Answer:
The price per t-shirts is $8.5.
Step-by-step explanation:
Let us assume the price per t- shirt = $m
The cost of 75 t-shirts = $637.50
Now, the total cost for 75 t-shirts = 75 x ( cost of 1 t-shirt)
⇒ $637.50 = 75 x ( m)
⇒ m = $637.50 / 75
or, m = 8.5
Hence, the price per t-shirts is $8.5.
An automobile travels past the farmhouse at a speed of v = 45 km/h. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 3.7 km past the intersection of the highway and the road?
Answer:
[tex]\frac{ds}{dt} = 39.586 km/h[/tex]
Step-by-step explanation:
let distance between farmhouse and road is 2 km
From diagram given
p is the distance between road and past the intersection of highway
By using Pythagoras theorem
[tex]s^2 = 2^2 +p^2[/tex]
differentiate wrt t
we get
[tex]\frac{d}{dt} s^2 = \frac{d}{dt} (4 + p^2)[/tex]
[tex]2s \frac{ds}{dt} =2p \frac{dp}{dt} [/tex]
[tex]\frac{ds}{dt} = \frac{p}{s}\frac{dp}{dt}[/tex]
[tex]\frac{ds}{dt} = \frac{p}{\sqrt{p^2 +4}} \frac{dp}{dt}[/tex]
putting p = 3.7 km
[tex]\frac{ds}{dt} = \frac{3.7}{\sqrt{3.7^2 +4}} 45[/tex]
[tex]\frac{ds}{dt} = 39.586 km/h[/tex]
The distance between the automobile and the farmhouse is increasing at the automobile's constant speed of 45 km/h, which is the same as the car's rate of change of distance as it moves away from the farmhouse.
Explanation:The question asks us to determine how fast the distance is increasing between an automobile and a farmhouse when the car is 3.7 km past a certain intersection, given the car's speed is 45 km/h. This is a problem that can be solved using the concepts of rates of change and kinematics.
Given that the car is moving in a straight line away from the farmhouse and there are no other factors altering the speed, the rate at which the distance between the car and farmhouse is increasing is constant and is equal to the speed of the car.
Since the car's speed is constant at 45 km/h, and it moves directly away from the farmhouse without any acceleration or deceleration, the rate at which the distance increases is exactly the car's speed. Therefore, when the car is 3.7 km past the intersection, the distance between the car and the farmhouse is still increasing at 45 km/h.
This straightforward problem shows that when an object moves away from a point at constant speed, the rate at which the distance between the object and the point increases is simply the speed of the object. This concept is very useful in solving more than 100 questions involving rates of change in kinematics, which is a part of classical mechanics.
In the game of Dubblefud, red chips, blue chips and green chips are each worth 2, 4 and 5 points respectively. In a certain selection of chips, the product of the point values of the chips is 16,000. If the number of blue chips in this selection equals the number of green chips, how many red chips are in the selection?A. 1
B. 2
C. 3
D. 4
E. 5
Final answer:
The problem is solved by finding the prime factorization of 16,000 and assigning the factors to the chips based on their point values, resulting in there being 3 red chips.
Explanation:
The student's question is a typical problem in combinatorics, a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. This problem also involves prime factorization as a method to solve for the number of chips.
To solve the problem, let us denote by R, B, and G the number of red, blue, and green chips respectively. Since the blue chips and green chips have the same quantity, the problem can be solved by first finding the prime factorization of the point value product of 16,000, which is 2⁷ × 5³, and then distributing these prime factors to match the point values of the chips.
Since the point values of blue and green chips are 4 (2²) and 5 respectively and their quantities are equal, we match the prime factors of 5 first. There are 3 factors of 5, so we assign one to each blue and green chip, resulting in 1 remaining.
Then, we match 2² or 4 to each blue and green chip, using 4 out of 7 factors of 2, leaving us with 3 factors of 2, which can be matched with 3 red chips since they have a value of 2 each.
Therefore, we have 3 red chips which is answer option C.
A fountain has two drainage valves. With only the first valve open, the fountain completely drains in 4 hours. With only the second valve open, the fountain completely drains in 5.25 hours. About how many hours will the fountain take to completely drain with both valves open?
You can use variables to model the situation and convert the description to mathematical expression.
The time that the fountain will take to get drained completely with both valves open is approximately 2.27 hours.
How to form mathematical expression from the given description?You can represent the unknown amounts by the use of variables. Follow whatever the description is and convert it one by one mathematically. For example if it is asked to increase some item by 4 , then you can add 4 in that item to increase it by 4. If something is for example, doubled, then you can multiply that thing by 2 and so on methods can be used to convert description to mathematical expressions.
