The equation for the translated function is the one in option D:
G(x) = (x - 6)² - 4
What is the equation of the new function?We start with the parent quadratic function:
F(x) = x²
We apply two translations, one of 6 units to the right which can be written as:
G(x) = F(x - 6)
And a translation of 4 units down, which is written as:
G(x) = F(x - 6) - 4
Replacing F(x) we will get:
G(x) = (x - 6)² - 4
The correct option is D.
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Identify an equation in standard form for a hyperbola with center (0, 0), vertex (6, 0), and focus (10, 0).
Answer:
Step-by-step explanation:
Given the table of values below from a quadratic function, write an equation of that function.
X | 0 | 1 | 2 | 3 | 4 |
———————————
Y | 4 | 10 | 12 | 10 | 4 |
this is a answer, thanks
Answer:
Step-by-step explanation:
Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 20 , upper bound is 30.
The point estimate of the population mean is 25 and the margin of error for the confidence interval is 2.04.
To determine the point estimate of the population mean, we take the midpoint of the interval.
Point estimate of population mean = (Lower bound + Upper bound) / 2 = (20 + 30) / 2 = 25
To calculate the margin of error, we need to know the level of confidence for the interval. Assuming a 95% confidence level, we can use the formula:
The margin of error = (Upper bound - Lower bound) / (2 * z-score)
Where the z-score for a 95% confidence level is 1.96.
Margin of error = (30 - 20) / (2 x 1.96) = 2.04
Hence, the population means point estimate is 25, and the confidence interval's margin of error is 2.04 percent.
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a) A mechanical engineer conducted experiments to investigate the effect of four different types of boxes on compression strength (lb). The sample means from five experiments for each type of box were 650, 750, 700, 650 (unit: lb). Compute the SSTr.
b) In a single-factor ANOVA problem involving 5 populations, the total number of observations is 20, SSTr = 12 and SST = 20. What is the MSTr, MSE, and test statistic f.
a) To calculate SSTr, we need to first find the overall mean of the samples and then use it to calculate the sum of squares due to treatments (SSTr).
The overall mean of the samples can be found by adding up all the sample means and dividing by the number of samples:
Overall mean = (650 + 750 + 700 + 650) / 4 = 687.5
Next, we can calculate SSTr using the formula:
SSTr = n * (sample mean - overall mean)^2
where n is the number of observations in each sample. In this case, n = 5 for each sample.
So for the first sample, SSTr = 5 * (650 - 687.5)^2 = 5362.5
For the second sample, SSTr = 5 * (750 - 687.5)^2 = 14062.5
For the third sample, SSTr = 5 * (700 - 687.5)^2 = 3062.5
For the fourth sample, SSTr = 5 * (650 - 687.5)^2 = 5362.5
Therefore, the total SSTr is:
SSTr = 5362.5 + 14062.5 + 3062.5 + 5362.5 = 27750
b) The degrees of freedom (df) for SSTr is k-1 where k is the number of groups/populations, and df for SSE is N-k where N is the total number of observations.
df(SSTr) = k - 1 = 5 - 1 = 4
df(SSE) = N - k = 20 - 5 = 15
The mean square for treatments (MSTr) is calculated as:
MSTr = SSTr / df(SSTr) = 12 / 4 = 3
The mean square for error (MSE) is calculated as:
MSE = SSE / df(SSE) = (20 - 5) / 15 = 1
Finally, we can calculate the F-test statistic as:
F = MSTr / MSE = 3 / 1 = 3
Therefore, the MSTr is 3, the MSE is 1, and the F-test statistic is 3.
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what can you say about the end behavior of the function f(x)=-4x6 -52
The end behavior of the given function is Both ends of the graph goes down(In the same direction),
The degree and the leading coefficient (The coefficient of the highest degree monomial) of a function shows the end behavior of its graph.
if Degree is even and leading coefficient is negative then the end behavior of the function f(x) is,
f(x)---> -∞, as x---> -∞
f(x)---> -∞, as x---> + ∞
That is, Both ends of the graph goes down(In the same direction)
Here, the given function is f(x)=-4x⁶+6x²-52
Degree = 6(even), leading coefficient = -4 (negative)
Thus, the end behavior of the given function is Both ends of the graph goes down(In the same direction),
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Another copy machine also has the ability to reduce image dimensions, but by a different percentage. This graph shows the results found when
copying a design x times. Use the graph to write the equation modeling this relationship.
