The difference quotient of f(8 + h) - f(8) / h = x³ + 3x²h + 3xh³ + h³ - 512 / h.
The difference quotient: what is it?When you hear the phrase "difference quotient formula," what comes to mind? Difference and quotient resemble the slope formula in appearance. Yes, the difference quotient formula does really provide the slope of a secant line drawn to a curve. What is a secant line? A line that joins any two points on a curve is known as the secant line.
Given that the function is f(x) = x³
The value of f(8 + h) = (x+ h)³ = x³ + 3x²h + 3xh³ + h³
The value of f(8) = (8)³ = 512
Substituting the value of f(8 + h) and f(8) we have:
f(8 + h) - f(8) / h = x³ + 3x²h + 3xh³ + h³ - 512 / h
Hence, the difference quotient of f(8 + h) - f(8) / h = x³ + 3x²h + 3xh³ + h³ - 512 / h.
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If a pair of linear equation in two variable have no solution then the pair of lines are
If a pair of linear equations in two variables have no solution then the pair of lines are inconsistent or parallel lines.
Solutions of a pair of linear equations:
Depending on the number of variables and type of equations there are 3 types of solutions to an equation.
1. Unique Solution (One solution):
When there is only one variable then the linear equation will have a Unique Solution or One solution. In the case of pair of linear equations, the linear equation will have a unique solution when the straight lines are intersected at a single point.
2. No solution:
When the given pair of linear equations is inconsistent equations the linear equation will have no solutions in common or we can say that the given lines are non-intersected lines. Usually, a pair of parallel lines will have no solutions.
3. Infinite Solutions (Many solutions):
When the given lines are overlapped on each other or coincident with each other the given lines will have Infinite Solutions. These intersected lines.
Therefore,
If a pair of linear equations in two variables have no solution then the pair of lines are inconsistent or parallel lines.
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for the function g(x) graphed here, find the following limits or explain why they do not exist. a. limx→−2g(x) b. limx→−1g(x) c. limx→0g(x) d.
The function g(x) graphed here, the limit of g(x) as x approaches -2 is -1, the limit of g(x) as x approaches -1 is 2, and the limit of g(x) as x approaches 0 is 3.
a. limx→−2g(x)
= g(-2)
= -1
Therefore, the limit of g(x) as x approaches -2 is -1.
b. limx→−1g(x)
= g(-1)
= 2
Therefore, the limit of g(x) as x approaches -1 is 2.
c. limx→0g(x)
= g(0)
= 3
Therefore, the limit of g(x) as x approaches 0 is 3.
d. There is no limit of g(x) as x approaches infinity because the graph of the function is not defined at infinity. However, there is no limit of g(x) as x approaches infinity because the graph of the function is not defined at infinity.
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A diagonal of a square mirror is 10 inches long. What is the length, in inches, of each side of the mirror?.
Toula calculates the cost of cementing the bottom and sides of a circular pond. When all costs are considered, the job will cost $175.85 per square metre of finished area. if the pond has a radius of 3 feet and a depth of 2 feet, how much will she charge for the job?
Step-by-step explanation:
First, we need to calculate the area of the pond by using the formula for the area of a circle: πr^2, where r is the radius. In this case, the radius is 3 feet.
3^2 = 9
π * 9 = 28.27 square feet
Now, we need to find the volume of the pond to find the finished area by using the formula for the volume of a cylinder: πr^2h, where r is the radius, h is the height/depth.
28.27 * 2 = 56.54 cubic feet
Finally, we need to convert the square feet to square meters and multiply it by the cost per square meter.
56.54 square feet * 0.092903 square meters/square foot = 5.24 square meters
5.24 square meters * $175.85/square meter = $924.99.
So, the cost of cementing the bottom and sides of the circular pond would be $924.99
Macy's starting balance in her checking account was $654.98. She wrote one check in the amount of $321.45 and made a
deposit that she forgot to write into her check register. At the end of the month, Macy's ending balance was $452.78. Find the
amount of the deposit.
$119.25
$114.38
$121.56
$116.74
Answer:
$119.25.
Step-by-step explanation:
Let's call the deposit amount "D".
