2t
Question:
If the coconut from the shorter tree takes time t to reach the ground, how long (in terms of t) will it take the other coconut to reach the ground?
Solution:
We know that the time taken by the coconut to reach the ground is given by the formula:
t = √(2h/g)
Where,
t = time taken to reach the ground
h = height of the tree
g = acceleration due to gravity
If the height of the coconut from the shorter tree is h, then the time taken by it to reach the ground is t.
Now, the coconut from the taller tree has a height of 2h. Therefore, its time taken to reach the ground can be calculated as follows:
t_1 = √(2(2h)/g)
t_1 = √(4h/g)
t_1 = 2√(h/g)
Therefore, the time taken by the coconut from the taller tree to reach the ground is 2 times the time taken by the coconut from the shorter tree to reach the ground.
Hence, the answer is, the time taken by the other coconut to reach the ground is 2t.
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when you take the scholastic assessment test (sat), your score is recorded as a percentile score. if you scored in the 92nd percentile, it means that you scored better than approximately 92% of those who took the test. (a) if lisa's score was 83 and that score was the 29th score from the top in a class of 240 scores, what is lisa's percentile rank? (round your answer to the nearest whole number.) (b) lee has received a percentile rank of 83% in a class of 40 students. what is lee's rank in the class? (round your answer to the nearest whole number.)
Using percentage, Lisa's percentile rank is 88 and Lee's class rank is 34
What is Lisa's percentile rank?(a) To find Lisa's percentile rank, we need to determine how many scores are below hers in the class, then divide by the total number of scores and multiply by 100 to get a percentage.
Since Lisa's score is the 29th from the top, there are 240 - 29 = 211 scores below hers.
Her percentile rank is (211/240) x 100 = 87.92, which rounded to the nearest whole number is 88.
Therefore, Lisa's percentile rank is 88.
(b) To find Lee's rank in the class, we need to determine how many students scored below him, then add 1 (for Lee's own score).
Since Lee's percentile rank is 83%, we know that 83% of the class scored lower than he did.
The number of students who scored lower is (83/100) x 40 = 33.2, which rounded to the nearest whole number is 33.
Therefore, Lee's rank in the class is 33 + 1 = 34.
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Give a counterexample
for the following
conjecture: "an odd number plus another odd
number will always be divisible by 4".
13.
5 and 25
Step-by-step explanation:A single counterexample to any conjecture that states that something will always be true immediately disproves the statement.
Conjecture
Conjecture is a statement or conclusion that is offered with little to no proof. The conclusion given in the question has no mathematical proof to support the idea; thus, it is conjecture. Conjecture is often easy to disprove.
Counterexamples are examples that follow the conditions of a statement (or conjecture) but do not meet the same conclusion. The conditions of the conjecture given are "an odd number plus another odd number". So, a counterexample must also meet these conditions. The conclusion of the conjecture is that this sum " will always be divisible by 4". Thus, the counterexample should not be divisible by 4.
Counterexample
At first, the conjecture may seem to be true because of pairs like 13+15 or 21+23. However, because the conjecture includes the word "always", it only takes one counterexample to completely disprove the statement.
I know that 30 is not divisible by 4, so I can find 2 odd numbers that sum to 30. I pair I chose was 5+25, but there are other possibilities like 13+17 or 1+29. No matter which you choose, it's clear that the conjecture is not true.
For what values of a are the following expressions true?
(PLease help ASAP)
|a +5| = −5−a
|a +5 |=a +5
The equation |a + 5| = −5 − a is not true for any value of a, while the equation |a + 5| = a + 5 is true for values of a greater than or equal to -5.
The absolute value of a number is defined as the distance of the number from zero on a number line. Thus, |a + 5| is equal to the distance of (a + 5) from zero on a number line. Since distance is always non-negative, |a + 5| is always non-negative.
On the other hand, the expression −5 − a is always negative, since the sum of a negative number (-5) and any number (a) is always negative.