Using above methodology to get to the solutionLet the fountain contains V amount of water
Let the first valve emits x liter of water per hour
Let the second valve emits y liter of water per hour
Then, from the given description, we have:
With only the first valve open, the fountain completely drains in 4 hours
or
[tex]x \times 4 = V\\\\x = \dfrac{V}{4}[/tex] (it is since V volume of water is drained when we let x liter of water drained for 4 hours, thus adding x 4 times which is equivalent of x times 4)
Similarly,
With only the second valve open, the fountain completely drains in 5.25 hours
or
[tex]y \times 5.25 = V\\y = \dfrac{V}{5.25}[/tex]
Let after h hours, the fountain gets drained, then,
[tex]x \times h + y \times h = V\\\\\dfrac{V}{4}h + \dfrac{V}{5.25}\times h = V\\\\\text{Multiplying both the sides with} \: \dfrac{4 \times 5.25}{V}\\\\5.25 \timses h + 4 h = 4 \times 5.25\\9.25h = 21\\\\h = \dfrac{21}{9.25} \approx 2.27[/tex]
Thus,
The time that the fountain will take to get drained completely with both valves open is approximately 2.27 hours.
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Final answer:
By adding the work rates of the two valves and then calculating the reciprocal of the combined rate, it is estimated that it will take approximately 2.27 hours for the fountain to drain with both valves open.
Explanation:
To find out how long it will take for the fountain to drain with both valves open, you can use the rate of work formula, which states that work is the product of rate and time. When dealing with two independent work rates that contribute to completing a single job, we can add their work rates together to find the combined work rate.
Valve 1's rate of work is draining the fountain in 4 hours, which means its rate is 1/4 fountain per hour. Valve 2's rate of work is draining the fountain in 5.25 hours, which means its rate is 1/5.25 fountain per hour.
To find the combined rate, we add these two rates together:
1/4 + 1/5.25 = 0.25 + 0.19 = 0.44 (approximately)
This combined rate is 0.44 fountain per hour. Now, to find the total time to drain the fountain with both valves open, we take the reciprocal of the combined rate:
1 / 0.44 = 2.27 hours (approximately)
Therefore, it will take approximately 2.27 hours for the fountain to drain with both valves open.
As students move to thinking about formulas it supports their conceptual knowledge of how the perimeter of rectangles can be put into general form. What formula below displays a common student error for finding the perimeter?
A) P = l + w + l + w
B) P = l + w
C) P = 2l + 2w
D) P = 2(l + w)
Answer:
B. [tex]P=l+w[/tex]
Step-by-step explanation:
Let l represent length of rectangle and w represent width of rectangle.
We have been given four formulas for the perimeter of rectangle. We are asked to choose the formula that displays a common student error for finding the perimeter.
We know that perimeter of a polygon is sum of all sides of the polygon. We also know that rectangle is polygon having two sets of equal sides.
The perimeter, P, of rectangle will be sum of its all sides that is:
[tex]P=l+w+l+w[/tex]
Combine like terms:
[tex]P=2l+2w[/tex]
Factor out 2:
[tex]P=2(l+w)[/tex]
Upon looking at our given choices, we can see that formula represented by option B displays a common student error for finding the perimeter.
A recent survey showed that in a sample of 100 elementary school teachers, 15 were single. In a sample of 180 high school teachers, 36 were single. Is the proportion of high school teachers who were single greater than the proportion of elementary teachers who were single? Use α = 0.01.
Answer:
By using hypothesis test at α = 0.01, we cannot conclude that the proportion of high school teachers who were single greater than the proportion of elementary teachers who were single
Step-by-step explanation:
let p1 be the proportion of elementary teachers who were single
let p2 be the proportion of high school teachers who were single
Then, the null and alternative hypotheses are:
[tex]H_{0}[/tex]: p2=p1
[tex]H_{a}[/tex]: p2>p1
We need to calculate the test statistic of the sample proportion for elementary teachers who were single.
It can be calculated as follows:
[tex]\frac{p(s)-p}{\sqrt{\frac{p*(1-p)}{N} } }[/tex] where
p(s) is the sample proportion of high school teachers who were single ([tex]\frac{36}{180} =0.2[/tex])p is the proportion of elementary teachers who were single ([tex]\frac{15}{100} =0.15[/tex])N is the sample size (180)Using the numbers, we get
[tex]\frac{0.2-0.15}{\sqrt{\frac{0.15*0.85}{180} } }[/tex] ≈ 1.88
Using z-table, corresponding P-Value is ≈0.03
Since 0.03>0.01 we fail to reject the null hypothesis. (The result is not significant at α = 0.01)
The proportion of single high school teachers (0.20) is greater than the proportion of single elementary school teachers (0.15). However, further statistical testing would be required to determine if this difference is significant.
Explanation:To answer the student's question regarding proportions, we first need to calculate the proportion of single teachers in both samples. For elementary school teachers, the proportion is 15 out of 100, or 0.15. For high school teachers, the proportion is 36 out of 180 or 0.20.
Now, to determine if the proportion of high school teachers who were single is statistically greater than the proportion of elementary school teachers, we would typically perform a hypothesis test for the difference in proportions. However, in this simplified comparison, we can see that the proportion of single high school teachers (0.20) is indeed greater than the proportion of single elementary school teachers (0.15).
It's important to note that this does not mean there is a significant difference, we would need to conduct a significance test (like a Z-test for two proportions at the α = 0.01 level ) to determine this.
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a motorboat takes 3 hours to travel 108 mi going upstream. the return trip takes 2 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?