Starting balance: $654.98
Check amount: -$321.45
Deposit: +D
Ending balance: $452.78
So, we can set up an equation:
$654.98 - $321.45 + D = $452.78
Solving for D:
$654.98 - $321.45 + D = $452.78
$333.53 + D = $452.78
D = $119.25
So the deposit amount was $119.25.
What i the monthly fee for Jeica’ plan if he doe not end any text meage? Show your work
Answer:
can u screenshot the question then ill solve it!!
Step-by-step explanation:
Bob bought 2 t-shirts at 249. 95 pesos each and a pair of pants for 675. 95. How much change did he get from his 1 000 peso bill?
1. What is asked?
2. What are given?
3. What is/are the hidden question?
4. What operations will you use to solve the problem?
5. What is number sentence?
6. What is the answer?
pasagot po tnx
I will mark you as brainliest if your answer is good :3
The solution is :
1. How much change did he get from 1500-pesos bill.
2. 2 t-shirt 249.95 pesos pants for 675.95 pesos .
3. 2 t-shirt and pair of pants.
4. 249.95 pesos + 675.95 pesos and subtraction 1500-pesos bill.
5. Seventy-four and twenty cents.
6. 74.20 pesos.
Arithmetic is one of the oldest and elementary branches of mathematics, originating from the Greek word ‘Arithmos’, which means ‘number’. Arithmetic involves the study of numbers, especially the properties of traditional operations, such as: Addition, Subtraction, Multiplication, Division.
The arithmetic operators based on these operations are ‘+’, ‘-’, ‘×’ and ‘÷’.
as given, Bobby bought:
2 shirts bought at 249.95 each
Pair of pants for 675.95
So,
How much change did he get from his 1500 peso bill ? (correction : 1500 peso)
2 shirts for 249.95 each.
249.95 * 2 = 499.9
Total spent on the shirts : 499.9
A pair of pants bought for: 675.95
Total money spent = 675.95 + 499.9 = 1175.85
In order to determine the final solution we have to subtract the total amount from 1500.
1500 - 1175.85 =324.15
He will getting a change of 324. 15 pesos .
Therefore, The solution is :
1. How much change did he get from 1500-pesos bill.
2. 2 t-shirt 249.95 pesos pants for 675.95 pesos .
3. 2 t-shirt and pair of pants.
4. 249.95 pesos + 675.95 pesos and subtraction 1500-pesos bill.
5. Seventy-four and twenty cents.
6. 74.20 pesos.
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Bob received 824.15 pesos as change from his 1 000-peso bill.
What is asked?
The question asks for the change Bob received from his 1 000-peso bill.
What are given?
The given information is that Bob bought 2 t-shirts at 249.95 pesos each and a pair of pants for 675.95.
What is/are the hidden question?
The hidden question is: How much did Bob spend in total?
What operations will you use to solve the problem?
To solve the problem, you need to find the total amount that Bob spent, and then subtract it from his 1 000-peso bill to get his change.
What is the number sentence?
The number sentence would be: change = 1 000 pesos - (2 * 249.95 pesos + 675.95 pesos)
What is the answer?
To find the answer, you can calculate the expression in the number sentence:
change = 1 000 pesos - (2 * 249.95 pesos + 675.95 pesos)
change = 1 000 pesos - (499.90 pesos + 675.95 pesos)
change = 1 000 pesos - 1175.85 pesos
change = 824.15 pesos
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A 24-hour digital clock shows the time in hours and minutes.
How many times in one day will it display all four digits 2, 0, 1 and 9 in some order?
Answer:
10
Step-by-step explanation:
All possible permutations except those that are not shown by the clock.
Eg:92:02
Given the following probabilities, compute all joint probabilities.
P(A) = .9
P(B | A) = .4
P(AC) = .1
P(B | AC) = .7
The joint probabilities are: P(A and B) = 0.36, P(A and C) = 0.1, P(AC and B) = 0.07.
