{3}} \][/tex]
Now, let's compute this.
[tex]\[ \binom{48}{3} = \frac{48!}{3!(48-3)!} = \frac{48 \times 47 \times 46}{3 \times 2 \times 1} = 17296 \][/tex]
So, the probability is:
[tex]\[ \text{Probability} = \frac{1}{17296} \][/tex]
Therefore, the probability that the candidate selects the route of three specific capitals out of 48 states is [tex]\( \frac{1}{17296} \)[/tex].
DBE is obtained by enlarging ABC. If the area of ABC is 3 square units, what is the area of DBE?
A. 27 square units
B. 24 square units
C. 12 square units
D. 9 square units
Answer:
A. 27 square units
Step-by-step explanation:
PLATO 2022 Lnhs
2767545 to the nearest ten
An airliner carries 100 passengers and has doors with a height of 76 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8in.
a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.
(Round to four decimal places as needed.)
First let us calculate for the z score using the formula:
z = (x – u) / s
where u is the mean height of men = 69 in, x is the height of the door = 76, and s is the standard deviation = 2.8 in
z = (76 – 69) / 2.8
z = 2.5
From the standard probability table, at z = -2.5 the probability P is:
P = 0.9938 = 99.38%
If a(x) = 3x + 1 and b(x)=square root of x-4 , what is the domain of (b*a)(x)?
A.(-infinity ,+infinity)
B.(0 , +infinity)
C.(1 , +infinity)
D.(4 , +infinity)