The area of the largest rectangle that can be inscribed in the right triangle is calculated to be 6 cm^2
As we know that for a rectangle of width W and length L the area is given as;
A = L × W
Now let's say that L is along the 8 cm side and W is along the 3 cm side, let's find a relation between these two.
When we inscribe the rectangle in the larger triangle the quotient between the catheti of the right triangle must be equal to the quotient of the catheti of the right triangle that is generated.
Therefore that new triangle will have legs equal to the;
8cm - L and W.
Then we have that:
8cm - L / W = 8 cm / 3 cm
L = 8 cm - 8/3 × W
Now we can write L in terms of W so that we can replace that in the area equation to get
A = (8cm - (8/3)×W)×W = 8cm×W - (8/3)×W^2
This is the equation we have to maximize, also this is a quadratic polynomial of negative leading coefficient, which means that the maximum is at the vertex.
As we know the vertex of the general quadratic polynomial;
a × x^2 + b × x + c
is at:
x = -b / 2a
Then, in this scenario, the vertex is at;
W = (-8 cm) / (2 × 8/3) = 1.5 cm
Then the length is given by:
L = 8cm - (8/3) × 1.5cm = 4 cm
Therefore the area of the largest rectangle that can be inscribed in the given right triangle is;
A = 1.5 cm × 4 cm = 6 cm^2
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Select the equation of the line that passes through (3, 4) and (-1, -4).
y= -1/2x+2
y=2x+2
y=2x-2
y=-1/2x+4
none of the above
The equation of the line that passes through (3, 4) and (-1, -4) will be y = 2x - 2. Then the correct option is C.
What is the equation of a line passing through two points?Let the equation of the line pass through (x₁, y₁) and (x₂, y₂).
Then the equation of the line is given as,
[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]
The points are (3, 4) and (-1, -4). Then the equation of the line is given as,
(y + 4) = [(4 + 4) / (3 + 1)](x + 1)
y + 4 = 2(x + 1)
y = 2x + 2 - 2
y = 2x - 2
The equation of the line that passes through (3, 4) and (-1, -4) will be y = 2x - 2. Then the correct option is C.
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Which of the following Which of the following statements is true about the graphs of f(x)=x and g(x)=f(x+7)?statements is true about the graphs of f(x)=x and g(x)=f(x+7)?
A statement which is true about the graphs of f(x) = x and g(x) = f(x + 7) is: C. g(x) is the graph of f(x) translated 7 units to the left.
What is a translation?In Mathematics, the translation a geometric figure or graph to the left simply means subtracting a digit from the value on the x-coordinate of the pre-image while the translation a geometric figure or graph downward simply means subtracting a digit from the value on the y-coordinate (y-axis) of the pre-image.
In Geometry, g(x + 7) or g(x) = f(x + 7) simply means shifting a graph 7 units to the left as illustrated by the graph shown in the image attached below.
In this context, we can reasonably infer and logically deduce that the parent function f(x) = x was shifted by 7 units to the left.
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Complete Question:
Which of the following statements is true about the graphs of f(x)=x and g(x)=f(x+7)?
A. g(x) is the graph of f(x) translated 7 units down.
B g(x) and f(x) have the same y-intercept.
C g(x) is the graph of f(x) translated 7 units to the left.
D g(x) is the graph of f(x) translated 7 units to the right.
if you were trying to store a list of single digit numbers in your short-term memory, about how many should you be able to store on average? group of answer choices 10 2 15 7
If one tries to store a list of single-digit numbers in his or her short-term memory, then he will be able to store 7 numbers on average.
What is short-term memory?
Short-term memory is a concept in psychology that defines the capacity for holding a small amount of information in mind for a short interval. The concept of short-term memory is given by Atkinson-Shiffrin. The information of short-term memory can be transformed into long-term memory through rehearsal.
According to several studies, short-term memory has a space limit of 7 ± 2 chunks of information. The fact is written by a cognitive psychologist of Harvard University named George A. Miller and was published in 1956 in psychological reviews.