What do you mean by Probability?Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. An event with probability 0 cannot occur, and an event with probability 1 is certain to occur. The probability of an event A is usually written as P(A) and is calculated as the number of favorable outcomes divided by the total number of possible outcomes in the sample space. For example, if a fair six-sided die is rolled, the sample space is the set of all possible outcomes (1, 2, 3, 4, 5, 6), and the probability of rolling a 4 is 1/6. Probability is used in many fields to make predictions, draw inferences, and make decisions under uncertainty.
We can use the formula for conditional probability to calculate the joint probabilities:
P(A and B) = P(B | A) * P(A) = .4 * .9 = .36
P(A and C) = P(A) * P(C | A) = .9 * P(C) / P(A) = .9 * .1 / .9 = .1
P(AC and B) = P(B | AC) * P(AC) = .7 * .1 = .07
So, the joint probabilities are:
P(A and B) = .36
P(A and C) = .1
P(AC and B) = .07
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Amar is saving for a trip to Costa Rica with his friends. He has estimated that his travel
expenses will cost between $1,800 and $2,400 total. He has $300 In savings, and he earns
$16.50 per hour at his job. He wants to know how many hours he must work to afford the
trip.
1.
Write an inequality to represent the situation.
The inequality for number of hours Aman worked
90.9 hour < h < 127.27 hour
What is Inequality?Mathematical expressions with inequalities are those in which the two sides are not equal. Contrary to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign.
Given:
Amar travel expenses will cost between $1,800 and $2,400 total.
He has $300 In savings, and he earns $16.50 per hour at his job.
let the number of hours be represented by h.
So, 300 + 16.5h = 1800
16.5h = 1500
h = 90.9
and, if the expense exceeds to $2400
300 + 16.5h = 2400
16.5h = 2100
h = 127.27
Thus, the inequality is
90.9 hour < h < 127.27 hour
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PLEASE ANSWER WILL GIVE BRAINLIEST!!!
The box plot shows the times for sprinters on a track team. A horizontal number line starting at 40 with tick marks every one unit up to 59. The values of 42, 44, 50, 54, and 56 are all marked by the box plot. The graph is titled Sprinters' Run Times, and the line is labeled Time in Seconds. Which value does 50% of the data lie above, and what is it called? 54, the upper quartile 44, the lower quartile 50, the median 42, the median
The value that 50% of the data, is 50 the median.
What is the Box Plot?An illustration of variation in a set of data is called a box and whisker plot. A histogram analysis usually gives an adequate display, but a box and whisker plot can add more detail while enabling the display of different sets of data on the same graph.
Given Data: 42, 44, 50, 54, and 56
The median can be referred to as the second quartile (Q2) or 50th percentile of the data set.
The median value shows that 50% of the data set is above it while 50% is below it also.
So, 50 is the median, and this means that 50% of the data lie above 50.
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30 points! Draw a line representing the "rise" and a line representing the "run" of the line. State the slope of the line in simplest form.
Slope of the line: ___
Answer:
m = 5/4
Step-by-step explanation:
Slope = rise/run or (y2 - y1) / (x2 - x1)
Pick 2 points (0, -3) (4,2)
We see the y increase by 5 and the x increase by 4, so the slope is
m = 5/4
One of the questions asks you to measure the expected acceleration given: Expected acceleration = (5/7)gsin(θ) where g=9.8 m/s2.
The expected acceleration of an object is based on the acceleration due to gravity, which is a constant value of 9.8 m/s2. To calculate the expected acceleration, we use the equation: Expected acceleration = (5/7)gsin(θ), where θ is the angle at which the object is moving.
This equation takes into account the angle of the object and uses the acceleration due to gravity to calculate the expected acceleration. In this equation, the acceleration due to gravity is multiplied by a fraction, which is determined by the angle at which the object is moving. Therefore, the higher the angle at which the object is moving, the greater the expected acceleration.
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problem 3: compute the general solution for: d^2y/dt2 − 2 dy/dt − 15y = e^4t
The general solution for d^2y/dt^2 - 2dy/dt - 15y = e^4t is a sum of a homogeneous solution (c1e^(r1t) + c2e^(r2t)) and a particular solution (Ate^4t).
What is differentiation ?
