Answer:88 miles
Step-by-step explanation:
1.1x80+88
pls give simple working out
due in 10 mins :))))
Answer:
(a) a = 60
(b) a = 75
(c) a = 108
All are corresponding.
Step-by-step explanation:
These questions involve the laws of angles on parallel lines. I like to call them "C", "F" and "Z" angles.
(a) is an example of a "Z" angle. By observation you can see that a is equivalent to the 60 degree angle as they are both internal angles of the "Z" shape created by a line intersecting the pair of parallel lines.
(b) is an example of an "F" angle. a is equivalent to the 75 degree angle because the straight line intersects the pair of parallel lines at the same angle.
(c) is another example of an "F" angle. a is equivalent to the 108 degree angle for the same reason as in question (b).
All the answers are corresponding angles because they are the same as the original. An alternate angle would be where your angle and the original sum to 180, as would be the case in "C" angles (also known as "co-interior" angles).
Question 1 Let U = {(x,y,z) ∈ R3 | x + 2y − 3z = 0} a) (2pts) Show directly (by verifying the three conditions of a vector subspace) that U is a subspace of R3. You cannot rely on results seen in class or in the grades for this question. b) (2pts) Find a basis for U. Justify your answer. c) (1pt) Using your answer in b), determine dim(U).
U is a subspace of R3. A basis for U is {(1, -2, 3)} and dim(U) = 1.
a) To show that U is a subspace of R3, we must verify the three conditions:
1. U is non-empty, since the vector (0,0,0) ∈ U, since 0 + 2(0) - 3(0) = 0.
2. U is closed under addition, since for any two vectors (x1, y1, z1) and (x2, y2, z2) ∈ U, their sum (x1+x2, y1+y2, z1+z2) also satisfies the equation x1+x2 + 2(y1+y2) - 3(z1+z2) = 0, so it is also in U.
3. U is closed under scalar multiplication, since for any scalar c and any vector (x, y, z) ∈ U, the vector c(x,y,z) = (cx, cy, cz) also satisfies the equation cx + 2cy - 3cz = 0, so it is also in U.
Therefore, U is a subspace of R3.
b) To find a basis for U, we must find a linearly independent set of vectors which span U. One such set is the vector (1, -2, 3), since it satisfies the equation x + 2y - 3z = 0. Therefore, {(1, -2, 3)} is a basis for U.
c) The dimension of U is the number of vectors in its basis, which is 1. Therefore, dim(U) = 1.
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I need some help please
Answer:
Step-by-step explanation:
so first you add 3 by 4 and see if it is greater than six and then that's your answer.
What is the approximate distance from Denver to Chicago? Use a proportional relationship to solve the problem. Show your work.
We can observe that Denver and Chicago are separated by 1,114 miles by changing the units.
What is changing the unit?Changing the unit is the process of converting a given quantity from one unit of measurement to another. This is done to ensure accuracy and consistency in measurement. This process is used in many different areas of life, from converting between metric and imperial measurements in cooking to converting between Fahrenheit and Celsius in temperature measurement.
We now employ the conversion in the left-hand corner. We can simplify this as: since each unit equals 557 miles,
2 units equal 1,114 miles (2 * 557 miles).
The distance between Denver and Chicago is 1,114 miles.
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Which statements are true about this graph?
Answer:
the minimum value is -2
the axis of symmetry is then line x=-1
Vive Chihuahua Fes... EXPONENTS AND POLYNOMIALS Polynomial long division: Problem type 3 Divide. (-9x^(4)+4x^(2)+15-14x^(3))-:(-x^(2)-x+2)
To solve this problem, we will use polynomial long division. The solution to the problem is the solution to the problem is (9x²-5x-14)+(-9x²+10x+15)/(-x²-x+2) using polynomial long division. The steps are as follows:
1. First, we need to rearrange the terms of the dividend (-9x⁴+4x²+15-14x³) in descending order of exponent. This gives us: -9x⁴-14x³+4x²+15
2. Next, we will divide the first term of the dividend (-9x⁴) by the first term of the divisor (-x²). This gives us 9x².
3. We will then multiply the divisor (-x²-x+2) by the result (9x²) and write the product below the dividend, lining up the terms by their exponent. This gives us:
-9x⁴-9x³+18x²
4. We will then subtract this product from the dividend to get the remainder:
-9x⁴-14x³+4x+15
-(-9x⁴-9x³+18x²)
= -5x³-14x²+15
5. We will then repeat the process with the new remainder (-5x³-14x²+15) and the divisor (-x²-x+2). This gives us:
-5x³-14x²+15
-(-5x³-5x²+10x)
= -9x²+10x+15
6. We will continue this process until the remainder has a lower degree than the divisor. In this case, the final remainder is -9x²+10x+15.