1 2 3 4 5
Enter the correct answer in the box by replacing the values of a and b.
Note that the graph of the function f(x) = a(b)^x is attached accordingly. Here a = 8 and b = 5
How is this so?f(x) = a(b^x)......1
from the graph we can see that
When x = 0, y = 8
f(0) = 8
8 = a(b^0)
8 = a x 1
a = 8
f(1) = 4
4 = a(b^1)
4 = ab
8b = 4
b = 0.5
Thus a = 8 and b = 5
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Full Question:
Another copy machine also has the ability to reduce image dimensions, but by a different percentage. This graph shows the results found when copying a design x times. Use the graph to write the equation modeling this relationship
Enter the correct answer in the box by replacing the values of a and b.
f(x) = a(b)^x
What is the probability that the throw of two dice yields a total of 5 or 8
Answer:
The probability of getting a sum of 5 or 8 when 2 dice are rolled once is 1/4.
Step-by-step explanation:
Your welcome ;)
the difference of two numbers is 0.77 . if the greater number is increased 5 times the difference becomes 77
Step-by-step explanation:
x - y = 0.77
5x - y = 77
let's subtract the first from the second equation :
5x - y = 77
- x - y = 0.77
---------------------
4x 0 = 76.23
x = 76.23/4 = 19.0575
x - y = 0.77
y = x - 0.77 = 19.0575 - 0.77 = 18.2875
45 teachers teach math and 9 teachers teach science. What is the ratio of science teachers to math teachers?
The ratio of science teachers to math teachers is 1:5.
The ratio of science teachers to math teachers can be found by dividing the number of science teachers by the number of math teachers.
The ratio of science teachers to math teachers = (Number of science teachers) / (Number of math teachers)
Here, the number of science teachers is 9 and the number of math teachers is 45.
The ratio of science teachers to math teachers = 9 / 45
Simplifying this ratio by dividing both the numerator and denominator by 9, we get:
The ratio of science teachers to math teachers = 1 / 5
Therefore, the ratio of science teachers to math teachers is 1:5.
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Complete the proof that angle UYV is congruent to angle WYX
The statement and reason that completes the proof that ∠UYV ≅ ∠WYX is:
Statement - Reason
∠UYV ≅ ∠WYX - Equivalent Substitution Postulate.
What is Equivalent Substitution Postulate?
Incorporating various postulates into math problems has been an essential aspect of mathematics education for decades.
One such idea is the Equivalent Substitution Postulate, which relates specifically to geometry. This postulate indicates that if two geometric figures have identical size and shape, they can effectively substitute each other in any mathematical equation or statement without changing its accuracy.
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What is the value of 2x3−2x2+2x−82x2−5x if x=−2 ?
The value of the given algebraic expression is -2.
To find the value of the expression 2x³ - 2x² + 2x - 8 / 2x² - 5x when x = -2, we simply substitute -2 for every instance of x in the expression:
2(-2)³ - 2(-2)² + 2(-2) - 8 / 2(-2)² - 5(-2)
= 2(-8) - 2(4) - 4 - 8 / 8 + 10
= -16 - 8 - 4 - 8 / 18
= -36 / 18
= -2
Therefore, the value of the expression 2x³ - 2x² + 2x - 8 / 2x² - 5x when x = -2 is -2.
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Tessa’s gpa for three semesters was 3.35 for 46 course units and for her fourth semester gpa was 3.74 for 12 course units. What is Tessa’s cumulative gpa for all four semesters
Tessa's cumulative GPA for all four semesters is 3.44.
To find Tessa's cumulative GPA, we first need to calculate the total number of course units she has taken. For the first three semesters, she took 46 course units. For the fourth semester, she took 12 course units. So, in total, she has taken 46 + 12 = 58 course units.
Next, we need to calculate her total grade points. To do this, we multiply her GPA for each semester by the number of course units she took in that semester, and then add up the results. For the first three semesters, her total grade points are:
3.35 x 46 = 154.1
For the fourth semester, her total grade points are:
3.74 x 12 = 44.88
So her total grade points for all four semesters are:
154.1 + 44.88 = 198.98
Finally, we divide her total grade points by the total number of course units she has taken:
198.98 / 58 = 3.44
Therefore, Tessa's cumulative GPA for all four semesters is 3.44.
