we can estimate the cost of a $50 textbook in 8 years to be approximately $32.78.
What is logarithms?
Logarithms are mathematical functions that are used to solve equations involving exponents. Specifically, a logarithm is the inverse operation of exponentiation. Just as exponentiation involves raising a base to a power, logarithms involve finding the power to which a base must be raised to produce a given value.
The logarithm of a number y with respect to a base b is written as logb y, and it is defined as the exponent to which the base b must be raised to produce the number y. In other words, if [tex]b^x = y[/tex], then logb y = x.
According to the question:
In Exercise 36(a), we found that the value of a dollar decreases by 26% over 5 years. We can use this information to answer the following questions:
(a) To estimate the cost of the same house in 10 years, we need to solve the equation:
0.74t = 105,000
where t is the time in years. Solving for t, we get:
[tex]t = log(105,000) / log(0.74) = 17.9[/tex]
So it will take approximately 17.9 years for the value of the dollar to decrease by 26% enough to make the $105,000 house cost the same as it does today. Therefore, we can estimate the cost of the same house in 10 years to be:
[tex]$105,000 * (1 / 0.74)^(10/17.9) = $211,775.22[/tex]
(b) To estimate the cost of a $50 textbook in 8 years, we need to find the value of k in the exponential function:
[tex]V(t) = V(0) * e^kt[/tex]
where V(0) is the initial value (in this case, $50), t is the time in years, and V(t) is the value after t years. We know that the value of a dollar decreases by 26% over 5 years, so we can use this information to find k:
[tex]0.74 = e^(k*5)[/tex]
Taking the natural logarithm of both sides, we get:
ln(0.74) = 5k
Solving for k, we get:
k ≈ -0.064
So the exponential function for the value of a dollar over time is:
[tex]V(t) = $50 * e^{-0.064t}[/tex]
To estimate the cost of the textbook in 8 years, we can plug in t = 8 into this function and solve:
[tex]V(8) = $50 * e^{-0.064*8} = $32.78[/tex]
Therefore, we can estimate the cost of a $50 textbook in 8 years to be approximately $32.78.
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According to the National Student Clearinghouse Research Center, women now comprise 60% of enrollment in universities. What is the probability in a random sample of 55 university students that more than 35 are women? Round your answer to 3 decimal places.
The probability that more than 35 students are women in a random sample of 55 university students is 0.256.
What is binomial distribution?The binomial distribution is a probability distribution that expresses the number of successes in a finite number of independent trials when the chance of success is constant and there are only two potential results (success or failure). Since a known chance of success for each trial is known, we may use it to determine the likelihood of a certain number of successes (or failures) in a set number of trials.
Given that p = 60% = 0.6, n = 55.
Using normal approximation we have:
SD = sqrt(np(1-p))
= sqrt(55 x 0.60 x 0.40) = 3.05
The z-score is:
z = (X - np) / SD)
z = (35 - (0.4)(0.6)) / 3.05
z = 11.39 = 0.256.
Hence, the probability that more than 35 students are women in a random sample of 55 university students is 0.256.
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help me please
A car was valued at $44,000 in the year 1992. The value depreciated to $15,000 by the year 2006.
A) What was the annual rate of change between 1992 and 2006?
r=---------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=---------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ?
value = $ -----------------Round to the nearest 50 dollars.
In the exponential decay, A) r = -0.0839 , B) r = -8.39% , C) Value=$11,800.
What is exponential decay?
The term "exponential decay" in mathematics refers to the process of a constant percentage rate reduction in an amount over time. It can be written as y=a(1-b)x, where x is the amount of time that has passed, an is the initial amount, b is the decay factor, and y is the final amount.
To find the annual rate of change between 1992 and 2006, we can use the formula:
r = [tex](V_2/V_1)^{1/n}-1[/tex]
where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.
=>r = -0.0839
Therefore, the rate of change between 1992 and 2006 is -0.0839.
To express the rate of change in percentage form, we can multiply the result from part A by 100:
=>r = -0.0839 x 100
=> r = -8.39%
Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.