1.Rate of the boat in still water is_________.
2.Rte of the current is__________.
Answer: The speed of the motorboat in still water is 45mph
The current rate is 9mph.
Step-by-step explanation:
Let u be the motorboat speed in still water and v be the current rate.
The effective speed going upstream is
Distance÷Time = 108÷3 = 36mph
It is the DIFFERENCE of the motorboat speed in still water and the rate of the current. It gives you your first equation
u - v = 36. (1)
The effective speed going downstream is
Distance÷Time = 108÷2 = 54mph
It is the SUM of the motorboat speed in still water and the rate of the current. It gives you your second equation
u + v = 54. (2)
Thus you have this system of two equations in 2 unknowns
u - v = 36, (1) and
u + v = 54. (2)
Add the two equations. You will get
2u = 36 + 54 = 90 ====> u = 90÷2 = 45mph
So, you just found the speed of the motorboat in still water. It is 45mph
Then from the equation (2) you get v = 54 = 45 = 9mph is the current rate.
Answer. The speed of the motorboat in still water is 45mph
The current rate is 9mph
Mrs Wright spent 2/9 of her paycheck on food and 1/3 on rent. She spent 1/4 of the remainder on transportation. She had $210 left. How much was Mrs. Wright's paycheck ?
Answer:
Mrs. Wright's paycheck is $630
Step-by-step explanation:
Let x = Mrs. Wright's paycheck.
Mrs. Wright spent 2/9 of her paycheck on food. This means that the amount of money spent on food is 2/9 × x = 2x/9
She spent 1/3 on rent. This means that the amount spent on rent is
1/3 ×x x = x/3
Amount she spent on food and rent is x/3 + 2x/9 =3x + 2x /9
= 5x/9
The remainder is her pay check - the amount that she spent on food and rent. It becomes
x - 5x/9 = (9x- 5x)/9 = 4x/9
She spent 1/4 of the remainder on transportation. It means that she spent 1/4 × 4x/9 = x /9 on transportation.
Amount left = 4x /9 - x/9= 3x/9
She had $210 left. Therefore,
210 = 3x/9
3x = 9×210 = 1890
x = 1890/3
x = $630
Determine the seating capacity of an auditorium with 35 rows of seats if there are 20 seats in the first row, 23 seats in the second row, 26 seats in the third row, 29 seats in the forth row, and so on.
Answer:
2485
Step-by-step explanation:
The number of seats in row n is given by the explicit formula for an arithmetic sequence:
an = a1 +d(n -1)
an = 20 +3(n -1)
The middle row is row 18, so has ...
a18 = 20 + 3(18 -1) = 71 . . . . seats
The total number of seats is the product of the number of rows and the number of seats in the middle row:
capacity = (71)(35) = 2485
The seating capacity is 2485.
A protected wilderness area in the shape of a rectangle is 3 kilometers long and 3.2 kilometers wide. The forest is to be surrounded by a hiking trail that will cost $11 comma 000 per mile to construct. What will it cost to install the trail? Note that 1 mile (mi)almost equals1.6 kilometers (km).
Answer: $66
Step-by-step explanation:
Area of a rectangular =length*breadth
= 3.2 km * 3.0 km = 9.6km(squared)
Note, 1.6km = 1 mile
Cost of a hiking trail = $11 per mile
Converting kilometer to kill = 9.6km/1.6km = 6 miles
Hence, cost of a hiking trail to surround the rectangular shaped wilderness = $11 * 6 = $66
In case the cost is $11,000 per mile; total cost of hiking trails to surround the wilderness equals $66,000
You can find the cost of the trail by first calculating the perimeter of the rectangular wilderness area in kilometers, converting this into miles, and then multiplying by the given cost per mile. The total cost to install the trail would be approximately $85,250.
Explanation:To calculate the cost of installing the hiking trail, we first need to find the perimeter of the rectangular forest, which will represent the length of the trail. In a rectangle, the perimeter is calculated as 2*(length + width).
Thus, the perimeter of the forest is 2*(3 km + 3.2 km) = 2*6.2 km = 12.4 km.
Next, we need to convert this into miles, as the cost is given in dollars per mile. We know that 1 mile is approximately equivalent to 1.6 km, so to convert km to miles, we can divide the distance in kilometers by 1.6.
The length in miles is therefore 12.4 km / 1.6 = 7.75 miles.
Finally, to find the total cost, we multiply the length of the trail in miles by the cost per mile: $11,000*7.75 miles = $85,250.
So, the cost to install the trail surrounding the wilderness area would be approximately $85,250.
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The mean height of women in a country (ages 20minus29) is 64.2 inches. A random sample of 50 women in this age group is selected. What is the probability that the mean height for the sample is greater than 65 inches? Assume sigmaequals2.58. Round to four decimal places.
Answer: 0.0142
Step-by-step explanation:
Given : The mean height of women in a country (ages 20 - 29) is 64.2 inches.
i.e. [tex]\mu=64.2[/tex]
Also, [tex]\sigma=2.58[/tex]
Sample size : n= 50
Let x denotes the height of women.