Hence, If one tries to store a list of single-digit numbers in his or her short-term memory, then he will be able to store 7 numbers on average.
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a binomial experiment consists of 10 independent trials. the probability of success in each trial is 0.43. give the variance of the random variable associated with this experiment. a) 1.5656 b) 4.3000 c) 0.2451 d) 2.4510 e) 2.0736 f) none of the above.
Part (d) is correct.
i.e. The variance of the random variable associated with this given experiment is 2.4510.
What is Binomial experiment?
It is an experiment with a set number of independent trials, each having the same chance of success and the two outcomes of success or failure. A binominal distribution can be used to describe the likelihood of a certain number of successes.
Given that,
No. of independent trials ( n ) = 10
Probability of success in each trail ( p ) = 0.43
∴ probability of failure in each trial ( q ) = 0.57
we can find the variance of a binomial experiment by using the formula i.e. npq.
∴ Variance = npq
= 10 × 0.43 × 0.57
= 2.4510.
Hence, part (d) matches with the answer.
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a university charges students $2,300 less for tuition each year to encourage them to stay in school. the tuition for freshman year is $34,000. write a regression equation for this formula, with y
a regression equation for this formula, where x represents the number of years enrolled and y represents the tuition.
y = 34000 - 2300x
Regression is a statistical technique that connects one or more independent (explanatory) variables to a dependent variable. If there is a relationship between changes in one or more of the explanatory factors and changes in the dependent variable, it can be shown using a regression model.
The regression equation for this formula would be:
Y = 34,000 - 2,300x
where Y is the tuition cost and x is the number of years in school.
The equation states that the tuition cost for any year in school (x) is equal to the freshman year tuition cost (34,000) minus 2,300 multiplied by the number of years in school (x).For example, if x = 4 (4 years in school), then the tuition cost for that year would be 34,000 - 2,300(4) = 25,400.
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Why do we take the square root of the mean squared error to calculate the root mean squared error?.
The square in RMSE is utilized because it consistently provides a positive number for error, preventing errors from canceling one another out, and grants higher weight to values further from the target function, emphasizing locations for which the estimator is unreliable. To reverse the results of squaring, use the square root.
What is RMSE?
The difference between values (sample or population values) predicted by a model or estimator and the values observed is typically measured using the root-mean-square deviation (RMSD) or root-mean-square error (RMSE).
The RMSD is the quadratic mean of the square root of the second sample moment of the discrepancies between expected values and observed values.
When computations are made outside of the data sample that was used for estimation, the deviations are referred to as errors (or prediction errors) instead of residuals.
The RMSD is used to combine the sizes of predictions' mistakes for different data points into a single indicator of predictive power. .
RMSD is a metric for accuracy that is used to compare the predicting mistakes of various models.
Hence, The square in RMSE is utilized because it consistently provides a positive number for error, preventing errors from canceling one another out, and grants higher weight to values further from the target function, emphasizing locations for which the estimator is unreliable. To reverse the results of squaring, use the square root.
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sonia can paint a room in 12 hours. roman can paint the same room in 15 hours. how long does it take for both sonia and roman to paint the room it they are working together
It will take 7 hours to paint a room when Sonia and Roman works together.
Given,
Time taken by Sonia to paint a room = 12 hours
Time taken by Roma to paint the same room = 15 hours
We have to find the time taken by both of them to paint the room if they work together.
Here,
Sonia takes 12 hours
Roman takes 15 hours
LCM of 12 and 15 = 60
12 × 5 = 60
15 × 4 = 60
Total number of works is 60
Now,
Number of works done by Sonia = 5 in an hour
Number of works done by Roman = 4 in an hour
So,
5 + 4 = 9 work can be done by them together in an hour
Then,
Time taken to complete 60 works;
60/9 = 6.66 ≈ 7 hours
That is,
In 7 hours Sonia and Roman paint a room together.
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a cheese merchant looks again at the data set about total cheese sales, in which the sample mean is 164.5 and the sample standard deviation is 103. find the rounded upper level of the 68% confidence interval estimate.
The rounded upper level of the 68% confidence interval estimate is
164.5 ± 3.260.