To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.
From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:
where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.
=>V = 11792.51
Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).
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The initial population of a bacterial culture is 2000, growing at a fixed rate. Table shows the population every hour for six hours.
A. Find the equation that models this relation. (round to one decimal place).
B. Use the equation to find the population of bacteria after 12 hours after 24 hours.
Answer: A. To find the equation that models this relation, we can plot the data and look for a trend.
From the graph, we can see that the data follows an exponential growth pattern. The equation for exponential growth is:
y = ab^x
where y is the final population, a is the initial population, b is the growth rate, and x is the time in hours.
Using the data from the table, we can find the value of b:
b = y/x
where y is the population after x hours.
For example, after 2 hours, the population is 5000:
b = 5000/2 = 2500
We can repeat this process for each time period and find the average value of b:
b = (2500 + 3125 + 3906.25 + 4882.81 + 6103.51 + 7629.38) / 6 = 4275.5
Now that we have the value of b, we can find the equation that models the relation:
y = 2000(4275.5)^x
Rounding to one decimal place:
y = 2000(4.3)^x
B. Using the equation, we can find the population after 12 and 24 hours:
y = 2000(4.3)^12 ≈ 8,898,335.7
y = 2000(4.3)^24 ≈ 7.42 x 10^13
Step-by-step explanation:
Find the measure of XY.
Y
●
74°
N
X
Check the picture below.
A company finds that if it charges 2 dollars for a cell phone, it can expect to sell 1,000 - 2x phones. The company uses the function
defined by r(x)=(1,000-2x) to model the expected revenue, in dollars, from selling cell phones at a dollars each.
1. Use graphing technology to graph the function r.
2. What do the z-intercepts mean in this situation?
3. Is 0≤x≤ 600 an appropriate domain for the function 7? Explain your reasoning.
4. At what price should the company sell their phones to get the maximum revenue? Explain your reasoning.
The z-intercept is at 500, the domain of the function is all non-negative real numbers, or [0, ∞) and the maximum revenue is 500
What does the z-intercepts interpret in this situation?1. The graph of the function is attached below;
2. The z-intercept is where the graph intersects the x-axis, which occurs when the revenue is 0. In this situation, the z-intercept represents the maximum price that the company can charge for the cell phone and still expect to sell some units. So, the z-intercept is $500, meaning if the company charges more than $500, it will not sell any cell phones and its revenue will be 0.
3. No, 0≤x≤ 600 is not an appropriate domain for the function. The company can charge any price, not just prices between 0 and 600 dollars. The domain of the function should be all non-negative real numbers, or [0, ∞).
4. The maximum revenue occurs at the point where the slope of the function changes from negative to positive, which is the vertex of the parabola. In this case, the function r(x) is a linear function, so there is no vertex. The maximum revenue occurs at the endpoint of the domain, which is x=500, since the company cannot charge more than $500 without reducing the number of phones sold to 0. Therefore, the company should charge $500 to get the maximum revenue.
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Martha Martha and Steven are in the same math class. The box plot shows the number of minutes that Martha and Steven spent studying for their final exam each day for a week. Steven 0 5 10 Study Time 15 20 Number of Hours 25 30 35 Which statement is best supported by the information in the box plots? 2 of 9 A 07:34
Answer:
Based on the given information in the box plots, the statement that is best supported is that Steven spent more time studying than Martha did during the week for their math final exam. This can be inferred because the top line of Steven's box plot is higher than the top line of Martha's box plot, indicating that Steven spent more time studying than Martha in the given time period. Additionally, Steven's whisker extends higher than Martha's whisker, further supporting the idea that Steven studied for a longer period of time than Martha.
Suppose on your 21st birthday you begin saving $500 quarterly into an account that pays 12% compounded quarterly. If you continue the savings until your 51st birthday (30 years), how much money will be in the account?
Please help!! Will mark brainliest
From the time you are 21 until the time you turn 51, if you put $500 away quarterly in an account that produces 12% compound interest, you will have around $1,299,994.09 in it by the end of the 30 years.