Then, the probability that the mean height for the sample is greater than 65 inches :-
[tex]P(x>65)=1-P(x\leq 65)\\\\=1-P(\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}\leq\dfrac{65-64.2}{\dfrac{2.58}{\sqrt{50}}})\\\\=1-P(z\leq2.193)\ \ [\because z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}]\\\\=1-0.9858463\ \ [\text{By using z-value table or calculator}]\\\\=0.0141537\approx0.0142[/tex]
Hence, the the probability that the mean height for the sample is greater than 65 inches = 0.0142
Ashley recently opened a store that uses only natural ingredients she wants to advertise your product by distributing bags of samples in your neighborhood it takes actually 2/3 of a minute to prepare one day it takes each of her friends 75% long to repair a bag. How many hours will it take Ashley in forever friendship repair 1575 bags of samples
Answer:
Ashley and her 4 friends would required 5.33 hours to repair 1575 bags of sample.
Step-by-step explanation:
75% is written as 0.75 as a decimal or 3/4 as a fraction.
Her friends take 75% longer, so multiply the time it takes Ashley by 1 + the percentage: 1.75 as a decimal or [tex]1 \frac{3}{4} \ \ or \ \ \frac{7}{4}[/tex] as a fraction.
[tex]\frac{2}{3} \times \frac{7}{4} =\frac{14}{12} = 1 \frac{2}{12} = 1 \frac{1}{6} minutes[/tex].
1 hour = 60 minutes.
Divide minutes in an hour by minutes per pack:
Ashley can pack: [tex]\frac{60}{\frac{2}{3}} = 90 \ bags\ per \ hour[/tex]
1 friend can pack: [tex]\frac{60}{1\frac{1}{6}}=\frac{360}{7} = 51 \frac{3}{7} \ bags \ per \ hour.[/tex]
multiply the amount 1 friend can pack by 4 friends: [tex]51 \frac{3}{7} \times 4 = \frac {1440}{7} = 205 \frac{5}{7} \ bags \ per \ hour[/tex]
Ashley and her friends can pack [tex]90 + 205 \frac{5}{7} = 295 \times \frac{5}{7} \ bags \ per \ hour.[/tex]
Now divide the total bags by bags per hour:
[tex]\frac{1575}{295 \frac{5}{7}} = 5.33 \ hours[/tex] ( Round answer as needed.)
Hence, Ashley and her 4 friends would required 5.33 hours to repair 1575 bags of sample.
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A neighborhood is trying to set up school carpools, but they need to determine the number of students who need to travel to the elementary school (ages 5–10), the middle school (ages 11–13), and the high school (ages 14–18). A histogram summarizes their findings:
Histogram titled Carpool, with Number of Children on the y axis and Age Groups on the x axis. Bar 1 is 5 to 10 years old and has a value of 3. Bar 2 is 11 to 13 years old and has a value of 7. Bar 3 is 14 to 18 years old and has a value of 4.
Which of the following data sets is represented in the histogram?
A. {3, 3, 3, 7, 7, 7, 7, 7, 7, 7, 4, 4, 4, 4}
B. {5, 10, 4, 11, 12, 13, 12, 13, 12, 11, 14, 14, 19, 18}
C. {5, 6, 5, 11, 12, 13, 12, 13, 14, 15, 11, 18, 17, 13}
D. {3, 5, 10, 11, 13, 7, 18, 14, 4}
Answer:
c
Step-by-step explanation:
Let point M be outside of △ABC. Point N is the reflected image of M about the midpoint of segment AB . Point K is the reflected image of N about the midpoint of segment BC , and point K is the reflected image of L about the midpoint of segment AC . Prove that point A is the midpoint of segment ML .
Explanation:
Define points D, E, F as the midpoints of AB, BC, and AC, respectively. Point D is the midpoint of both AB and MN, so AMBN is a parallelogram, and side AM is parallel to and congruent with side NB.
Point E is the midpoint of both BC and NK, so BNCK is a parallelogram with side NB parallel and congruent to side CK, and by the transitive property of congruence, also to segment AM.
Point F is the midpoint of both AC and KL, so AKCL is a parallelogram with side CK parallel and congruent to side LA. By the transitive properties of congruence and of parallelism, sides AM, NB, CK, and LA are all congruent and parallel. Since AM and LA are congruent to one another and parallel, and share point A, point A must be their midpoint.
− 2 3 + 20 24 −1.75=minus, start fraction, 3, divided by, 2, end fraction, plus, start fraction, 24, divided by, 20, end fraction, minus, 1, point, 75, equals
The simplified expression is -19/12.
First, the least common multiple (LCM) of the denominators, which in this case is 24.
Now, convert the fractions to have a denominator of 24:
-2/3 = -16/24
20/24 remains the same
-1.75 can be written as -42/24
Now we can add and subtract the fractions:
-16/24 + 20/24 - 42/24 = -38/24
Finally, we can simplify the fraction:
-38/24 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2.
-38/24 ÷ 2/2 = -19/12
Therefore, the simplified expression is -19/12.