Differentiation is a method of finding the derivative of a function. Differentiation is a process, in Maths, where we find the instantaneous rate of change in function based on one of its variables.
The general solution for the second-order homogeneous linear ordinary differential equation d^2y/dt^2 - 2dy/dt - 15y = 0 can be found using the characteristic equation:
r^2 - 2r - 15 = 0
The characteristic equation has two roots, which are the eigenvalues of the corresponding homogeneous linear system. Let r1 and r2 be the roots of the characteristic equation, then the general solution for the homogeneous equation is:
y(t) = c1e^(r1t) + c2e^(r2t)
where c1 and c2 are arbitrary constants.
y(t) = c1e^(r1t) + c2e^(r2t) + Ate^4t
The general solution for d^2y/dt^2 - 2dy/dt - 15y = e^4t is a sum of a homogeneous solution (c1e^(r1t) + c2e^(r2t)) and a particular solution (Ate^4t).
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.A polling organization announces that the proportion of American voters who favor congressional term limits is 64%, with a 95%
confidence margin of error of 3%. Which of the following statements is a correct interpretation of 95% confidence?
A. If the poll were conducted again in the same way, there is a 95% chance that the fraction of voters favoring term limits in the
second poll would be between 61% and 67%.
B. There is a 95% probability that the true percent of voters favoring term limits is between 61% and 67%.
C. If the poll were conducted again the same way, there is a 95% probability that the percent of voters favoring term limits in the
second poll would be within 3% of the percent favoring term limits in the first poll.
D. Among 95% of the voters, between 61% and 67% favor term limits.
E. None of the above.
Which shows how to find the value of this expression when x = negative 2 and y = 5?.
The value of this expression is 576/365
What is Expression?
A phrase, also known as an expression in some circumstances, is a group of words or a single word that functions as a grammatical unit. For instance, the English word "the very joyful squirrel" is a noun phrase which contains the adjective phrase "extremely happy". Phrases can be made up of a single word or a whole sentence.
(3x³y⁻²)² = 576/365
Given,
x = -2
y = 5
Applying property of exponents to simplify the expression.
Property: [tex](a^{b} )^{c}[/tex] =[tex]a^{bc}[/tex]
So, we have : (3x³y⁻²)²
= 3²x⁶y⁻⁴
= 9x⁶y⁻¹
Property : a⁻ᵇ
= 1/aᵇ
= 9x⁶ / y⁴
Now, plugging in values of x and y
= 9 (-2)⁶ / 5⁴
= 9*64/625
= 576 / 365
This phrase has a value of 576/365.
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The value of this expression is 576/365
What is Expression?
A phrase, also known as an expression in some circumstances, is a group of words or a single word that functions as a grammatical unit. For instance, the English word "the very joyful squirrel" is a noun phrase which contains the adjective phrase "extremely happy". Phrases can be made up of a single word or a whole sentence.
(3x³y⁻²)² = 576/365
Given,
x = -2
y = 5
Applying property of exponents to simplify the expression.
So, we have : (3x³y⁻²)²
= 3²x⁶y⁻⁴
= 9x⁶y⁻¹
Property : a⁻ᵇ
= 1/aᵇ
= 9x⁶ / y⁴
Now, plugging in values of x and y
= 9 (-2)⁶ / 5⁴
= 9*64/625
= 576 / 365
This phrase has a value of 576/365.
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what is the probability mass function, p(x = x), for x? (round to two decimal places.)
The marginal probability mass function px(x) is:
Px(0) = 0.26
Px(1) = 0.48
Px(2) = 0.26
We have to determine the marginal probability mass function px(x).
We determine the total of each value of y(0, 1, 2) at x = 0 to evaluate the value of Px(0).
The values of y(0, 1, 2) at x = 0 is:
Px(0) = Py(0) + Py(1) + Py(2)
Px(0) = 0.10 + 0.11 + 0.05
Px(0) = 0.26
We determine the total of each value of y(0, 1, 2) at x = 1 to evaluate the value of Px(1).