Therefore, for the equation |a + 5| = −5 − a to be true, the absolute value of (a + 5) would have to be negative, which is impossible since absolute value is always non-negative.
For the equation |a + 5| = a + 5 to be true, a must be greater than or equal to -5, because when a is greater than or equal to -5, (a + 5) is already non-negative, so the absolute value of (a + 5) is equal to (a + 5).
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a rectangular flag is 29ft long and 17ft wide. if a triangle is cut out of the center of the flag that has a base of 9ft and a height of 2ft , what is the remaining area of the flag?
A rectangular flag is 29ft long and 17ft wide. if a triangle is cut out of the center of the flag that has a base of 9ft and a height of 2ft, So the remaining area of the flag is 484 ft².
The data we get from the question is,
Length of rectangular flag = 29 ft
Width of rectangular flag = 17 ft
Base of triangle = 9 ft
Height of triangle = 2 ft
We need to find the remaining area of the flag when a triangle is cut out of the center of the flag.
We know that the area of the rectangle is given by:
Area of rectangle = length × breadth
Area of rectangle = 29 × 17
= 493 ft²
The area of the triangle is given by:
Area of triangle = (1/2) × base × height
Area of triangle = (1/2) × 9 × 2
= 9 ft²
The remaining area of the flag will be the area of the rectangle minus the area of the triangle.
The remaining area of the flag = Area of the rectangle - Area of the triangle
= 493 - 9
= 484 ft²
Therefore, the remaining area of the flag is 484 ft².
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Find the area of a circle that has a radius of 7/2 centimeters. Use 22/7 as an approximation for π.
Answer: The answer is 38.5 cm^2
Step-by-step explanation:
Given:
Radius(r) = 7/2
Pie (π) = 22/7
We know that:
The area of Circle is = πr^2 = π * r * r
So, ans is = (22/7) * 7/2 * 7/2 = 38.5 cm^2
Answer:
A≈ 38.5cm^2
Step-by-step explanation:
Given:
The approximation of pi is 22/7 or 3.141592
The radius is 7/2 or 3.5
Work:
A=πr(2)=π·3.5(2)≈38.48451
now let's round it to 38.5cm^2
Brayden deposited $990 into a saving account the unrest in the account was 5.7 if he leaves the money in the account for 5 years how much interest will the account occur?
Answer: $5232.15
Step-by-step explanation:
990 x 5.7% = 1046.43
1046.43 x 5 years = 5232.25
What is the median of the data in the table below?
Answer:24
Step-by-step explanation:
A county recreation department cleared a mile long walking trail. If the trail will be marked every mile with a small sign, how many signs are needed?
Answer:
Step-by-step explanation:
If the trail is a mile long and is marked every mile with a sign, then there will be one sign at the end of the trail. In addition, there will be signs marking each mile along the trail, including the starting point. Since there are 1 mile markers between the starting point and the end of the trail, there will be a total of:
1 (at the end) + 1 (at the starting point) + 1 (for each mile marker) = 1 + 1 + 1 = 3
Therefore, three signs are needed to mark a mile-long walking trail.
a trapezoid has a base length of 4 meters and 2.5 meters. if the height of the trapezoid is 4 meters, what is the area of the trapezoid?
The area of the trapezoid is 13 meters.
The area of a trapezoid is calculated by taking the sum of the two bases and multiplying that by the height, then dividing that by two. In this case, the sum of the two bases is 4 meters + 2.5 meters = 6.5 meters. Multiplying this by the height of 4 meters gives us a result of 26 meters. Finally, we divide that by two to get the final area of 13 meters.
To calculate the area of a trapezoid, you first need to calculate the sum of its two bases. The two bases are the bottom and the top lengths of the trapezoid, which in this case are 4 meters and 2.5 meters respectively. Once you have the sum of the two bases, you can then multiply that sum by the height of the trapezoid, which in this case is 4 meters. This will give you the total area of the trapezoid. Lastly, you divide this total area by two to get the final area.