One owl hoots every 3 hours, and a second owl hoots every 8 hours, and a third owl hoots every 12 hours. If they all hoot together at midnight of the first day, how many times on the hour in 3 days will at least two owls not give a hoot.
The number of times on the hour in 3 days at least two owls will not hoot obtained using number theory are 69 times.
What is number theory?Number theory is the study of integers and functions of arithmetic.
The number of hours in 3 days = 72 hours
Number theory indicates that the multiples of the periods of hooting are;
The multiples of 8 are; 8, 16, 24, 32, 40, 48, 56, 64, 72
The multiples of 3 are; 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72
The multiples of 12 are; 12, 24, 36, 48, 60, 72
The common multiples of 3 and 8 are 24, 48, 72
The common multiples of 8 and 12 are 24, 48, 72
The number of times in 72 hours when at least 2 owls are hooting are therefore;
The times when only one owl is hooting are
8, 16, 32, 40, 56, 64, = 6 times only the second owl is hooting
3, 6, 9, 12, 15, 18, 21, 27, 30, 33, 36, 39, 42, 45, 51, 54, 57, 60, 63, 66, 69 = 21 times when only the first owl hoots
The times when non of the owls hoot are;
72 - 24 - 6 = 42 times
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A catapult is used to launch a boulder. The height of the boulder h can be modeled by the function 60x40 ft², where t is the time in seconds after the boulder is launched. Assuming that the boulder doesn't hit anything, how many seconds after launch will the boulder hit the ground?
Four seconds after the boulder is hurled, it will hit the ground.
When the boulder hits the ground, its height will be 0. So, we need to solve the equation:
[tex]h(t) = -16t^2 + 64t = 0[/tex]
We can factor out -16t from the expression:
h(t) = -16t(t - 4)
Setting each factor to zero, we get:
-16t = 0 or t - 4 = 0
Solving for t, we get:
t = 0 or t = 4
Since t = 0 represents the time when the boulder is launched, we can ignore it. Therefore, the boulder will hit the ground 4 seconds after it is launched.
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What does the acronym VAT represent?
Answer:
Value-Added Tax
Step-by-step explanation:
Value-added tax (VAT) is a consumption tax on goods and services that is levied at each stage of the supply chain where value is added, from initial production to the point of sale.
Given the function y = 3x^2 - 12x + 9. place a point on the coordinate grid to show each x-intercept of the function. Place a point on the coordinate grid to show the minimum value of the function.
The required graph shows the x-intercepts and the minimum value.
To find the x-intercepts of the function y = 3x² - 12x + 9, we set y = 0 and solve for x:
3x² - 12x + 9 = 0
Dividing both sides by 3 gives:
x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
So, the x-intercepts are x = 1 and x = 3.
To find the minimum value of the function, we can use the formula:
x = -b / 2a
Here a = 3 and b = -12.
Substitute these values gives:
x = -(-12) / 2(3) = 2
So the minimum value occurs at x = 2.
To find the corresponding y-value, we can plug in x = 2 into the original function:
y = 3(2)² - 12(2) + 9 = -3
So the minimum value is y = -3.
Here is a graph showing the x-intercepts and the minimum value.
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math project : I need an interesting project idea for mathematics that can relate 3 different topics, I should write a full research paper related on the topic this are some examples related to the project
You must state a question that you would like to answer. This must be a specific question within your topic and should be explored thoroughly to create a complete paper.
Examples:
(i) How can we use Mathematical/Calculus-based tools to study the spread of COVID-19?
(ii) Designing a new Mathematical/Calculus-based model to analyze the spread of COVID-19
(iii) How many entrances should there be at Expo to accommodate all visitors?
(iv) How much water does the UAE need in order to sustain its ever changing population? (i.e. comparing water usage vs. water production)
Literature review:
You must show using multiple references and sources of the current literature on your given topic. This does NOT imply that information is simply copied from the internet but rather you must present a comprehensive review and summary of the latest research on your topic. It is suggested that you chose a specific aspect of your topic in order to include all required elements.
Examples:
(i) Review of the existing Mathematical/Calculus-based models used to analyze the spread of COVID-19
(ii) Review of existing Mathematical/Calculus-based models and calculations regarding risk insurance.