When does a quarterly monthly occur?Four quarters, numbered Q1, Q2, Q3, & Q4, make up an entire year. There are 3 months in each quarter. About 91 days make up each quarter. Quarterly is defined as once every quarter, or around every 91 days.
The calculation for future value of the an annuity can be used to resolve this issue:
FV = PMT x ((1 + r/n)nt - 1) / (r/n), where FV is just the future value of a annuity, PMT is indeed the payment amount (in this case, $500), r is indeed the interest rate (the value of which in this case is 12% or 0.12), n refers to the number of occasions a year the interest is greatly exacerbated (which in this case is four for quarterly cumulative), and t is the duration of the annuity .
When we enter the data, we get the following result: FV = 500 x (1.03120 - 1) / 0.03 = $1,299,994.09
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Add 11.7+19.1 what the answer
Answer:
➠ 11.7 + 19.1
➠ 30.8
Answer: 30.8
1. Add 11 and 19
2. Add .7 and .1
3. Put the results together
Please mark brainliest! Hope this helped!
0.1(9) repeating as a decimal.
Answer:
0.2
Step-by-step explanation:
You want to know the value of 0.199...repeating 9s.
ValueLet the value be represented by x:
x = 0.1999...
Then 10x is ...
10x = 1.9999...
The difference is ...
10x -x = 1.9999... -0.1999... = 1.8
9x = 1.8 . . . . . simplify
x = 1.8/9 = 0.2
The value of the repeating decimal is 0.2.
__
Additional comment
The repeating decimal 0.99... is equivalent to 9/9 = 1. The repeating portion of 0.1999... has the value 0.1, Added to the first non-repeating digit, the sum is ...
[tex]0.1\overline{9}=0.1+0.0\overline{9}=0.1+0.1=0.2[/tex]
What is 2-2x5+7 hurry!!!!!!!!!!!!
Answer: 15 / fifteen
Step-by-step explanation:
2*5 = 10 - 2 = 8 + 7 = 15
The value of the given expression is -1.
What is PEDMAS rule?PEMDAS is an acronym used to mention the order of operations to be followed while solving expressions having multiple operations. PEMDAS stands for P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.
The given numerical expression is 2-2×5+7.
Here, first multiply 2 and 5, that is
2-10+7
Next add 2 and 7, that is
9-10
Finally subtract 9 and 10, that is
9-10= -1
Therefore, the value of the given expression is -1.
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The police lieutenant in charge of the traffic division reviews the number of traffic citations issued by each of the police officers in his division. He finds that the mean number of citations written by each officer is 23.2 citations per day, with a standard deviation of 3.1. Assume that the distribution of the number of tickets issued is approximately bell-shaped. The coefficient of variation for the number of citations is
Answer:
[tex] \sf \: 13.36\%[/tex]
Step-by-step explanation:
The coefficient of variation (CV) is a measure of relative variability, which expresses the standard deviation as a percentage of the mean. It can be calculated as follows:
[tex] \sf \: CV = \frac{standard \: deviation}{mean} \times 100%[/tex]
In this case, the mean number of citations is 23.2, and the standard deviation is 3.1. Thus, the coefficient of variation is:
[tex] \sf \: CV = \frac{3.1}{23.2} \times 100% [/tex]
[tex] \sf = 13.36\%[/tex]
Therefore, the coefficient of variation for the number of citations is approximately 13.36%.
Determine the length of side AB and BC in triangle ABC.
AB=9.68,BC=7.04
AB=8.8,BC=6.4
AB=9.152,BC=6.656
AB=7.92,BC=5.76
In triangle ABC, the values of AB and BC are 9.68 and 7.04 respectively i.e. A.
What is a triangle's definition?
A triangle is a geometric form that has three straight sides and three angles. It is a fundamental geometric form that is frequently utilised in mathematics, science, and everyday life.
Triangles are distinguished by three sides and three angles. The sum of each triangle's three angles is always 180 degrees. Triangles are classified according to the length of their sides and the measurement of their angles.