Given:
a cheese merchant looks a again at the data set about total cheese of sales, in which the sample mean is 164.5 and the sample of standard deviation is 103.
μ = x =164.5
σ = s = 103
n = 987
z score at 68 % :
= 0.9945
Confidence interval CI = x + z * s/[tex]\sqrt{n}[/tex]
= 164.5 + 0.9945 * 103 / √987
= 164.5 ± 3.260
Therefore The rounded upper level of the 68% confidence interval estimate is 164.5 ± 3.260.
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Please help I’m timed
If g(n) varies inversely with n and g(n)= 10 when n=3 then find the value of n when
g(n)= 3
Round final answer to the tenths place. If answer is a whole number then put a zero
in the tenths place before entering your answer. Work must be shown for this
problem in order to receive any credit.
The value of n when the function g(n) assumes a value of 3 is of:
n = 10.
What is a proportional relationship?In the case of an inverse proportional relationship, as is the case in this problem, the output variable y is obtained as the division of the constant of proportionality k by the input variable x, as follows:
y = k/x.
Considering the variables in this problem, the equation is given as follows:
g(n) = k/n.
g(n)= 10 when n=3, hence the constant is obtained as follows:
10 = k/3
k = 3 x 10
k = 30.
Hence the equation is:
g(n) = 30/n.
When the input variable n when g(n) = 3 is obtained as follows:
3 = 30/n
3n = 30
n = 30/3
n = 10.
(we replaced the output variable g(n) by it's value and solved for the input variable)
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based on a sample of size 41, a 95% confidence interval for the mean score of all students, μ, on an aptitude test is from 60 to 66. find the margin of error.
To find the margin of error we have to use the formula,
Margin of error = Z * ơ / √n ,
Where
z = critical factor
ơ = population standard deviation
n = sample size
Now we have to find confidence interval
Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.
The confidence interval of a statistical data can be written as.
x+/-zr/√n
x+/-ME
Where margin of error ME = zr/√n
Given that;
Mean = ц
Standard deviation r = 25
Number of samples n = 41
Confidence interval = 95%
z(at 95% confidence) = 1.96
Xbar = 60.............(1)
Xbar + E = 66.........(2)
from equating both the equations we get
E= 6
and for finding the value of margin of error we know that its the half of the E
Margin of error = E/2
=6/2
=3
Therefore , the Margin of error is 3
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two poles are connected by a wire that is also connected to the ground. the first pole is 20 ft tall and the second pole is 10 ft tall. there is a distance of 30 ft between the two poles. where should the wire be anchored to the ground (in ft, from the shorter pole) to minimize the amount of wire needed?
The wire should be anchored to the ground to minimize the amount of wire needed 20 ft from the left pole.
What is the quadratic equation?
ax2 + bx + c = 0 is a quadratic equation, which is a second-order polynomial equation in a single variable. a 0. It has at least one solution because it is a second-order polynomial equation, which is guaranteed by the algebraic fundamental theorem. The answer could be simple or complicated.
Let x (ft) be the distance from the base of the left pole to the anchor point. Since 2 poles has a distance of 30 ft, the distance from the anchor point to the right pole is 30 - x.
The wire length from the top of the left pole to the anchor point is
[tex]L^{2} _{l} = 20^{2} + x^{2} = x^{2} + 400[/tex]
Similarly for the 2nd part of the wire
[tex]L^{2} _{t} = 10^{2} + (30 - x^{2})[/tex]
= 100 + 900 - 60x + x²
= x²+ 1000 - 60x
We want to minimize the length of the wires, or [tex]L^{2} _{l} + L^{2} _{r}[/tex]
[tex]\frac{x}{\sqrt{{x}^2 + 400} } + \frac{x - 30}{\sqrt{{x}^2 + 400}} = 0[/tex]
we can take the first derivative of this and set it to 0
x = 20
Hence, the wire should be anchored to the ground to minimize the amount of wire needed 20ft.
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Bicycle Parade There was a parade of bicycles and tricycles. Jackie counted the number of wheels; there were 23 wheels. Tony counted the riders; 10 children were riding. How many bicycles were there and how many tricycles?