The values of y(0, 1, 2) at x = 1 is:
Px(1) = Py(0) + Py(1) + Py(2)
Px(1) = 0.17 + 0.23 + 0.08
Px(1) = 0.48
We determine the total of each value of y(0, 1, 2) at x = 2 to evaluate the value of Px(2).
The values of y(0, 1, 2) at x = 2 is:
Px(2) = Py(0) + Py(1) + Py(2)
Px(2) = 0.06 + 0.14 + 0.06
Px(2) = 0.26
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The complete question is:
The joint probability mass function of random variable X and Y is given in the following table:
y
X 0 1 2
0 0.10 0.11 0.05
1 0.17 0.23 0.08
2 0.06 0.14 0.06
Find the marginal probability mass function px(x). Round the answers to two decimal places.
Px(0) =
Px(1) =
Px(2) =
In a pizza retaurant, 72% of the cotumer order pepperoni pizza. Chooe 5 cotumer at random. Find The probability that all 5 cotumer order pepperoni pizza. The probability that at leat 1 doen’t order pepperoni pizza
The probability that all 5 customers order pepperoni pizza is 0.2457 and the probability that at least 1 customer doesn't order pepperoni pizza is 0.7543.
We can calculate the probability that all 5 customers order pepperoni pizza by multiplying the probability that each individual customer orders pepperoni pizza.
The probability of each customer ordering pepperoni pizza is 72%. Therefore, the probability that all 5 customers order pepperoni pizza is (0.72)^5 = 0.2457.
We can calculate the probability that at least 1 customer doesn't order pepperoni pizza by subtracting the probability that all 5 customers order a pepperoni pizza from 1.
Hence, The probability that at least 1 customer doesn't order pepperoni pizza is 1 - 0.2457 = 0.7543.
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An equation is shown below.
√−x=x+6 Which of these statements are true? Choose all that are correct. A.x=−9 satisfies the equation.
в. This equation has exactly one solution. c. This equation has the same solution set as−x=x2+12x+36. D. This equation has the same solution set as−x=x2+36. E.x=−4
Given quadratic equation has same solution set as -x=x²+12x+36 i.e. C.
Quadratic equations: what are they?A second-degree algebraic equation in x is a quadratic equation. Ax2 + bx + c = 0 is the quadratic equation in its standard form, with a and b acting as coefficients, x acting as the variable, and c acting as the constant term.
The coefficient of x2 must not be zero (a 0) for an equation to qualify as a quadratic equation. The x2 term comes first, followed by the x term, and finally the constant term when stating a quadratic equation in standard form. Instead of being written as fractions or decimals, the numerical values a, b, and c are frequently expressed as integral values.
Now,
Given equation is √-x=x+6
-x = (x+6)²
-x = x² +36+12x
x² + 11x + 36 = 0
As it is same quadratic equation as given in option C.
Hence,
Given quadratic equation has same solution set as
-x = x² + 12x + 36.
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Can anyone help me solve this problem step by step?
Step-by-step explanation:
to solve an equation with a variable we want to transform the equation, so that in the end we have the variable alone on one side of the equation, and everything else on the other side.
so, when we look at
(n + 8)/2 - 4 = 16
what blocks us from reading the value of n ?
several things on the left side.
I would start with the "- 4".
to get rid of that from the side of the variable we need to add 4. and to keep the equality true, we need to add this on both sides :
(n + 8)/2 - 4 + 4 = 16 + 4
and we get
(n + 8)/2 = 20
what blocks us now ?
I would say the fraction, this division by 2.
how do we get rid of that ?
by multiplying everything by 2 :
2×(n + 8)/2 = 2×20
and we get
n + 8 = 40
still, something is blocking us from reading the value of n : the "+ 8".
how do we get rid of that ?
by subtracting 8 from both sides :
n + 8 - 8 = 40 - 8
and we get
n = 32
that's it. we are finished. goal achieved.
how to develop the shape of the hole in the cone when it intersects with a cylinder
The generatrix of a cylinder is intersected by a plane parallel to its axis to determine the shape of the hole in the cylinder, while the generatrix of a cone is intersected by a plane via its vertex.