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A recipe for a pan of brownies requires 1 ½ cups of milk, 2/3 cups of sugar, and 1 1/5 cups of oil. How many cups of ingredients will there be in the mixing Bowl
Answer:
Step-by-step explanation:
A recipe for a pan of brownies requires 1 ½ cups of milk. Converting the 1 1/2 to improper fraction, it becomes 3/2 cups of milk
The recipe also requires 2/3 cups of sugar
It also requires 1 1/5 cups of oil. Converting to improper fraction, it becomes 6/5 cups of sugar
Total number of cups of ingredients in the mixing Bowl will be sum of the amount if milk, sugar and oil. It becomes
3/2 + 2/3 + 6/5 = (45 + 20 + 36)/30
= 101/30 cups of ingredients
Bruce went to the hardware store and bought 66 yellow ropes. The total length of the ropes was 5,586.5 meters. To the nearest hundredth of a meter, how long was each rope? meters
Each rope was 84.64 meters long.
Step-by-step explanation:
Given,
No. of ropes bought = 66
Total length = 5586.5 meters
Length of one rope = [tex]\frac{Total\ length}{No.\ of\ ropes\ bought}\\[/tex]
[tex]Length\ of\ one\ rope=\frac{5586.5}{66}\\Length\ of\ one\ rope=84.64\ meters[/tex]
Hence,
Each rope was 84.64 meters long.
Keywords: Division.
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The average cost of tuition and room and board at a small private liberal arts college is reported to be $8,500 per term, but a financial administrator believes that the average cost is higher. A study conducted using 350 small liberal arts colleges showed that the average cost per term is $8,745. The population standard deviation is $1,200. Let α = 0.05. What is the critical z value for this test?
Answer:
H₀: µ ≤ $8,500; H₁: µ > $8,500
z= +1.645
Step-by-step explanation:
From the given problem As average cost of tuition and room and board at a small private liberal is less than the financial administrator As hypothesis is true.
As standard deviation is $ 1,200
α = 0.05
H₀: µ ≤ $8,500
if the null hypothesis is true then value for critical z is +1.645.
The critical z-value for a one-tailed test at a significance level (α) of 0.05 is approximately 1.645. This statistic is used in hypothesis testing to make informed financial decisions about tuition costs.
Explanation:We are asked to find the critical z-value for a one-tailed test at a significance level (α) of 0.05. In this case, the financial administrator believes the average cost is higher, thus the test is one-tailed to the right. Looking up the z-table, we know that for a one-tailed test with a significance level of 0.05, the critical z-value is 1.645 (approximate).
The information about rising tuition costs and student loan debt demonstrates the real-world importance of using statistical analysis to understand trends and make informed financial decisions.
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If five times the measure of an angle is decreased by 30, the result is the same as when two times the measure of the angle is increased by 18. What is the measure of the angel
Answer:
16
Step-by-step explanation:
Let "a" represent the measure of the angle. Then ...
5a -30 = 2a +18
3a = 48 . . . . . . . add 30-2a
a = 16 . . . . . . . . . divide by 3
An angel is beyond measure, but this angle has measure 16.
Dividing in scientific notation
Answer: it’s easy your teacher should’ve taught this in 9th
Step-by-step explanation:
Answer:
The answer to your question is 2 x 10¹⁴
Step-by-step explanation:
Process
1.- Divide the whole numbers
[tex]\frac{4}{2} = 2[/tex]
2.- Apply rule of exponents
8 - (-6) = 8 + 6 = 14
3.- Write the answer
2 x 10¹⁴
Bill Casler bought a $8000, 9-month certificate of deposit (CD) that would earn 8% annual simple interest. Three months before the CD was due to mature, Bill needed his CD money, so a friend agreed to lend him money and receive the value of the CD when it matured.
(a) What is the value of the CD when it matures?
(b) If their agreement allowed the friend to earn a 10% annual simple interest return on his loan to Bill, how much did Bill receive from his friend? (Round your answer to the nearest cent.)
Final answer:
The value of the CD when it matures is $8480, calculated using simple interest. Bill received $8366.02 from his friend, who extended a loan expecting a 10% annual simple interest return.
Explanation:
To calculate the value of the CD when it matures, we use the formula for simple interest: Interest = Principal × Rate × Time. In this case, the Principal is $8000, the Rate is 8% per annum, and since the CD is for 9 months, the Time is ⅓ year. The simple interest earned on the CD is calculated as:
Interest = $8000 × 0.08 × ⅓ = $8000 × 0.08 × ⅔ = $480.
So, the mature value of the CD is the principal plus the interest, which is $8000 + $480 = $8480.
To determine how much Bill received from his friend with a 10% annual simple interest rate, the friend would expect to get the mature value of the CD in 3 months, which is one-quarter of a year. The agreement suggests that the friend treats the loan to Bill as an investment, thus expecting the same return as the interest he could have earned elsewhere at his desired 10% rate for the time frame. So, the calculation would be:
8480 = Principal × (1 + (10% × ⅔)), solving for Principal gives Bill:
Principal = 8480 ÷ (1 + (0.10 × ⅔)) = $8366.02 (rounded to the nearest cent).