Convert these unlike fractions to equivalent like fractions and add them. You must use the lcd to get the answer correct. If possible, reduce the final sum. 14512.
Unlike fraction 1/7 is converted to equivalent like fraction 2/14 and the sum of 2/14 + 3/14 = 5/14.
As given in the question,
Given fraction is equal to:
1/7=? + 3/14=?
Here there are two fractions
1/7 and 3/14
LCD( least common denominator) of 7 and 14 is equal to 14.
Now make them like fractions
1/7 is equivalent to
( 1/7) × ( 2/2) = 2/ 14
Now 2/14 and 3/14 are like fractions
Sum of like fractions is:
2/14 + 3/14
= ( 2+ 3) / 14
= 5/ 14
Therefore, the conversion of unlike fraction 1/7 to like fraction is 2/14. And sum of 2/14 + 3/14 = 5/14.
The complete question is:
Convert these unlike fractions to equivalent like fractions and add them. You must use the LCD to get the answer correct. If possible, reduce the final sum. 1/7=? + 3/14=?
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suppose x is a normal random variable with exam image and exam image find p(3.6 < x < 53.5). a) exam image b) exam image c) exam image d) exam image e) exam image f) none of the above.
After using the test statistic z, P(3.6<X<53.5) = 0.967.
In the given question, suppose X is a normal random variable with mean 35 and sdev 10.
We have to find P(3.6<X<53.5).
Fro the given question,
Mean x = 35
sdev s = 10
Z = X - mu / sd
P(3.6<X<53.5) = P ( 3.6-35/10 < Z < 53.5-35/10 )
P(3.6<X<53.5) = P ( -31.4/10 < Z < 18.5/10 )
P(3.6<X<53.5) = P ( -3.14 < Z < 1.85 )
P(3.6<X<53.5) = P ( Z < 1.85) - P ( Z < -3.14)
left tail of both
P(3.6<X<53.5) = 0.9678 - 0.0008
P(3.6<X<53.5) = 0.967
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The right answer is:
Suppose X is a normal random variable with mean 35 and sdev 10. Find P(3.6<X<53.5)
A school staff meeting had 60 teachers in attendance, 30% of whom were first-year teachers. How many first-year teachers were in the meeting?
a woman decides to purchase a $23,800 car and makes a down payment of $3,500. she arranges to borrow the remainder from a bank that will charge her 4.5% interest. she will make monthly payments for 4 years. a. find her monthly payment amount.
The monthly payment amount a woman has to pay for the car is $462.91.
Here we have to find the monthly payment amount.
The cost of car = $23,800
Down payment = $3,500
Rate(r) = 4.5%
time(t) = 4 years
Periodic interest(i) = 0.045/12
Formula to calculate monthly payment:
A = P i( 1 + i[tex])^{n}[/tex] / ( 1 + i[tex])^{n}[/tex] - 1
Loan amount = 23800 - 3500
= $ 20300
A = 20300 ( 0.045 /12 ( 1 + 0.045/12[tex])^{48}[/tex])/ (1 + 0.045/12[tex])^{48}[/tex] - 1
= 20300× 0.004488 / 0.196814
= 462.91
Therefore the monthly payment is $462.91.
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Is the converse of the following conditional True or False? "If a polygon is a triangle, then it has exactly three sides."
The converse of the following conditional True which is If a polygon is a triangle, then it has exactly three sides."
What is the triangle?In terms of geometry, a triangle is a three-sided polygon with three edges and three vertices. The triangle's interior angles add up to 180°.
A polygon has precisely three sides if it is a triangle. The opposite is also accurate.
Contrarily, a polygon is a triangle if it has precisely three sides. Both of the statements can be joined to create a biconditional because they are both true.
Thus, the converse of the following conditional True which is If a polygon is a triangle, then it has exactly three sides."
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Help me solve this question!
The volume of foam needed for the 45 cones is
v = 16,956 cm³
How much foam is needed to make 45 cones?