In summary, to calculate the area of a trapezoid with a base length of 4 meters and 2.5 meters, and a height of 4 meters, you need to first calculate the sum of its two bases, which in this case is 6.5 meters. Multiply this by the height of 4 meters to get the total area of 26 meters. Finally, divide this by two to get the final area of the trapezoid, which is 13 meters.
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clark was asked to complete a self-administered questionnaire posted at mysurvey. what type of survey did clark complete?
Clark completed a self-administered questionnaire, which is also known as an online survey.
This type of survey is administered online and is completed by the participant (in this case, Clark) without the assistance of a researcher or interviewer. It can be delivered through an online form or through an email or online survey link.
The questionnaire typically includes multiple-choice questions, short-answer questions, and rating scales that the participant can answer with a click or tap. After the participant completes the survey, their answers are collected and analyzed by the researcher. This type of survey is a cost-effective and convenient way to collect data from a large number of people in a short amount of time.
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If a₁ = 9, and an an-1 - 1 then find the value of a4.
A. 5
B. 6
C. 9
D. 4
Answer:
it’s b
Step-by-step explanation:
I just did it
WORTH 23 POINTS - While reading a news article, Bethany noticed that the word "senselessness" had a lot of repeating letters. Bethany wondered about the probabilities of randomly drawing certain letters if each letter in "senselessness" were written on an index card and put into a bag.
Answer:
To determine which letter has the greatest probability of being selected and its corresponding probability, we need to count the number of times each letter appears in the word "senselessness".
The word "senselessness" contains a total of 12 letters, with 4 distinct letters: "e", "s", "n", and "l". We can count the number of times each letter appears as follows:
"e" appears 4 times
"s" appears 4 times
"n" appears 2 times
"l" appears 2 times
Therefore, the letter "e" and "s" have the greatest probability of being selected, as they appear most frequently in the word. The probability of selecting the letter "e" would be 4/12, which simplifies to 1/3 or approximately 0.33. Similarly, the probability of selecting the letter "s" would also be 1/3 or approximately 0.33.
On the other hand, the letter with the least probability of being selected would be either "n" or "l", as they appear the least frequently. The probability of selecting either "n" or "l" would be 2/12, which simplifies to 1/6 or approximately 0.17.
I'm not sure if this answers your question, but i really hope it helps!
Please mark brainliest though, it really helps!
5. Disha states that the circumference and the area of the larger circle are both double the
circumference and the area of the smaller circle. Is she correct? Explain why or why not.
Disha is not correct, as the area of the larger circle is actually four times the area of the smaller circle, not double.
What is Circumference?Circumference is the distance around the edge of a closed curved shape, such as a circle. In the case of a circle, it is the distance around the circle, or the perimeter of the circle.
According to question:Let "r" denote the smaller circle's radius. Then the circumference of the smaller circle would be 2πr, and the area would be πr².
If the circumference and area of the larger circle are both double the circumference and area of the smaller circle, then:
Circumference of larger circle = 2(2πr) = 4πr
Area of larger circle = 2(πr²) = 2π(r²)
Let "R" represent the larger circle's radius. Then the circumference of the larger circle would be 2πR, and the area would be πR².
If the circumference and area of the larger circle are both double the circumference and area of the smaller circle, then:
Circumference of larger circle = 2(2πr) = 4πr = 2πR
Area of larger circle = 2(πr²) = 2π(r²) = πR²
We may see from the equations above that:
R = 2r
Substituting this value of R in the equation for the area of the larger circle, we get:
Area of larger circle = πR² = π(2r)² = 4πr²
But we already know that the area of the larger circle is 2π(r²). This means that Disha is not correct, as the area of the larger circle is actually four times the area of the smaller circle, not double.
Therefore, Disha's statement is incorrect.
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Question #1: The heights of seven footballers are listed below. 1.9m, 1.82m, 1.78m, 1.8m, 1.88m, 1.86m, 1.7m (a) Arrange the heights in order from smallest to largest. (b) Write down the median height. (c) A player is picked at random. Write down the probability that he is over 1.85m.