We select cutting planes that intersect both surfaces along their generatrixes in order to construct the intersection points of a cone and a cylinder. A cylinder's generatrix is intersected by a plane parallel to its axis, and a cone's generatrix is intersected by a plane via its vertex.
Since the line passing through the cone's vertex is parallel to the axis of the cylinder, the system of planes that intersect both surfaces in their generatrixes is a pencil-shaped arrangement of planes.
The cone and the cylinder are both intersected by every plane of that pencil in two different generatrixes.
In contrast to the vertices of the surfaces, the junction curve has four points that are intersections of these four lines.
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how much time fo students at your school spend on the internet? you collect sata from the 32 members of your ap statistics class and calculate the mean amount of time that these students spent on the internet yesterday
Calculating the mean amount of time that these students spent on the internet yesterday it is not appropriate because sample is a convenience sample to determine condition for population mean.
In statistics, in addition to the mode and median, the mean is one of the measures of central tendency. Simply put, the mean is the average of the values in the given set. It indicates that values in a particular data set are distributed equally. The three most frequently employed measures of central tendency are the mean, median, and mode.
The total values provided in a datasheet must be added, and the sum must be divided by the total number of values in order to determine the mean. When all of the values are organized in ascending order, the Median is the median value of the given data. While the number in the list that is repeated a maximum of times is the mode.
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PROBLEM SOLVING
The area of a door is given by the expression 7x- 14. The door's length is 7 feet.
7. Write an expression to represent the 8. Evaluate your expression for for x = 6,
width of the door.
to find the door's actual width.
9. Find the area of the door.
10. Find the perimeter of the door.
Answer:
see below
Step-by-step explanation:
area of a door = length * width = 7* width = 7x-14
so width = ( 7x-14 ) / 7 = x-2
when x=6 width = 4
area = 4*7 = 28
perimeter = (4+7)*2 = 22
Rewrite the expression (3^3/43^3/8)^4/9 as an exponent with a single base, 3x
The expression when rewritten is 3^1/2
How to rewrte the expressionFrom the question, we have the following parameters that can be used in our computation:
(3^3/4 * 3^3/8)^4/9
Apply the law of indices
So, we have
(3^9/8)^4/9
Simplify the exponent
3^1/2
Hence, the solution is 3^1/2
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The expression can be rewritten as 3^1/2 = √3
How to rewrite the expression:
We want to simplify the expression below:
(3^3/4 * 3^3/8)^4/9
Remember that we can add these exponents, so we get:
3/4 + 3/8 = 9/8
Then we can rewrite this as:
(3^9/8)^4/9
Takin the product between the exponents we will get:
(3^9/8)^4/9 = 3^(4/8) = 3^1/2
the solution is 3^1/2
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Nathan is 1.25 meters tall. At 1 p.m., he measures the length of a tree's shadow to be
24.65 meters. He stands 20.1 meters away from the tree, so that the tip of his shadow
meets the tip of the tree's shadow. Find the height of the tree to the nearest
hundredth of a meter.
The height of the tree is 6.77 meters.
What are similar triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are identical, their respective sides are equal in number and their corresponding angles are congruent.
Given that, the height of the person is 1.25 meters, the length of the tree's shadow is 24.65 meters, and the distance between the person and the tree is 20.1 meters.
Let the height of the tree be x.
Note that the scenario makes two similar triangles.
Since the ratio of the side lengths of similar triangles is proportional, it follows:
(24.65 - 20.1)/1.25 = 24.65/x
3.64 = 24.65/x
x = 24.65 / 3.64
x = 6.77
Hence, the height of the tree is 6.77 meters.
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suppose that one customer who participated in the study is chosen at random. what is the probability that the customer had a high level of satisfaction and used the company more than five times per month?
The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the favorable outcome is that the customer had a high level of satisfaction and used the company more than five times per month. The total number of possible outcomes is the total number of customers in the study, which is 700.
Therefore, the desired probability can be calculated as:
P(High satisfaction and more than 5 times per month) = 70 / 700
And the answer is P(High satisfaction and more than 5 times per month) = 70 / 700 = 1/10.
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Relative to ground level, a black garden ant digs -20 7/9 feet and a red harvester ant digs -20 39/50 feet.Which ant is closer to ground level?