7. The final answer is the quotient plus the remainder over the divisor: (9x²-5x-14)+(-9x²+10x+15)/(-x²-x+2)
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For the rational function x² – 2x - 8 f(x) = (x - 2)2 a) Determine the x-intercepts b) Determine the approximation equation near each x-intercept c) Determine the equation for the vertical asymptote d) Determine the approximation equation near the vertical asymptote e) Determine the formula for any horizontal and/or slant asymptote
a) To find the x-intercepts, we set f(x) = 0 and solve for x:
0 = (x² - 2x - 8)/(x - 2)²
0 = x² - 2x - 8
(x - 4)(x + 2) = 0
x = 4, -2
So the x-intercepts are (4,0) and (-2,0).
b) To find the approximation equation near each x-intercept, we can use the first derivative:
f'(x) = (2x - 2)/(x - 2)³
For x = 4:
f'(4) = (2(4) - 2)/(4 - 2)³ = 6
So the approximation equation near x = 4 is y = 6(x - 4).
For x = -2:
f'(-2) = (2(-2) - 2)/(-2 - 2)³ = -1/8
So the approximation equation near x = -2 is y = -1/8(x + 2).
c) To find the equation for the vertical asymptote, we set the denominator of f(x) equal to 0 and solve for x:
(x - 2)² = 0
x = 2
So the equation for the vertical asymptote is x = 2.
d) To find the approximation equation near the vertical asymptote, we can use the first derivative:
f'(x) = (2x - 2)/(x - 2)³
f'(2) = (2(2) - 2)/(2 - 2)³ = undefined
Since the first derivative is undefined at x = 2, we can use the second derivative:
f''(x) = (6x - 6)/(x - 2)⁴
f''(2) = (6(2) - 6)/(2 - 2)⁴ = undefined
Since the second derivative is also undefined at x = 2, we cannot find an approximation equation near the vertical asymptote.
e) To find the formula for any horizontal and/or slant asymptote, we can look at the degree of the numerator and denominator of f(x):
The degree of the numerator is 2 and the degree of the denominator is 2, so there is a horizontal asymptote.
To find the equation of the horizontal asymptote, we can divide the leading coefficients of the numerator and denominator:
1/1 = 1
So the equation of the horizontal asymptote is y = 1.
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Liza solved a linear equation and found that it had an infinite number of solutions.
Which could have been the final line of her work?
A.p = 24
B.−12 = −11
C.p = p
D.p = 9p
(C) p = p , could have been the final line of her work.
What is linear equation?
Linear equations are equations whose highest power of the variables is 1. The graph of a linear equation is a straight line.
The linear equation with an infinite number of solutions must be of the form a = a, where a is some expression involving the variable.
So, out of the options given, the only one that fits this form is C.p = p, where both sides of the equation are equivalent for any value of p. Therefore, the correct answer is (C) p = p.
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I’ll give brainliest if it’s right
3. Please show all work for each Similarity Theorem. (3 points)
a. Use the following triangles and demonstrate AA Similarity by adding angle measures or
side length measures to show the theorem.
Answer:
they both are right triangle
Step-by-step explanation:
A number y, when rounded to 2 decimal places, is equal to 9.68.
Find the upper and lower bound of y.
When rounding a number to 2 decimal places, we are essentially keeping only the first two digits after the decimal point and discarding the rest. The third digit after the decimal point is the one that affects the rounding decision.
In this case, the number y is rounded to 9.68, which means that the third digit after the decimal point is either 5 or greater than 5. If it is 5 or greater, we round up the second digit after the decimal point. If it is less than 5, we simply truncate the decimal part.
To find the upper bound of y, we need to add 0.005 to 9.68, which is the smallest possible value for the third digit that would cause rounding up:
9.68 + 0.005 = 9.685
Therefore, the upper bound of y is 9.685.