Amy is planning the seating arrangement for her wedding reception. Each round table can sit 12 guests. The head table can sit the bride and groom with the 6 wedding attendants. If Amy expects 198 to 270 guests to attend her wedding, including the attendants, what is the range for the number of round tables she will need for her reception?
A. 22 to 28
B. 17 to 23
C. 19 to 23
D. 16 to 22
Answer:
D
Step-by-step explanation:
16*12 is 192. Plus the 6 for the bride and groom, and you get a total of 198.
Amy will need between 16 to 22 round tables for her wedding reception, accounting for a total guest range of 198 to 270, and considering that the head table seats 8 people. We calculate the number of tables by dividing the remaining guests by 12 and rounding up.
Explanation:Let's solve the problem of determining the number of round tables Amy will need for her wedding reception. First, we know that the head table can sit the bride, groom, and their 6 attendants, totaling 8 people. Next, we calculate the number of guests that can be seated at the round tables, by subtracting the 8 people at the head table from the total number of guests:
Minimum guests (198) - head table (8) = 190 guestsMaximum guests (270) - head table (8) = 262 guestsEach round table can sit 12 guests, so we divide the numbers of remaining guests by 12 to find the range of tables needed:
190 guests ÷ 12 guests/table ≈ 15.83 tables262 guests ÷ 12 guests/table ≈ 21.83 tablesSince we can't have a fraction of a table, we round up because each table can only seat whole numbers of guests. Therefore, Amy will need at least 16 tables for the minimum number of guests and at most 22 tables for the maximum number of guests.
The range for the number of round tables Amy will need for her reception, therefore, is 16 to 22 tables.
After reviewing the options, we can see that Option D matches our calculated range and is the correct choice for Amy’s wedding reception planning.
A hypothesis test is conducted at the 0.05 level of significance to test whether or not the population correlation is zero. If the sample consists of 25 observations and the correlation coefficient is 0.60, what is the computed value of the test statistic? Round to two decimal places.
The computed value of the test statistic is 4.06 when using a sample size of 25 observations and a correlation coefficient of 0.60. Since this value is greater than 2.045 (the critical value for a two-tailed test at an alpha level of 0.05), we conclude that the population correlation is significantly different from zero.
Explanation:In the context of this question, the student is conducting a hypothesis test to examine whether the population correlation is significantly different from zero. Given that the student has a sample size of 25 observations with a correlation coefficient of 0.6, we can calculate the test statistic. Our first step is to find the t value using the formula: t = r*sqrt[(n-2)/(1-r²)], where n is the sample size and r is the correlation coefficient.
Substituting the given values, t=0.6*sqrt[(25-2)/(1-0.6²)].
Solving this gives a t-value that can be rounded to 4.06. Since this value is greater than the critical value of 2.045 for a two-tailed test at an alpha level of 0.05, we reject the null hypothesis and conclude that the sample correlation coefficient is statistically significant, meaning it is unlikely that the correlation in the population is zero.
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A watch manufacturer believes that 60% of men over age 50 wear watches. So, the manufacturer took a simple random sample of 275 men over age 50 and 170 of those men wore watches. Test the watch manufacturer's claim at a=.05.
Answer:
The total number of men is 32. Of these men, 25 wore a watch. So, the number of men who didn’t wear a watch is 7, because 32 − 25 = 7.
Step-by-step explanation:
Find the distance between points P (-1,5) and Q (3,4)
Answer:
Step-by-step explanation:
the formular for distance between two points is: [tex]\sqrt{(x2-x1)^2+(y2-y1)^2} \\\\\sqrt{4^{2}+1^{2} } \\\sqrt{17}\\ = 4.12[/tex]
Answer: Distance = 4.123
Step-by-step explanation:
The given coordinates are P(-1,5) and Q (3,4)
- 1 = x1 = the horizontal coordinate(along the x axis) at P
3 = x2 = the horizontal coordinate (along the x axis) at Q
5 = y1 = vertical coordinate( along the y axis) at P
4 = y2 = vertical coordinate(along the y axis) at Q
The distance between points P and Q is expressed as square root of the sum of the square of the horizontal distance and the square of the vertical distance. It becomes
Distance = √(x2 - x1)^2 + (y2 - y1)^2
Distance = √(3 - - 1)^2 + (4 - 5)^2
Distance = √ 4^2 + (-1)^2
Distance = √16+1 = √17
Distance = 4.123
Please help with this question I really need it right now please show work and please if you don't understand this question please don't answer . Question number 12
Answer:a is the correct option
Step-by-step explanation:
The table is divided into columns for month, sales and expense
Total number of sales for four months is determined by adding up the sales for each month. It becomes
35.75+65.34+12.15+49.68= $162.92
Total number of sales for four months is determined by adding up the expenses for each month. It becomes
43.18+52.24+41.09+59.50=196.01
Profit or loss = total sales - total expenses
Therefore 162.92-196.01 = - $33.09
It's a loss because it is negative
From a sample of 9,750 Ajax trucks, 273 developed transmission problems within the first two years of operation. What is the probability that an Ajax truck will develop transmission problems within the first two years?