Answer:
Here's an interesting project idea for mathematics that can relate three different topics:
Topic 1: Fractals
Topic 2: Chaos Theory
Topic 3: Differential Equations
Research Question: Can fractals be used to model chaotic systems described by differential equations?
In this project, you can explore the concept of fractals and their applications in modeling complex systems. You can also delve into chaos theory and differential equations to understand how they are used to describe chaotic systems. By combining these three topics, you can investigate whether fractals can provide a better understanding of chaotic systems by modeling them more accurately.
Your research paper can cover the following areas:
Introduction: Provide an overview of fractals, chaos theory, and differential equations, and explain their relevance to the research question.
Fractals: Discuss the properties of fractals and how they can be used to model complex systems. Provide examples of fractals in nature and technology.
Chaos Theory: Explain the concept of chaos and how it is described by differential equations. Discuss the importance of chaos theory in understanding complex systems.
Differential Equations: Provide an overview of differential equations and their applications in physics, engineering, and other fields. Explain how differential equations are used to model chaotic systems.
Combining the three topics: Explain how fractals can be used to model chaotic systems described by differential equations. Provide examples of fractals used in modeling chaotic systems and compare the results to traditional methods.
Conclusion: Summarize the findings of your research and discuss the implications of using fractals to model chaotic systems.
Overall, this project can be a challenging and rewarding exploration of the interplay between three different mathematical topics.
Step-by-step explanation:
The volume of this rectangular prism is 199.68 cubic yards. What is the value of z?
Answer:
z=199.68
Step-by-step explanation:
Please help!!!!!!!!!!!!
Answer:
48
Step-by-step explanation:
total cost will be the answer
Find the equation of the line perpendicular to y=1/2x-1 and passes through the point (5,6)
The equation of line perpendicular to "y = (1/2)x - 1" and passing through point (5,6) is y = -2x + 16,
The equation of the line is y = (1/2)x - 1, is in slope-intercept form "y = mx + b", where m is slope,
To find the slope of the given line, we see that it is in the form y = mx + b, where m = 1/2.
We know that, slope of a line perpendicular to this line will be the "negative-reciprocal" of this slope.
So, the slope of the perpendicular line will be -2.
Now we use the point-slope form of the equation of a line to find the equation of the perpendicular line passing through the point (5, 6). The point-slope form is : y - y₁ = m(x - x₁),
where (x₁, y₁) is given point and "m" is slope of the line;
Substituting the values, we get:
⇒ y - 6 = -2(x - 5),
⇒ y = -2x + 16,
Therefore, the equation of perpendicular line is y = -2x + 16.
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Jevonte surveyed 400 of the students in his school about their favorite color. 95% said their favorite color was red. How many students' favorite color was red?
Answer: 380
Step-by-step explanation:
95% = 95/100
95/100 x 400
= 380
Answer: 380 students
Step-by-step explanation:
Question 1: There are two agents (A and B) in the economy. They demand for two goods:
1 and 2. Agent A's utility is UA (x₁, x₂) = ln x₁ + 2ln x2; agent B's utility is UB (x1, x₂) =
2 lnx₁ + Inx2. We also know agent A's endowment is (1,2), which means that she has
one unit of good 1 and two units of good 2. In addition, agent B's endowment is (2,1).
Suppose the prices for goods 1 and 2 are p₁ and p2, respectively.
1. Derive agent A's demands for goods 1 and 2: x4 and x.
2. Derive agent B's demands for goods 1 and 2: x and x2.
3. Derive agent A's excess demands for goods 1 and 2: z and z.
4. Derive agent B's excess demands for goods 1 and 2: z
and 22.
5. Derive aggregate excess demands for goods 1 and 2: Z₁ and Z₂.
6. Obtain the Walrasian equilibrium prices for goods 1 and 2: p₁ and p2.
7. Comment on your findings.
For points A and B in the economy:
x₁ = 2p₂/p₁ and x₂ = p₂/p₁.
x₁ = 2p₁ / p₂ and x₂ = p₁ / p₂.
2 - p₂ / p₁
1 - p₁ / p₂
2 - p₂ / p₁ + 1 - p₁ / p₂
The market clears for both items at the Walrasian equilibrium, and the aggregate excess demands are zero.