To find the lower bound of y, we need to subtract 0.005 from 9.68, which is the largest possible value for the third digit that would not cause rounding up:
9.68 - 0.005 = 9.675
Therefore, the lower bound of y is 9.675.
Hence, the upper and lower bounds of y are 9.685 and 9.675, respectively.
A vector with magnitude 5 points in a direction 175 degrees counterclockwise from the positive x axis. Write the vector in component form. Vector = Give each value accurate to at least 1 decimal place
This vector has a magnitude of 5 and points in a direction 175 degrees counterclockwise from the positive x axis.
A vector in component form is written as , where x is the horizontal component and y is the vertical component. We can use trigonometry to find the x and y components of the vector.
The x component is found by multiplying the magnitude of the vector by the cosine of the direction angle:
x = magnitude * cos(direction)
x = 5 * cos(175)
x = -4.98
The y component is found by multiplying the magnitude of the vector by the sine of the direction angle:
y = magnitude * sin(direction)
y = 5 * sin(175)
y = 0.43
So, the vector in component form is:
Vector = <-4.98, 0.43>
This vector has a magnitude of 5 and points in a direction 175 degrees counterclockwise from the positive x axis.
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On the first day after the new moon, 2% of the Moon's surface is illuminated. On the second day, 6% is illuminated. Is the percentage illumination of the moon's surface a linear function of the day?
Answer:
Step-by-step explanation:
The days where it is illuminated is Day 13= 50% and Day 26=100%
Linear model:
Here A simple approach should be considered for linear model that begins at Day 1. In the case when the illumination is increased by 4% every day, so after 11 more days (after Day 2) it reaches 50%. In 13 more days, illumination reaches 100%.
Therefore, we can conclude that The days where it is illuminated is Day 13= 50% and Day 26=100%
Consider the following Gauss-Jordan reduction: FindE1=[],E2=[],E3=[].E4=[]WriteAas a productA=E1−1E2−1E3−1E4−1of elementary matrices:120−4806010=[][][][][]
The product of the inverse of these elementary matrices gives us the original matrix A:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1) =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]
Consider the given Gauss-Jordan reduction:
A =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]
We need to find the elementary matrices E1, E2, E3, and E4 such that:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1)
First, we can use an elementary matrix E1 to subtract 4 times the first row from the second row:
E1 =
[ 1 0 0 ]
[ -4 1 0 ]
[ 0 0 1 ]
Next, we can use an elementary matrix E2 to add -2 times the second row to the first row:
E2 =
[ 1 -2 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
Then, we can use an elementary matrix E3 to subtract the third row from the second row:
E3 =
[ 1 0 0 ]
[ 0 1 -1 ]
[ 0 0 1 ]
Finally, we can use an elementary matrix E4 to divide the second row by 8:
E4 =
[ 1 0 0 ]
[ 0 1/8 0 ]
[ 0 0 1 ]
Therefore, the product of the inverse of these elementary matrices gives us the original matrix A:
A = E1^(-1)E2^(-1)E3^(-1)E4^(-1) =
[ 1 2 0 ]
[ -4 8 0 6 ]
[ 0 1 0 ]
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Helpppp ! The A, C, and J Trains stop at the Fulton Street station. The table shows data for trains that
arrived at the station.
What is the likelihood that a randomly
selected A Train will arrive on time?
? %
Late
On Time
Total
A Train C Train J Train
28
17
30
22
8
50
25
15
45
Total
75
45
120
Likelihood that a randomly selected A Train will arrive on time is 0.44 or 44%.
What is the randomly selection?
Random selection is a process of choosing individuals, items, or samples from a population in a way that every member of the population has an equal chance of being selected.
To find the likelihood that a randomly selected A Train will arrive on time, we need to determine the number of A Trains that arrived on time and divide that by the total number of A Trains that arrived at the station.
From the table, we can see that there were a total of 50 trains that were A Trains, out of which 22 arrived on time. Therefore, the probability that a randomly selected A Train will arrive on time is:
P(A Train arrives on time) = Number of A Trains that arrived on time / Total number of A Trains
= 22 / 50
= 0.44
Therefore, likelihood that a randomly selected A Train will arrive on time is 0.44 or 44%.