We know that the volume of a cone of diameter F and height H is:
V = pi*(D/2)²*H/3
Where pi = 3.14
Here we know that:
D = 12cm
H = 10cm
And we want to find the volume of 45 of these cones, so the total volume of foam needed is:
v = 45*V = 45*(3.14*(12cm/2)²*10cm/3
v = 16,956 cm³
That is the volume of foam needed.
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The longest side of a triangle is a cm. The second side is b cm shorter than the first side. The third side is half as long as the second side.
Write an expression, in its simplest form, for the perimeter of the triangle.
Answer:
2.5a - 1.5 b or
5/2a - 3/2b
Step-by-step explanation:
a + (a - b) + 0.5(a - b) =
a + a - b + 0.5a - 0.5b =
2a - b + 0.5a - 0.5b =
2a - 1.5 b + 0.5a =
2.5a - 1.5 b
Math Question Attached
The slope of the straight line is 4/5
Slope of a LineThe slope of a line is a measure of the ratio between the rise and run of that line. The slope of a line is an integral quantity used to calculate the equation of a line
In general, to find the slope of a line, we need to have the values of any two different coordinates on the line.
The slope of a line basically the change in y-coordinate with respect to change in x-coordinate.
This is easily done with a formula given as
m = y₂ - y₁ / x₂ - x₁
Taking data points from the graph attached;
A = (5, 2)B = (0, -2)Substituting the values into the slope equation;
m = -2 - 2 / 0 - 5
m = -4 / - 5
m = 4/5
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Exponential Growth
Graph: y = 2*
20
-1
O
X
1
2
y
Growth
b>1
Answer:
it's 20 if u look closely
a sample of 17001700 computer chips revealed that 366% of the chips fail in the first 10001000 hours of their use. the company's promotional literature states that 399% of the chips fail in the first 10001000 hours of their use. the quality control manager wants to test the claim that the actual percentage that fail is less than the stated percentage. is there enough evidence at the 0.020.02 level to support the manager's claim? step 1 of 7 : state the null and alternative hypotheses. answer
After solving, the value of Test statistic z is 1.449.
In the given question,
A sample of 1700 computer chips revealed that 36% of the chips fail in the first 1000 hours of their use.
The company's promotional literature states that 39% of the chips fail in the first 1000 hours of their use.
Then we have to find the value of z statistic.
Hypothesized proportion (p) = 36%=0.36
Sample proportion (p') = 39% = 0.39
Sample size (n) = 1700
Hence,
Test statistic z
z= (0.76-0.78)/√(0.78* 0.22/900)
z= (-0.02)/√(0.78* 0.000244)
z= -0.02/√0.00019032
z= -0.02/0.0138
z = -1.449
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Eliza solves the equation (x-7)^2 = 25 and gets X = 12. Winnie solves the same equation and finds a different solution. What is the other solution to the problem?
Answer: 5^2
Step-by-step explanation:
5 x 5 = 25
and (x-7)^2 = 25
Which expression is equivalent to ? Assume . m=0 n=0
7m-(7n-18m) is an expression which is equivalent to (2mn)^4/6m^-3 n^-2 Assume m=0 n=0
What is a expression in maths?
An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division. The structure of an expression is: Expression is (Number/variable, Math Operator, Number/variable)
for example 4m +5 where 4 is the coefficient of m and 5 is the constant term in other word 4m and 5 are the term of the expression and m is the variable in the expression and addition is the operator.
m=0, n=0.
8m^7n^6/3 10m^7n^6/3 8m^16n^12/3 m^4n^6/3
-7m-(3n-(8m-4n-10m)
-7m-(3n-(8m-4n+10m))
-7m-(3n-(18m-4n))
-7m-(3n-18m+4n)
-7m-(7n-18m)
complete question is
Which expression is equivalent to (2mn)^4/6m^-3 n^-2? Assume m=0 n=0
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Barbara ell iced tea for $1. 49 per bottle and water for $1. 25 per bottle. She wrote an equation to find the number of bottle he need to ell to earn $100
The equation representing the number of bottle Barbara need to sell to earn $100 is; 1.49x + 1.25y = 100.