Now,
As given ΔDEF ≈ΔABC
that means ratio of all sides will be equal
i.e. AB/DE=BC/EF=AC/DF
then AB/8.8=6.71/6.1
AB=1.1*8.8
AB=9.68
and BC/6.4=6.71/6.1
BC=1.1*6.4
BC=7.04
Hence,
the values of AB and BC are 9.68 and 7.04 respectively.
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A Small Monster collector is visiting the Knightrola region, bringing 30 Small Monster Containment Devices with them. The devices are highly reliable and can effectively be assumed to always work. How many different types of Small Monsters can the collector capture under the following conditions? a. The collector has access to Grass, Space Electric Scooter, Mask, Citrus, and Rubber Duck type Small Monsters 7 types b. The above, and intends to capture at least one of each of those types c. Both of the above, and intends to capture at least five Space types d. All of the above and intends to capture at most four Electric Scooter types e. All of the above, and intends to capture at most three Rubber Duck type
The different types of Small Monsters combinations can the collector capture under that collector .lunder the following conditions are a) 593775 ; b) 100947 ; c) 27132 ; d) 30752 ; e) 26334.
For these questions as we are dealing with combination and repetition, we will be using the rule nC(r-1) where n = the number of total items and r = the number of ways we can choose.
We have provide that Small Monster collector is visiting the Knightrola region, bringing 30 Small Monster Containment Devices with them. We have to determine the different types of Small Monsters can the collector capture under the following conditions :
a) Here n = 30,
Number of monstor access by him = 7 {Grass, Space Electric Scooter, Mask, Citrus, and Rubber Duck}
So, different types of Small Monsters can the collector capture = ³⁰C₇₋₁ = ³⁰C₆
= 593775
b) Here, he captures at least one of each of those 7 places. So, we have to distribute 23 ( 30 -7) among 7 places
= ²³C₆ = 100947
c) Here provided that atleast one to all plus atleast five Space types captured, so we have to distribute 30 - 11 = 19 devices among 7 places, = ¹⁹C₆ = 27132
d) All above conditions plus at most four Electric Scooter types captured, so ³⁰⁻¹¹C₅ + ³⁰⁻¹²C₅+ ³⁰⁻¹³C₅ + ³⁰⁻¹⁴ C₅
= 30752
e) All of the above and capture at most three Rubber Duck type, First, we will subtract the combination where we capture n = 20-3 (at most 3 rubber ducks) from the combination we have so ³⁰⁻¹⁴C₅ + ³⁰⁻¹⁵C₅ + ³⁰⁻¹⁶C₅ = 26334. Hence, required value is 26334.
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Write an equation for a linear function whose graph has the given characteristics. Horizontal, passes through (9, −45)
Answer:
y = -45
Step-by-step explanation:
y=mx+b
because the graph is horizontal m = 0
y=0x+b --> y=b
next sub in points
y=b (9,-45)
-45=b
therefore equation is y=-45
Question 3 a) What is the theoretical probability of rolling a sum of 8? b) What is your experimental probability of rolling a sum of 8? c) What are the odds of rolling a sum of 8?
The theoretical probability of rolling a sum of 8 is 1/11, the experimental probability is 5/36 and the odds is 5 : 31
a) What is the theoretical probability of rolling a sum of 8?When two dice are rolled, the possible sum are
Sum = 2 to 12
In the above sum, we have
n(8) = 1
Total = 11
This means that
P(sum of 8) = 1/11
b) What is your experimental probability of rolling a sum of 8?For the experimental probability, we have
n(Sum of 8) = 5
Total = 36
So, we have
P(sum of 8) = 5/36
c) What are the odds of rolling a sum of 8? In (b), we have
P(sum of 8) = 5/36
This means that
Odds = 5 : 36 - 5
Odds = 5 : 31
Hence, the odds of rolling a sum of 8 is 5 : 31
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Write and solve an equation to represent the hanger.