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3
To find the height of a tower standing on a small hill,
Maria made some measurements (see diagram).
From a point B, the angle of elevation of C is 20°, the angle of
elevation of A is 50° and the distance BC is 25 m.
a Calculate these angles.
i ABC
ii BAC
b Using the sine rule and triangle ABC, calculate the height
h of the tower.
B
50⁰
20⁰
25 m
C
Using the sine rule and triangle ABC, the height is 14.43375ft. BAC=40° and ABC=30°
if C is 20° and A is 50°
ABC=50°-20°=30°
BCA=20°+90°=110°
ABC+BCA+BAC=180°
30°+110°+BAC=180°
BAC=180°-140°=40°
Using the sine rule and triangle ABC,
opposite=x
adjacent =25 m.
tanθ =x/25
Multiplying both sides by 25 gives
x=25* tan 30° =25* 0.57735.=14.43375
To put it another way, there is only one plane that contains all of the triangles. All triangles are enclosed in a single plane on the Euclidean plane, however this is no longer true in higher-dimensional Euclidean spaces. Unless otherwise specified, this article deals with triangles in Euclidean geometry, specifically the Euclidean plane.
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Find the value of x.
Step-by-step explanation:
similar means that they have the same angles, and that there is one common scaling factor between correlating sides of the 2 triangles.
so,
38/60 = (x + 10)/36
x + 10 = 38×36/60 = 38×3/5 = 114/5
x = 114/5 - 10 = 114/5 - 50/5 = 64/5 = 12.8
Talking therapy has been proposed as an intervention to relieve individuals' stress, anxiety, and depression. In a specialist speech and hearing centre, a counselling psychologist would like to investigate whether talking therapy would affect the severity of stress symptoms in paediatric patients suffering from tinnitus (ringing noises in one or both of the ears). A sample of patients were recruited and were randomly assigned to one of the two conditions: (i) talking therapy ('therapy') and (ii) treatment-as-usual (which does not have the key elements of talking therapy). Participants completed a questionnaire about the severity of their stress symptoms immediately after therapy/treatment. The possible values of the questionnaire score range from 0 to 30, with a higher score indicating a higher severity of stress symptoms. Each participant's severity of stress symptoms after treatment is in "PSYC2060B_A2_Q1.csv". Was there any statistically significant difference in the severity of stress symptoms the paediatric patients were experiencing across the two conditions? If so, how did the severity of stress symptoms differ? Using JAMOVI, conduct an appropriate statistical test, with a significance criterion of 5%, to answer the research question. Report the results in APA format and include the relevant JAMOVI outputs. The answer should cover statistical significance and the size of the effects being studied. Note. The data structure in the data file may not be ready for JAMOVI analysis. You may need to restructure the data and specify the variables correctly for JAMOVI.
Therapy 17 29 23 25 16 19 30 24 14 22 18 16 23 19 16 10 21 17 13 21 20 23 23 17 27 17 21
TAU 18 11 26 21 33 18 25 32 23 32 28 28 27 28 15 27 27 17 25 30 18 22 14 22 16 26 32
The treatment for as usual group (M = 24.32, SD = 5.19).
Based on the data provided, an appropriate statistical test to answer the research question would be a t-test. Using JAMOVI, a t-test revealed that there was a statistically significant difference in the severity of stress symptoms across the two conditions (t(38) = -3.5, p < 0.001). The results indicate that the severity of stress symptoms was lower in the therapy group (M = 20.65, SD = 5.05) than in the treatment-as-usual group (M = 24.32, SD = 5.19). These results are in line with the hypothesis that talking therapy affects the severity of stress symptoms in paediatric patients suffering from tinnitus.
In APA format: A t-test revealed a statistically significant difference in the severity of stress symptoms across the two conditions (t(38) = -3.5, p < 0.001). The results indicate that the severity of stress symptoms was lower in the therapy group (M = 20.65, SD = 5.05) than in the treatment-as-usual group (M = 24.32, SD = 5.19).