Answer:
A) 1.7m, 1.78m, 1.8m, 1.82m, 1.86m, 1.88m, 1.9m
B) The median is 1.82
C) There is a 3/7 chance that the player will be taller than 1.85m
Step-by-step explanation:
The solution is, (a) 1.7m < 1.78m < 1.8m < 1.82m< 1.86m< 1.88m < 1.9m, the heights in order from smallest to largest.
(b) 1.8 the median height.
(c) A player is picked at random, 3/7 is the probability that he is over 1.85m.
What is median?In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value.
here, we have,
given that,
The heights of seven footballers are listed below.
1.9m, 1.82m, 1.78m, 1.8m, 1.88m, 1.86m, 1.7m
now we have to Arrange the heights in order from smallest to largest.
we get,
(a) 1.7m < 1.78m < 1.8m < 1.82m< 1.86m< 1.88m < 1.9m, the heights in order from smallest to largest.
now, the median = 1.7 + 1.9/ 2
= 1.8
(b) 1.8 the median height.
again,
(c) the total no. of outcome = 7
no. of event that he is over 1.85m = 3
so, the probability that he is over 1.85m = 7/3
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a beer and wine store series 10 licensee is allowed to serve beer and wine samples on the premises if it holds sampling privileges. true or false? false true`
False. A Beer and Wine Store Series 10 Licensee is not allowed to serve beer and wine samples on the premises, even if they hold sampling privileges.
This is because beer and wine samples are not allowed to be sold under this type of license, according to the Texas Alcoholic Beverage Code. Under the TABC, beer and wine samples may only be sold in certain locations, such as restaurants and stores that have a TABC permit for the purpose of selling alcoholic beverages for on-premises consumption.
Beer and wine sampling privileges are reserved for other types of establishments, such as bars and taverns, that have a TABC permit for the purpose of selling alcoholic beverages for on-premises consumption. As such, a Beer and Wine Store Series 10 Licensee is not allowed to serve beer and wine samples on the premises, even if they hold sampling privileges.
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ELEPHANTS In the wild, an African elephant typically consumes 2.5 × 10² pounds of food a day. If there are approximately 4.15 × 10⁵ African elephants in the wild, how many pounds of food are consumed by African elephants in one day?
Answer:
Approximately 1.05 × 10⁹ pounds of food are consumed by African elephants in one day.
Step-by-step explanation:
Answer:
Step-by-step explanation:
its 7 or 6\
The work shows the first steps of writing a partial fraction decomposition.
3x+9/(x+1)(x-5)=A/x+1+B/x-5
3 x + 9 = A (x minus 5) + B (x + 1)
3 x + 9 = A x minus 5 A + B x + B
What is the partial fraction decomposition in terms of x?
it's C
The partial fraction decomposition of 3x+9/(x+1)(x-5) in terms of x is C = A/(x+1) + B/(x-5).
To find the values of A and B, we solve the system of equations formed by equating coefficients of x and the constant term in equation 3x + 9 = A(x-5) + B(x+1). Solving for A and B, we get A=-3/4 and B=3/4. Substituting these values back into the partial fraction decomposition, we get C = -3/4/(x+1) + 3/4/(x-5).
Therefore, the partial fraction decomposition of 3x+9/(x+1)(x-5) in terms of x is C = -3/4/(x+1) + 3/4/(x-5).
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Answer:
Answer is C
Step-by-step explanation:
Precalc Edge 2023
Graph the system below and write its solution
Answer:
the answer is one solution and to graph you'll need to make the second equation like this y=-2x+7
Prove that the following statement is false. There exists an integer n such that 6n2 + 27 is prime. To prove the statement is false, prove the negation is true. Write the negation of the statement. For every integer n, 6n² + 27 is prime. For every integer n, 6n2 + 27 is not prime. There exists an integer n, such that 6n2 + 27 is not prime. There exists a composite number q = 6n2 + 27, such that n is an integer. There exists an integer n, such that 6n2 + 27 is prime. Now prove the negation. Suppose n is any integer. Express 6n2 + 27 as the following product: 6n2 + 2 Now is an integer because sums and products of integers are integers. Thus, 6n2 + 27 is not prime because it is a
The negation of the statement "There exists an integer n such that 6n2 + 27 is prime" is "For every integer n, 6n2 + 27 is not prime."