The equation for the hanger is 2x + 2x + 2 = 11 and x = 9/4
What are algebraic expressions?Algebraic expressions are defined as mathematical expressions that are composed of variables, constants, factors, coefficient and terms.
They are also made up of arithmetic operations, such as;
SubtractionAdditionBracketParenthesesMultiplicationDivisionFrom the diagram shown, we have that;
2x, 2x + 2 = 11
x + 2 + x + 2 + x as the remaining expression for the hanger
Then, this can be represented as;
2x + 2x + 2= 11
collect the like terms and add the terms
4x =11 - 2
4x = 9
Make 'x' the subject
x = 9/4
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9-3: Lesson Ouiz Two number cubes whose sides are numbered 1 through 6 are rolled on a table. The two numbers showing are added. If you repeat this process 300 times, how many times would you expect the two cubes to add to exactly 7?
Answer:
300 I think because when you repeat it 300 times it will give an exactly 7 300 times
Which statement about credit reports is not true?
he asked me what my name was into direct speech
A real estate broker decides to lease a car for 36 months. Suppose the anual interest rate is 7.8%, the negotiated price is $49,000, there is no trade-in, and the down payment is $4,000. Find the monthly lease payment (in dollars). Assume that the residual value is 49% of the MSRP of $51,500. (Round your answer to the nearest cent.)
Answer:
To find the monthly lease payment, we need to use the formula:
Monthly lease payment = (Net capitalized cost - Residual value) / Lease term in months
where:
Net capitalized cost = negotiated price - down payment
Residual value = percentage of MSRP x MSRP
Lease term in months = 36
First, we need to calculate the net capitalized cost:
Net capitalized cost = $49,000 - $4,000 = $45,000
Next, we need to calculate the residual value:
Residual value = 49% x $51,500 = $25,235
Now we can plug in the values and solve for the monthly lease payment:
Monthly lease payment = ($45,000 - $25,235) / 36 = $548.76
Therefore, the monthly lease payment is approximately $548.76.
Draw the image of traingel ABC under a dilation whose center is p and scale factor is 1/2
A graph of the image of ABC under a dilation whose center is P and scale factor is 1/2 is shown in the image below.
What is dilation?In Geometry, a dilation can be defined as a type of transformation which typically changes the size of a geometric shape, but not its shape.
In this exercise, we would use an online graphing calculator to plot the image of ABC after a dilation by a scale factor of 1/2 centered at P.
Based on the image (see attachment), we can logically deduce that each vertex is 1/2 times as far from center P as the original vertex and each segment is 1/2 times as long as the original:
Side length A'C' = 8.2/2 = 4.1
Side length A'B' = 5.7/2 = 2.85
Side length B'C' = 12.2/3 = 6.1
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4. Myron Security, Inc., had total sales for 1 year of $945,860. Their advertising expenses were $57,370. a. Estimate the percent advertising expenses were of total sales.
The calculated value of the percent advertising expenses were of total sales is 6.07%
How to estimate the percent advertising expenses were of total sales.To estimate the percent advertising expenses were of total sales, we need to calculate the ratio of advertising expenses to total sales and then convert it to a percentage.
The ratio of advertising expenses to total sales is:
$57,370 / $945,860 = 0.0607
To convert this ratio to a percentage, we can multiply it by 100:
0.0607 x 100 = 6.07%
Therefore, we can estimate that the advertising expenses were approximately 6.07% of the total sales.
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how many solutions does the equation have
4x−6y=−24
2x−3y=−6
and graph solution
The system of equation the 4x−6y = −24 and 2x−3y = −6 have no solution.
What is system of equation?In mathematics, a system οf equatiοns, alsο knοwn as a set οf simultaneοus equatiοns οr an equatiοn system, is a finite set οf equatiοns fοr which we sοught cοmmοn sοlutiοns. In systems οf equatiοns, variables are related in a specific way in each equatiοn. i.e., the equatiοns can be sοlved simultaneοusly tο find a set οf values οf variables that satisfies each equatiοn.