Define the term arithmetic equation?In order to find the next item in the pattern, a specific number is added to each item in the series. The number added, also known as the common difference, must be exactly the same for every phrase in the series.For the given question-
Price of each iced tea bottle = $1.49.Price of each water bottle = $1.25.Let the number of iced tea bottle be 'x'.
Let the number of water bottle be 'y'.
Total price = 100
Thus, the equation for the total cost becomes-
1.49x + 1.25y = 100.
Therefore, the equation representing the number of bottle Barbara need to sell to earn $100 is; 1.49x + 1.25y = 100.
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The Rodriguez family ate One-third of a pan of brownies, and they plan to divide the remaining brownies into two equal parts. They will take one part to their block party and the other part to share with co-workers. How much of the whole pan of brownies will each place get?
A model has 2 shaded parts and 1 unshaded part.
Answer:
1
Step-by-step explanation:
The equation to represent this would be (1 - (1 / 3)) / 2
1 - (1 / 3) = 2 / 3
2 / 3 / 2 = 1 / 3
Calculate the margin of error for a ampling ditribution that ha a ize of 36 for a 93% confidence level. The population tandard deviation i known to be 12
The margin of error for a sampling distribution is found as 3.34.
Explain the term margin of error?The margin of error, sometimes known as the confidence interval, provides information on how closely your survey results will likely reflect the opinions of the general community. Keep in mind that conducting surveys requires you to strike a balance by using a smaller group (the survey respondents) to represent a much broader one.Margin of error = z × σ/√n
n = sample size (36)σ = population standard deviation (12) z = z-scoreThus, put the values;
z-score for 93% confidence level = 1.67
Margin of error = 1.67 ×12/√36
Margin of error = 3.34
Thus, the margin of error for a sampling distribution is found as 3.34.
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assuming the population standated deviation is known, which of the following statements is true about the standard deviation of the sample mean x
Assuming the population standard deviation is known and the standard deviation of the sample mean x : option (b) is correct : the standard error of the sample mean decreases.
Central Limit theory:In probability theory, the central limit theorem (CLT) states that the distribution of a sample variable approximates a normal distribution (i.e., a “bell curve”) as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population's actual distribution shape.
Standard deviation:Standard deviation is a calculation of the dispersion or variation in a set of numbers. If the standard deviation is a small number, it means the data points are close to their average value. If the deviation is large, it means the numbers are spread out, further from the mean or average.
There are different ways to write out the steps of the population standard deviation calculation into an equation. A common equation is:
σ = ([Σ(x - u)2]/N)1/2
Where:
σ is the population standard deviation
Σ represents the sum or total from 1 to N
x is an individual value
u is the average of the population
N is the total number of the population
Sample Mean:A sample mean is an average of a set of data . The sample mean can be used to calculate the central tendency, standard deviation and the variance of a data set. The sample mean can be applied to a variety of uses, including calculating population averages.
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a helicopter is lifting an 800-pound unit at a rate of 200 feet per minute. how much horsepower, of work energy, is the helicopter using in the process?
4.84 HP is the amount of work energy used.
Given info,
A helicopter is lifting an 800-pound unit at a rate of 200 feet per minute.
1 horsepower = 550 foot-pounds per second
⇒ 800 x 200 = 160,000 foot-pounds per minute
= 160,000/60 foot pounds per second.
Horsepower = 160,000/60 * 550 = 428/33
= 4.848 HP
Therefore, the horsepower of work energy used is 4.84 HP
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Pam, a frequent business traveler, is considering signing up for a hotel rewards program. Hotel Elegance is currently offering 748 points for signing up, plus 86 points for every night Pam books at the hotel. Alternatively, Hotel Paradiso is offering a sign-up bonus of 734 points, plus 87 points per night. If Pam books a certain number of nights, the points she earns with either hotel will be the same. How many nights will that be? How many points will she have?
Answer:
Step-by-step explanation: x is the number of days she is booked.
List an equation
748+86x=734+87x
x=14
14 nights and plugging x into the equation 1952 points