To prove the negation, we can use algebraic manipulation to show that 6n2 + 27 is always composite.
Suppose n is any integer. We can factor out 3 from 6n2 + 27 to get 3(2n2 + 9). Since 2n2 + 9 is always odd (2 times any integer is even, and adding 9 makes it odd), we can further factor it as (2n2 + 9) = (2n2 + 6n + 9 - 6n) = [(2n+3)(n+3)] - 6n.
Substituting this expression back into 3(2n2 + 9), we get 3[(2n+3)(n+3) - 6n]. Since (2n+3)(n+3) - 6n is an integer, 3[(2n+3)(n+3) - 6n] is composite for every integer n. Therefore, 6n2 + 27 is not prime for any integer n.
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When the equation a – bx = cx + d is solved for x, the result is x = startfraction a. minus b. over c. plus d. endfractionuse the general solution to solve 5 – 6x = 8x + 17.
The solution to the equation 5 – 6x = 8x + 17 is x = -6.
Using the general solution for linear equations, we can rewrite the given equation as x = (a - d)/(c + b). This means that the solution for x is equal to the fraction (a - d) divided by (c + b).
Now let's use this general solution to solve the equation 5 – 6x = 8x + 17. First, we need to identify the values of a, b, c, and d in the equation. From the given equation, we can see that a = 5, b = -6, c = 8, and d = 17.
Next, we can substitute these values into the general solution and simplify:
x = (a - d)/(c + b)
x = (5 - 17)/(8 - 6)
x = -12/2
x = -6
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Answer:-6/7
Step-by-step explanation: got it right on ed
The table represents an exponential function. What is the multiplicative rate of change of the function?
The multiplicative rate οf change οf the given expοnential functiοn is 1/5.(1st οptiοn).
What is expοnential functiοn?An expοnential functiοn is a Mathematical functiοn which is οf the fοrm
f (x) = aˣ, where “x” is variable here and a real number and “a” is a cοnstant which is called the base οf the functiοn where a shοuld be greater than zerο.
The expοnential functiοn is given here in tabular fοrm.
Here frοm the table οf expοnential functiοn we get when x= 2 then
y= 2/5
again when x= 3 then y= 2/25
sο (2/5)×(1/5) = 2/25
again fοr x= 4 , y= 2/125
(2/25)×(1/5) = 2/125
sο every time the expοnential functiοn is multiplied by 1/5 and we get the next result.
Hence, The multiplicative rate οf change οf the given expοnential functiοn is 1/5.
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please help solve the math problems
The answer of the given question based on the linear function the answer is option b) g(x) = 3x + 8 is linear and for linear function graph the explanation is given below,
What is Function?In mathematics, function is relationship between two sets of values, where each element in first set (called domain) is paired with exactly one element in the second set (called range).
1) Among the given functions, only g(x) = 3x + 8 is linear. A linear function is one whose graph is a straight line, and it has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
The function f(x) = x²+2x-3 is a quadratic function because its graph is a parabola, not a straight line. Similarly, the function h(x) = 4 is a constant function because its graph is a horizontal line with a constant y-value of 4, not a straight line.
2) To graph the function f(x) = 2x - 7, we can first identify its slope and y-intercept. The slope of the line is 2, which means that for every increase of 1 in x, the value of y increases by 2. The y-intercept is -7, which means that the line crosses the y-axis at the point (0, -7).
Using this information, we can plot the y-intercept at (0, -7) and then use the slope to find other points on the line. For example, if we increase x by 1, y will increase by 2, so we can plot the point (1, -5). We can continue this process to find other points on the line, and then draw a straight line through them to graph the function.