Multiplying the second equation by 2, we get:
4x - 6y = -24
4x - 6y = -12
Lets graph these equations in a coordinate plane and see.
As we can see that the graph line do not intersect and is parallel to each other we can conclude that,
The system of equation the 4x−6y = −24 and 2x−3y = −6 have no solution.
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bobby is painting a figure on a banner for a homecoming parade. the figure he is painting consists of an isosceles triangle, a rectangle, and a semicircle, as shown how do i sovle it
Using the perimeter formula, the total length of sparkling lights he can add to the banner is 40.56ft.
Define perimeter?The perimeter of a shape is its whole length. It is any border or outline's length that defines a two-dimensional geometric shape. Different figures may have the same perimeter depending on their sizes.
The same wire can be used to create a square if each of its four sides is the same length.
Here in the question,
We have to find the perimeter of the whole figure.
Now the total perimeter =
Sum of sides of the triangle + sum of sides of quadrilateral and the sum of the circumference of the semi-circle.
Perimeter = 10 + 5 + 3 + 10 + πr
= 28 + 3.14 × 8/2
= 28 + 12.56
= 40.56ft.
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The complete question is:
Bobby is creating a banner for the homecoming parade. The banner consists of an isosceles triangle, a rectangle, and a semicircle. He wants to add sparkling lights around the trim of the banner.
What is the total length of sparkling lights he can add to the banner?
Select the correct answer from each drop-down menu.
Points A, B, and C form a triangle. Complete the statements to prove that the sum of the interior angles of AABC is 180°.
C
Reason
m21=m24 and m23 = m25
m24+ m22+ m25= 180°
mz1+m22+ m/3 = 180°
Statement
Points A, B, and C form a triangle.
given
Let DE be a line passing through B and parallel to AC. definition of parallel lines
23 25 and 21 24
angle addition and definition of a straight line
substitution
Using congruent angles , we have been able to prove that the sum of the interior angles of ΔABC is 180°.
How to find the angle proof?We are given that:
A, B, and C form a triangle.
The sum of the interior angles of ΔABC is 180°.
We want to find the complete the statements to prove that the sum of the interior angles of ΔABC is 180°
From the attached diagram, we have that:
A triangle ABC with angle 1, 2 and 3.
Line DE and AC are parallel and then BA is a traversal line.
Thus, the alternate interior angle is:
∠1 = ∠4
∠3 = ∠5
Therefore, the angle formed on line is:
∠4 + ∠2 + ∠5 = 180
∠1 + ∠2 + ∠3 = 180
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khaled plants to go to the animal shelter to adopt a cat and then take it to the grooming shop. the shelter is located at (-3,-5) and the grooming shop is located at (5,-1). find the location of lhaled house if the distance from his house to the shelter and the distance from his house to the grooming shop has a ratio of 1:3
After answering the presented question, we can conclude that coordinates Therefore, Khaled's house is located at approximately (-4.75, -1.5).
what are coordinate ?In mathematics, a coordinate is a number or combination of numbers that represents the location of a point or object in space. Coordinates are frequently used to define a point or object's position in a certain coordinate system, such as the Cartesian or polar coordinate systems. In the Cartesian coordinate system, a point is found by its distance from the origin along the x-axis and its distance from the origin along the y-axis. These distances are represented by two integers, the x-coordinate and the y-coordinate. The point's location in the plane is specified by the two coordinates.
Distance from Khaled's house to the animal shelter:
[tex]d_1 = \sqrt((x - (-3))^2 + (y - (-5))^2) = \sqrt((x + 3)^2 + (y + 5)^2)[/tex]
Distance from Khaled's house to the grooming shop:
[tex]d_2 = \sqrt((x - 5)^2 + (y - (-1))^2) = \sqrt((x - 5)^2 + (y + 1)^2)\\d_1/d_2 = 1/3\\\sqrt((x + 3)^2 + (y + 5)^2) / \sqrt((x - 5)^2 + (y + 1)^2) = 1/3\\((x + 3)^2 + (y + 5)^2) / ((x - 5)^2 + (y + 1)^2) = 1/9\\8x + 38y + 4 = 0\\[/tex]
So the coordinates of Khaled's house are:
x = -4.75
y = -1.5
Therefore, Khaled's house is located at approximately (-4.75, -1.5).