The red harvester ant is closer to ground level.
What is Relative distance?The relative distance is defined as the product of the relative distance multiplied by the specified time. The relative distance unit is the same as the length unit.
Distance is typically defined as the amount of space between two locations that can be measured. In geography, a few different types of distance are used to explain not only the physical distance between two points but also the variations in distinct racial and ethnic groups, cultural traditions, political ideals, and social mores. A unique unit of measurement known as relative distance is used in geography to explain a variety of societal aspects.
As it has dug deeper (more negative) than the black garden ant. The number -20 39/50 is more negative than -20 7/9.
Therefore, The red harvester ant is closer to ground level.
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Sara's past year at work pushed her to the limit, and she needs to recharge. She wants to splurge by spending 13 of his savings on a monster vacation. Her financial planner suggests that she spend less than $2500 on the vacation. Let x be the amount of money Sara has in savings. a) Write an inequality that best represents this situation. b) Solve the inequality for x. c) What does the value of x in part b represent in this situation.
An inequality that best represents this situation is: x - 1300 > 2500.
How to solve for the inequality?a) Let x be the amount of money Sara has in savings. We know that the amount she wants to spend on vacation is $1300, so we have x - 1300 > 2500.
b) To solve for x, we can add 1300 to both sides: x > 3800.
c) The value of x in this situation represents the amount of money Sara has in savings.
The inequality x > 3800 means that Sara must have more than $3800 in savings in order to spend $1300 on a vacation while following the financial planner's suggestion to spend less than $2500.
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the function y(x) solves the initial value problem below. dy dx = x6 (y 2)2 y(0) = −1
The solution to the ODE dy/dx = x^6 * (y - 2)^2 with initial condition y(0) = -1 is a complicated expression involving the antiderivative of a complex function, which may not have a closed-form solution.
What is function ?
function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
1) Rewrite the ODE into a separable form:
dy/dx = x^6 * (y - 2)^2
2) Integrate both sides of the equation:
∫(dy/dx)dx = ∫x^6 * (y - 2)^2 dx
3) Solve for y on one side of the equation and integrate the other side:
y = 2 + √((C - ∫x^6 * (y - 2)^2 dx)/x^6)
where C is an arbitrary constant of integration.
4) Use the initial condition y(0) = -1 to find the value of C:
-1 = 2 + √((C - ∫0^0 * (y - 2)^2 dx)/0^6)
5) Solve for C:
C = (1 - 2)^2 = 1
6) Substitute C back into the solution for y:
y = 2 + √((1 - ∫x^6 * (y - 2)^2 dx)/x^6)
7) The solution y(x) is a complicated expression that involves the antiderivative of a complex function. It may not have a closed-form solution, but it can be approximated numerically using numerical methods.
The solution to the ODE dy/dx = x^6 * (y - 2)^2 with initial condition y(0) = -1 is a complicated expression involving the antiderivative of a complex function, which may not have a closed-form solution.
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Create a rational function that has the
given lines as asymptotes.
1. y = 0, x = −3, x = 0, and x = 2
2. y = 3/2, x = -1, and x= 1/2
The rational functions with the given asymptotes are presented as follows:
1. y = 1/(x³ + x² - 6x).
2. y = 1.5x²/(x² + 0.5x - 0.5).
What are the asymptotes of a function f(x)?The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.
The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.
For item 1, due to the vertical asymptotes, the denominator is given as follows:
x(x + 3)(x - 2) = x(x² + x - 6) = x³ + x² - 6x.
As the horizontal asymptote is of zero, the numerator has a lower degree than the denominator, hence it can be constant, and the function is given as follows:
y = 1/(x³ + x² - 6x).
For item 2, the denominator is given as follows:
(x + 1)(x - 0.5) = x² + 0.5x - 0.5.
Due to the horizontal asymptote of 3/2, the degree of the numerator must be the same as the denomiator, with a coefficient 3/2 that of the denominator, hence the numerator is given as follows:
3x²/2 = 1.5x².
Meaning that the function is:
y = 1.5x²/(x² + 0.5x - 0.5).
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