How to determine demand of goods?1. To derive agent A's demands for goods 1 and 2, maximize her utility subject to her budget constraint:
max ln x₁ + 2ln x₂
s.t. p₁x₁ + p₂x₂ ≤ p₁(1) + p₂(2)
Using Lagrange multiplier method:
1/x₁ = p₁/(2p₂)
1/x₂ = p₁/p₂
Solving for x₁ and x₂:
x₁ = 2p₂/p₁
x₂ = p₂/p₁
So, agent A's demands for goods 1 and 2 are x₁ = 2p₂/p₁ and x₂ = p₂/p₁.
2. To derive agent B's demands for goods 1 and 2, maximize her utility subject to her budget constraint:
max 2ln x₁ + ln x₂
s.t. p₁x₁ + p₂x₂ ≤ p₁(2) + p₂(1)
Using Lagrange multiplier method:
2/x₁ = p₁/p₂
1/x₂ = p₂/p₁
Solving for x₁ and x₂:
x₁ = 2p₁/p₂
x₂ = p₁/p₂
So, agent B's demands for goods 1 and 2 are x₁ = 2p₁ / p₂ and x₂ = p₁ / p₂.
3. Agent A's excess demands for goods 1 and 2 are:
z₁ = 1 - x₁ = 1 - 2p₂ / p₁
z₂ = 2 - x₂ = 2 - p₂ / p₁
4. Agent B's excess demands for goods 1 and 2 are:
z₁ = 2 - x₁ = 2 - 2p₁ / p₂
z₂ = 1 - x₂ = 1 - p₁ / p₂
5. The aggregate excess demands for goods 1 and 2 are:
Z₁ = z₁ᵃ + z₁ᵇ = 1 - 2p₂ / p₁ + 2 - 2p₁ / p₂
Z₂ = z₂ᵃ + z₂ᵇ = 2 - p₂ / p₁ + 1 - p₁ / p₂
6. At Walrasian equilibrium, Z₁ and Z₂ are both equal to zero. So, solve the system of equations:
1 - 2p₂ / p₁ + 2 - 2p₁ / p₂ = 0
2 - p₂ / p₁ + 1 - p₁ / p₂ = 0
Solving for p₁ and p₂:
p₁ = 2
p₂ = 1
So, the Walrasian equilibrium prices for goods 1 and 2 are p₁ = 2 and p₂ = 1.
7. At the Walrasian equilibrium, the market clears for both goods and the aggregate excess demands are zero. This means that the prices are set such that both agents are willing to buy exactly what is being sold in the market.
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✔️z-=4z-2. Solve this problem. It should be the square root of z.
The equation is solved to give 16z² - z - 4 = 0
How to determine the valueNote that algebraic expressions are defined as expressions that are composed of terms, variables, coefficients, constants and also factors.
The expressions are also identified with mathematical operations, such as;
BracketParenthesesAdditionSubtractionMultiplicationDivisionFrom the information given, we have that;
√z= -4z - 2
Find the square of both sides, we get;
z = 16z² - 4
collect the like terms
16z² - z - 4
Equate to zero
16z² - z - 4 = 0
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You want to purchase a new car in 6 years and expect the car to cost $12,000. Your bank offers a plan with a guaranteed APR of 6.5% if you make regular monthly deposits. How
← much should you deposit each month to end up with $12,000 in 6 years?
T
You should invest $ each month
(Round the final answer to the nearest cent as needed. Round all intermediate values to seven decimal places as needed
Answer:
To calculate the monthly deposit required, we can use the formula for future value of an annuity, which is:
FV = Pmt x (((1 + r)^n - 1) / r)
where FV is the future value, Pmt is the monthly payment, r is the monthly interest rate, and n is the number of months.
In this case, we want to find the monthly payment required to achieve a future value of $12,000 in 6 years, or 72 months. The monthly interest rate is the annual percentage rate (APR) divided by 12, so:
r = 6.5% / 12 = 0.00541666667
Substituting these values into the formula, we get:
12,000 = Pmt x (((1 + 0.00541666667)^72 - 1) / 0.00541666667)
Solving for Pmt, we get:
Pmt = 12,000 / (((1 + 0.00541666667)^72 - 1) / 0.00541666667)
≈ $164.41
Therefore, you should deposit $164.41 each month to end up with $12,000 in 6 years
Samuel runs around a 100 meter track.He uses the linear equation y=200+5x to describe how far he runs after he has already jogged two laps. Which of the following best describes the slope of the linear equation?