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ompounding goes from _ value to _ value.
Answer:
Compounding goes from present value to future value.
. From a rectangular metal sheet of size 20 cm by 30 cm, a circular sheet as big as possible is cut. Find the area of the remaining sheet.
Can someone please help me with this? find the value of x and write answer in simplest radical form. I understand the concept but I don’t know how to differentiate each side to find out if it’s sin, cos, or tan. Please explain how you got the answer.
The value of x is 5√3
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
The longest sides is the hypotenuse and the side opposite the acute angle is the opposite, the other side is adjascent.
Sin(tetha) = opp/ hyp
cos(tetha) = adj/hyp
tan(tetha) = opp/adj
therefore Tan 60 = x/5
√3 = x/5
x = 5√3
therefore the value of x is 5√3
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Where are the vertices of triangle A’B’C’ located? Show your work or explain your steps. (4points)
The vertices of ∆A′B′C′ are A′(-6, 5), B′(-6, 8), and C′(-10, 5) upon the application of the translation rule.
What is the translation rule in geometry?
In geometry, a translation is a transformation that moves every point of a figure the same distance in the same direction. In the case of a triangle, a translation rule describes how to move each vertex of the triangle to a new location to create a new triangle with the same size and shape, but in a different position.
A translation rule for a triangle is given by a pair of numbers (a, b), which indicate the amount of horizontal and vertical movement of the vertices. Specifically, the rule is given by the formula:
(x, y) → (x + a, y + b)
This formula means that we move every point in the triangle a unit to the right or left (depending on the sign of a) and b units up or down (depending on the sign of b).
The translation rule (x, y) → (x − 3, y + 4) means that we move every point in the original triangle left by 3 units and up by 4 units. So, the translation is described as a leftward shift of 3 units and an upward shift of 4 units.
To find the vertices of ∆A′B′C′, we apply the translation rule to each vertex of ∆ABC:
Vertex A: (x, y) → (x - 3, y + 4)
(-3, 1) → (-3 - 3, 1 + 4) = (-6, 5)
So, vertex A of ∆A′B′C′ is located at (-6, 5).
Vertex B: (x, y) → (x - 3, y + 4)
(-3, 4) → (-3 - 3, 4 + 4) = (-6, 8)
So, vertex B of ∆A′B′C′ is located at (-6, 8).
Vertex C: (x, y) → (x - 3, y + 4)
(-7, 1) → (-7 - 3, 1 + 4) = (-10, 5)
So, vertex C of ∆A′B′C′ is located at (-10, 5).
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The complete question is as follows:
Peyton drew a right triangle with sides 5
cm, 12
cm, and 13
cm long. Carter drew a similar triangle to Peyton’s. Which of the following can be the measurements of Carter’s triangle?
Option B. The length of the hypotenuse of Carter's triangle is 13cm.
Since Carter's triangle is similar to Peyton's triangle, the ratio of corresponding side lengths will be the same. Let's use x to represent the length of the hypotenuse of Carter's triangle.
We know that the ratio of the lengths of the hypotenuse and the shorter leg of Peyton's triangle is 13:5. So, the ratio of the lengths of the hypotenuse and the shorter leg of Carter's triangle must also be 13:5.
Thus, we have x/5 = 13/5, which simplifies to x = 13.
Therefore, the length of the hypotenuse of Carter's triangle is 13cm.
Therefore, the only possible length for Carter's hypotenuse is 13cm, and the correct answer is A.
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Peyton drew a right triangle with sides 5cm, 12cm, and 13cm long. Carter drew a similar triangle to Peyton’s. Which of the following can be the measurements of Carter’s triangle?
A. 15cm
B. 13cm
C. 11cm
D. 10cm