Answer:
0.028
Step-by-step explanation:
Let sample space, S, be the trucks.
Hence, n(S)= number of trucks=9750
Let E be the event of the truck failing within the first two years of operation.
Hence, n(E)=273
Therefore by classical definition of probability, the probability of occurrence of the event E, denoted by P(E), is
P(E)=[tex]\frac{n(E)}{n(S)}[/tex]=[tex]\frac{number of trucks failed within two years of operation}{Total number of trucks}[/tex]
⇒P(E)=[tex]\frac{273}{9750}[/tex]=0.028
The probability that an Ajax truck will develop transmission problems within the first two years is 2.8%, calculated by dividing the number of trucks with transmission problems (273) by the total number of trucks (9,750).
To calculate the probability that an Ajax truck will develop transmission problems within the first two years, we can use the formula for simple probability:
P(event) = Number of favorable outcomes / Total number of possible outcomes
Given that 273 out of 9,750 Ajax trucks developed transmission problems, the probability P is calculated as:
P(transmission problems) = 273 / 9,750
P(transmission problems) = 0.028 or 2.8%
The probability is 0.028, which means there is a 2.8% chance that an Ajax truck will develop transmission problems within the first two years of operation.
A 17-year-old high school senior suddenly developed a high fever and chills, headache, stiff neck, and vomiting. His parents called the doctor, who told them to bring their son to the Emergency Department immediately. What disease does this boy probably have?
Answer:
Meningitis
Step-by-step explanation:
Meningitis is a inflammation of the membrane surrounding your brain and spinal cord.
In most cases it is caused by viral infection.
Some cases it improve without treatment in few days
Symptoms include:
sudden high feverHeadacheSevere HeadacheSeizuresSkin rashConstant CryingStiffness in a baby's bodyAs 17 year old had some symptoms such as headache,stiff neck, chills and vomiting o might be he have been suffered from Meningitis.
Barry and his wife, Mary, have accumulated over $3.5 million during their 50 years of marriage. They have three children and five grandchildren. How much money can they gift to their children in 2017 without any gift tax liability?
Answer:
$208,000
Step-by-step explanation:
For 50 years of marriage, Barry and Mary accumulated over $3.5million.
They have 3 children and 5 grandchildren which gives a total of 8.
You are allowed to give up $13,000 each year to a person without incurring tax liability.
This means Barry and Mary can give $13,000 to anyone each
For the 3 children and 5 grandchildren, Barry and Mary can gift $(13000 * 8) each
= $104,000 each
Therefore, Barry and Mary can gift $208,000 (104,000*2) to their children in 2017 without gift tax liability
A manufacturer of precision measuring instruments claims that the standarddeviation in the use of the instruments is less than 0.00002 millimeter. An analyst,who is unaware of claim, uses the instrument eight times and obtains a samplestandard deviation of 0.00001 millimeter. Assume normal population(i) Confirm using a test procedure and α level of 0.01 that there is insufficient evidence to support the claim that the standard deviation of the instruments is less than 0.00002 millimeters.(ii) Find the P-value for this test? How does it compare with α(greater than or less than)? Does it make sense with respect to your conclusionin part (i)?
Answer:
i) There is insufficient evidence to support the claim that the standard deviation of the instruments is less than 0.00002 millimeters.
ii) [tex] p_v= P(\chi^2_{7} <1.75)=0.0276[/tex]
Step-by-step explanation:
[tex]s=0.00001[/tex] represent the sample standard deviation
[tex]\alpha =0.01[/tex] represent the significance level for the test
[tex]\sigma_o =0.00002[/tex] represent the value that we want to test
n=8 represent the sample size
A chi-square test can be used to test "if the standard deviation of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test".
Null and alternatve hypothesis
The system of hypothesis on this case would be given by:
Null Hypothesis: [tex]\sigma \geq 0.00002[/tex]
Alternative Hypothesis: [tex]\sigma <0.00002[/tex]
The statistic to test this is given by:
[tex]T=(n-1)(\frac{s}{\sigma_0})^2 [/tex] (1)
Part i) Calculate the statistic
If we replace into formula (1) we got this:
[tex]T=(8-1)(\frac{0.00001}{0.00002})^2 =1.75 [/tex] (1)
The critical region for a for a lower one-tailed alternative is given by
[tex]T< \chi^2_{1-\alpha/2,N-1}[/tex]
The degrees of freedom are given by
[tex] df=n-1=8-1=7[/tex]
And if we use the Chi Square distribution with 7 degrees of freedom we see that [tex] \chi^2_{1-0.01/2,7}=1.239[/tex]
And our critical region would be [tex]T< 1.239[/tex] so on this case we can conclude that we fail to reject the null hypothesis. So it's not enough evidence to conclude that the population standard deviation is less than 0.00002 mm.
Part ii) Calculate the p value
In order to calculate the p value we can do this:
[tex] p_v= P(\chi^2_{7} <1.75)=0.0276[/tex]
If we compare this value with the significance level (0.01) we see that [tex]p_v>\alpha[/tex]
This agrees with the conclusion since when the p values is greater than the significance level we FAIL to reject the null hypothesis, same conclusion as part i).