Samuel runs 5 meters per second best describes the slope of the linear equation
Given that Samuel runs around a 100 meter track.
He uses the linear equation y=200+5x to describe how far he runs after he has already jogged two laps.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
y=200+5x
y=5x+200
Slope is 5
Hence, Samuel runs 5 meters per second best describes the slope of the linear equation
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Here is a Venn Diagram with sets B, D, and the universal set U . Shade (BUD)’ on the Venn diagram.
The Venn diagram is attached in the solution.
Given that, a Venn Diagram with sets B, D, and the universal set U, we need to Shade (BUD)’ on the Venn diagram,
We know that, the complement of a set is the set that includes all the elements of the universal set that are not present in the given set.
Therefore, (BUD)’ = U - (BUD)
therefore, we will shade all the portion except (BUD),
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A recipe that makes 16 cookies calls for 1/4 cup of sugar and 2/3 cup of flour. Janelle wants to proportionally increase these amounts to get a new recipe using one cup of sugar. Using the new recipe, how much flour should she use and how many cookies can she make with the new recipe?
Answer:
2 2/3 cups flour
Step-by-step explanation:
32 cookies
by my calculations
2. Two numbers are such that when the larger number is divided by the smaller number, both the quotient and the remainder are equal to 2. If five times the smaller number is divided by the larger number, both the quotient and the remainder are also equal to 2. Find the two numbers.
Answer:
Let's assume that the smaller number is x and the larger number is y.
From the problem, we can write:
y = 2x + 2 (since the quotient and remainder when y is divided by x are both equal to 2)
5x = 2(y - 2) (since the quotient and remainder when 5x is divided by y are both equal to 2)
Substituting the first equation into the second equation, we get:
5x = 2((2x + 2) - 2)
5x = 4x + 4
x = 4
Substituting x = 4 into the first equation, we get:
y = 2x + 2 = 10
Therefore, the two numbers are 4 and 10
Brooke has money in an account that earns 10% interest compounded annually.
She earns $1.47 in 2 years, how much money did Brooke initially invest?
The money Brooke initially invest is $1.21.
How to calculate how much money did Brooke initially investUsing the formula for compound interest formula:
A = P(1 + r/n)^(nt)
where:
A = final amount after t years
P = initial investment
r = annual interest rate
n = number of times interest is compounded per year
t = number of years
Substituting the given values:
1.47 = P(1 + 0.10/1)^(1*2)
1.47 = P(1.10)^2
1.47 = P(1.21)
P = 1.47/1.21
P = 1.2148...
Rounding to the nearest cent, Brooke initially invested $1.21.
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A line sgement has endpoints at (-3,2) amd (42,32). What is the x-coordinate of the point that is 2/3 the distance from (-3, 2) to (42, 32)
Answer:x = midpoint_x - (midpoint_x - (-3)) * (sqrt(5)/6) = 19.5 - (22.5 * sqrt(5))/6 = 19.5 - 3.75sqrt(5)
Step-by-step explanation:To find the x-coordinate of the point that is 2/3 of the distance from (-3, 2) to (42, 32), we first need to find the coordinates of that point.
The distance between the two endpoints of the line segment is:
d = sqrt((42 - (-3))^2 + (32 - 2)^2) = sqrt(45^2 + 30^2) = 15sqrt(45)
The distance from (-3, 2) to the desired point is:
(2/3)d = (2/3)sqrt(45^2 + 30^2)
Let's call the desired point (x, y). We can use the midpoint formula to find the coordinates of the midpoint of the line segment:
midpoint = ((-3 + 42)/2, (2 + 32)/2) = (19.5, 17)
The midpoint is halfway between the two endpoints, so the distance from (-3, 2) to the midpoint is:
sqrt((19.5 - (-3))^2 + (17 - 2)^2) = sqrt(22.5^2 + 15^2) = 15sqrt(5)
To find the coordinates of the desired point, we can use similar triangles. The ratio of the distance from (-3, 2) to the midpoint to the distance from (-3, 2) to the desired point is:
15sqrt(5) / (2/3)sqrt(45^2 + 30^2) = 15sqrt(5) / 30sqrt(45) = sqrt(5)/6
The ratio of the distance from (42, 32) to the midpoint to the distance from (42, 32) to the desired point is the same:
15sqrt(5) / (2/3)sqrt(45^2 + 30^2) = sqrt(5)/6