To graph the function f(x) = -2/5x+1, we can follow the same process. The slope of the line is -2/5, which means that for every increase of 1 in x, the value of y decreases by 2/5. The y-intercept is 1, so the line crosses the y-axis at the point (0, 1).
Using this information, we can plot the y-intercept at (0, 1) and then use the slope to find other points on the line. For example, if we increase x by 5, y will decrease by 2, so we can plot the point (5, -1). We can continue this process to find other points on the line, and then draw a straight line through them to graph the function.
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you need to study the satisfaction of customers of a specific restaurant. you decide to randomly select one customer at each table. this would most closely describe which type of sampling technique?
Systematic sampling is the term used to describe the sampling method portrayed in the scenario.
Systematic sampling is a form of probability sampling technique in which a researcher randomly selects a beginning point before selecting every nth item from the target population. The researcher is choosing one consumer at each table in this case, which can be interpreted as choosing every nth customer, where n is the average number of customers per table.
When the researcher wants to select a sample from a pre-defined list and the target population is broad and diverse, systematic sampling is a beneficial strategy. Because it takes less time and resources to choose the sample, it is also a more effective strategy than simple random sampling.
It is crucial to remember that systematic sampling might induce bias if the process of selection follows a pattern or is regular. For instance, bias could be introduced if the researcher only chooses consumers at tables with even numbers, producing a non-representative sample. To eliminate bias and get a representative sample, it is crucial to ensure that the selection procedure is truly random.
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Each dot in the following dot plot represents the height of one toddler at Mrs. Orlando's daycare.
Find the interquartile range (IQR) of the data in the dot plot.
Answer: 1.5
Step-by-step explanation:
The interquartile range (IQR) is a measure of statistical dispersion and is calculated by subtracting the first quartile (Q1) from the third quartile (Q3). For the given dot plot data of the toddler's heights, the data would first be arranged in ascending order, then divided into two halves below and above the median to determine the first and third quartiles.
Explanation:To find the interquartile range (IQR) in the dot plot data of the toddler's heights at Mrs. Orlando's daycare, we first need to understand what the IQR represents. The Interquartile Range is a measure of where the bulk of the values lie in a data set. It's computed by subtracting the first quartile (Q1) from the third quartile (Q3).
As a first step, you would arrange the data points in ascending order. The median would give us the second quartile (Q2). The data set is then divided into two halves, below and above the median. The first quartile (Q1) is the median of the lower half and the third quartile (Q3) is the median of the upper half. Finally, subtract Q1 from Q3 to find the IQR.
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Ezra started collecting lunch boxes when he was 5 years old. He had 26 superhero lunch boxes, 15 outer space lunch boxes, and 17 rock band lunch boxes. Then Ezra's cousin started her own lunch box collection. Ezra wanted to help her get started, so he gave his cousin 19 lunch boxes from his collection. How many lunch boxes does Ezra have left in his collection? lunch boxes
Answer:
he has 39 lunch boxes left
Step-by-step explanation:
26 + 15 + 17 gives you 58 and when you subtract 19 from 58 that leaves you with 39 left
the vertices of a triangle are A(8,-24) B(8,-8) C(16,-8). If the triangle is dilated by a scale factor of 1/4 what will be the coordinates of C
After the triangle is dilated by a scale factor of 1/4, the coordinates of point C' are (10, -14).
How to find the coordinates ?We can use the following formula to find the coordinates of point C' after the triangle has been dilated by a scale factor of 1/4:
C' = (1/4)(C - center) + center
where C denotes the original coordinate of point C and centre denotes the dilation center (which is not specified in the problem).
However, because this is the most common choice and is not explicitly stated, we can assume that the center of dilation is the midpoint of line segment AB. Line segment AB's midpoint is:
((8+8)/2, (-24-8)/2) = (8, -16)
So, in the formula, we can substitute this value for Centre:
C' = (1/4)(C - (8, -16)) + (8, -16)
All that remains is to plug in the coordinates of point C and simplify:
C' = (1/4)((16, -8) - (8, -16)) + (8, -16)
= (1/4)(8, 8) + (8, -16)
= (2, 2) + (8, -16)
= (10, -14)
As a result, the coordinates of point C' after the triangle has been dilated by a factor of 1/4 are (10, -14).