A wise old troll wants to make a small hut. Roofing material costs five dollars per square foot and wall materials cost three dollars per square foot. According to ancient troll customs the floor must be square, but the height is not restricted.
(a) Express the cost of the hut in terms of its height h and the length x of the side of the square floor. ($)
(b) If the troll has only 960 dollars to spend, what is the biggest volume hut he can build? (ft^3)
Answer:
a) Cost (h,x) = 12*x*h + 5*x²
b)
V = V(max) = 355.5 ft³
Dimensions of the hut:
x = 9.48 ft (side of the base square)
h = 3.95 ft ( height of the hut)
Step-by-step explanation:
Let x be the side of the square of the base
h the height of the hut
Then the cost of the hut as a function of "x" and "h" is
Cost of the hut = cost of 4 sides + cost of roof
cost of side = 3* x*h then for four sides cost is 12*x*h
cost of the roof = 5 * x²
Cost(h,x) = 12*x*h + 5*x²
If the troll has only 900 $
900 = 12xh + 5x² ⇒ 900 - 5x² = 12xh ⇒(900-5x²)/12x = h
And the volume of the hut is V = x²*h then
V (h) = x² * [(900-5*x²)]/12x
V(h) = x (900-5x²) /12 ⇒ V(h) = (900*x - 5*x³) /12
Taking derivatives (both sides of the equation):
V´(h) = (900 - 10* x²)/12 V´(h) = 0
900 - 10*x² = 0 ⇒ x² = 90 x =√90
x = 9.48 ft
And h
h = (900-5x²)/12x ⇒ h = [900 - 90(5)]/12*x ⇒ h = 450/113,76
h = 3.95 ft
And finally the volume of the hut is:
V(max) = x²*h ⇒ V(max) = 90*3.95
V(max) = 355.5 ft³
A hut will consist of four walls and one roof. The figures needed evaluates to:
The cost of the hut expressed in terms of its height h and the length x of the side of the square floor is [tex]12hx + 5x^2 \text{\:(in dollars) }[/tex]The biggest volume hut that can be build with 960 dollars at max is 426.67 sq. ft approxHow to find the volume of cuboid?Let the three dimensions(height, length, width) be x, y,z units respectively.
Then the volume of the cuboid is given as
[tex]V = x \times y \times z \: \rm unit^3[/tex]
How to obtain the maximum value of a function?To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.
Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.
For this case, we're specified that:
Cost of roofing material = $5 / sq. footCost of wall material = $3 / sq. footThe side length of floor = side length of roof = [tex]x \: \rm ft[/tex]The height of the room = [tex]h \: \rm ft[/tex]Four walls are attached to sides of floor. Thus, their one edge's length = length of side of floor = [tex]x \: \rm ft[/tex]
Thus, we get:
Area of four walls = [tex]4 \times (h \times x)[/tex] sq. ftThus, cost of four walls' material = [tex]3 \times 4 \times h \times x = \$ 12hx[/tex]
Area of roof = [tex]side^2 = x^2 \: \rm ft^2[/tex]Thus, cost of roofing material for this roof = [tex]5 \times x^2 = \$5x^2[/tex]
Thus, cost of hut = cost for walls + cost for roof = [tex]12hx + 5x^2 = x(12h+5x) \: \rm (in \: dollars)[/tex]
The volume of the hut is: [tex]x\times x\times h =x^2.h \: \rm ft^3[/tex]
If troll has got only $960, then,
[tex]12hx + 5x^2 \leq 960\\\\\text{Multiplying x on both the sides}\\\\12x^2h + 5x^3 \leq 960x\\\\x^2h \leq \dfrac{960x-5x^3}{12}[/tex]
Let we take [tex]f(x) =\dfrac{960x-5x^3}{12}[/tex]
Then, taking its first and second derivative, we get:
[tex]f(x) =\dfrac{960x-5x^3}{12}\\\\f'(x) = 80 -1.25x^2\\\\f''(x) = -2.5x[/tex]
Putting first derivative = 0, we get critical points as:
[tex]80-1.25x^2 = 0\\\\x = 8 \text{\:(positive root as x denotes side length, thus a non-negative quantity)}[/tex]
At x = 8, the second derivative evaluates to:
[tex]f''(8) = -2.5(8) < 0[/tex]
Thus, we obtain maxima at x = 8.
Thus, we get the maximum value of function when x = 8.
Since we have:
[tex]V = x^2h \leq f(x)[/tex] (V is volume of the hut)
and [tex]max(f(x)) = f(8) = \dfrac{960(8) - 5(8)^3}{12} = 640-213.3\overline{3} \approx 426.67[/tex]
Thus, max(V) = 426.67 sq. ft approximately.
Thus, the figures needed evaluates to:
The cost of the hut expressed in terms of its height h and the length x of the side of the square floor is [tex]12hx + 5x^2 \text{\:(in dollars) }[/tex]The biggest volume hut that can be build with 960 dollars at max is 426.67 sq. ft approxLearn more about maxima and minima here:
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