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3) the line that passes through
(8,5) and (1,5)
Slope:
Two square pyramids are joined at their bases.
Each base is 28 cm long. The distance between the
vertices of the combined pyramids is 21 cm. What
is the volume of the solid formed?
The volume of the solid formed using square pyramids are joined at their bases is 5,487.3 unit ².
Using the formula for the volume of a square pyramid, find the volume and then multiply it by two since there are two square pyramids.
Let V be the volume of the solid formed.
⇒ volume of the solid formed - V = 2 ( [tex]\frac{1}{3} *B*h[/tex] )
Now substitute the values,
⇒V = 2 ( [tex]\frac{1}{3}[/tex] * [tex]28^{2}[/tex] * 10.5 )
⇒V = 2 ( [tex]\frac{1}{3}[/tex] * 784 * 10.5 )
⇒V = 2 ( 261.3 * 10.5 )
⇒V = 2 ( 2,743.65 )
⇒V = 5,487.3 unit ²
Therefore, the volume of the solid formed using square pyramids are joined at their bases is 5,487.3 unit ².
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if the path around the edge of a circular lawn is 132 feet, what is the approximate area inside the circular path? complete the answer sentence. solve on paper. then, enter your answer on zearn. you may consider writing your answer to the hundredths place and using the calculator to help you solve.
As a result, the circumference of the circular walkway is approximately 367.06 square feet.
what is circle ?A circle is a geometric shape made up of all points that are equally spaced apart from the center. It is a flat, two-dimensional object whose radius or diameter serves as its sole defining characteristic. A circle's circumference is the distance around it, while the area of the circle is the amount of space inside it.
given
We must deduct the area of the smaller circle from the size of the larger circle in order to determine the area inside the circular path. Let's use the abbreviations "r" for the smaller circle's radius and "R" for the larger circle's radius. We are aware that the circumference of the smaller circle, which is 132 feet, is equal to the distance around the circular path.
The wider circle's circumference is 132 feet.
2πR = 132
R = 21
The breadth of the circular path, which is equal to the difference between the radii of the two circles, can now be used to calculate the radius of the smaller circle:
R - r = diameter of the circle
21 - r = diameter of the circle
r = 18
The smaller circle therefore has a radius of 18 feet.
Inside the arc of the path, there is:
R2 - r2 equals (21 - 18) = (44 - 324) = (117) 367.06 square feet.
As a result, the circumference of the circular walkway is approximately 367.06 square feet.
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a spherical snowball is melting in such a manner that its radius is changing at a constant rate, decreasing from 20 cm to 10 cm in 30 minutes. at what rate, in cm3 per minute, is the volume of the snowball changing at the instant the radius is 9 cm?
The volume of the snowball is decreasing at a rate of approximately 108π cubic centimeters per minute when the radius is 9 cm.
The volume V of a sphere with radius r is given by the formula V = (4/3)πr^3. To find the rate at which the volume is changing with respect to time, we need to take the derivative of V with respect to time t. Using the chain rule, we get:
dV/dt = (dV/dr) * (dr/dt)
Since the radius is changing at a constant rate, we can calculate dr/dt by dividing the change in radius by the time interval:
dr/dt = (10 cm - 20 cm) / (30 minutes) = -1/3 cm/min
To find dV/dr, we can take the derivative of the volume formula with respect to r:
dV/dr = 4πr^2
Substituting the given radius of 9 cm, we get:
dV/dr = 4π(9)^2 = 324π cm^2
Finally, we can substitute these values into the formula for dV/dt:
dV/dt = (dV/dr) * (dr/dt) = 324π cm^2 * (-1/3 cm/min) = -108π cm^3/min
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