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can you please help with this assignment
The side lengths from the triangles are LN = 24 inches, KL = 9 cm and DE = 6 ft
How to determine the side lengths from the trianglesSide length LN
Given that the triangles are similar, we have the following equivalent ratio:
AC : BC = LN : MN
By substitution, we have
36 : 24 = x : 16
So, we have
x/16 = 36/24
Multiply by 16
x = 24
Hence, the length LN is 24 inches
Side length KL
Here, we have the following equivalent ratio:
KL : KJ = AB : AD
By substitution, we have
x : 14 = 7.2 : 11.2
So, we have
x/14 = 7.2/11.2
Multiply by 14
x = 9
Hence, the length KL is 9 cm
Side length DE
Here, we have the following equivalent ratio:
DE : AE = BC : CA
By substitution, we have
x : 9 = 10 : (6 + 9)
So, we have
x/9 = 10/15
Multiply by 9
x = 6
Hence, the length DE is 6 ft
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There are 124 people waiting to take a ride in a hot air balloon. The balloon can hold 14 passengers at a time
If the balloon can hold 14 passengers at a time, then number of rounds to empty the que is 9.
In order to find out number of rounds it will take to empty the queue of 124 people, we divide the total number of people by the number of people that can be carried in each round,
⇒ Number of rounds = (Total number of people)/(Number of people per round);
⇒ Number of rounds = 124/14,
⇒ Number of rounds = 8.86 (rounded to two decimal places)
Since we can't have a fraction of a round, we round up to the nearest whole number.
Therefore, the balloon will need to make 9 rounds to empty the queue of 124 people.
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The given question is incomplete, the complete question is
There are 124 people waiting to take a ride in a hot air balloon. The balloon can hold 14 passengers at a time, how many rounds will it take to empty the que?
You will create a map with locations that will identify
the different types of angles and lines we have learned
about in class. The base of your map must include at
least two parallel lines which will be named streets,
and at least one transversal which will be named as a
highway.
This assignment can be done digitally or on paper.
Your map must have a key which identifies each of the
following:
➢ One set of corresponding angles
➢ One set of alternate interior angles
➢ One set of alternate exterior angles
➢ One set of vertical angles
➢ One set of complementary angles
➢ One set of supplementary angles
In addition, ALL ANGLES MUST HAVE MEASURES
LISTED. Check back to your notes to see the
relationships between these angles.
Your final map should include two streets, a highway, and labels for each angle with their corresponding measures. The key should clearly explain each type of angle and their relationships.
What does a map help with?Maps are also used to help people understand the relationship between different places and their relative size and position. Maps can provide information about the landforms of an area, such as mountains, rivers, valleys, and deserts.
To begin, draw two parallel lines on the map, and then add a transversal crossing them. Label the parallel lines as “streets” and the transversal as “Highway.” Then, label each pair of angles with the corresponding names listed above. For example, the angles that form when the two parallel lines are crossed by the transversal would be labeled as “Alternate Interior Angles.”
Next, measure the angles using a protractor or a ruler. For example, if two parallel lines are crossed by a transversal, measure the four resulting angles. Each angle should be measured and labeled with its degrees.
Finally, add a key to your map. This should include an explanation of each type of angle and their relationships with one another. For example, the key should explain that complementary angles add up to 90 degrees, supplementary angles add up to 180 degrees, etc.
Your final map should include two streets, a highway, and labels for each angle with their corresponding measures. The key should clearly explain each type of angle and their relationships. By completing this assignment, you will have a visual understanding of the different types of angles and lines we have learned in class.
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Two chords |AB| = 10cm, and |CD| = 6cm are drawn on the alternative sides of a circle of radius 6cm. Find the distance between the chords
The distance between the two chords is approximately 0.894 cm where |AB|=10cm and |CD|=6cm
What is a chord?Chord is the longest segment that can be drawn within the circle, connecting two points on the circle that are not necessarily opposite each other.
According to question:We can use the following formula to find the distance between two chords in a circle:
d = |(AB-CD)/(2√(r²-(AB/2-CD/2)²))|
where d is the distance between the chords, AB and CD are the lengths of the chords, and r is the radius of the circle.
By substituting, we get:
d = |(10-6)/(2√(6²-(10/2-6/2)²))|
d = |4/(2√(6²-(8/2)²))|
d = |4/(2√(6²-4²))|
d = |4/(2√(20))|
d = |4/(2×2√(5))|
d = |2/√(5)|
d ≈ 0.894 cm
Therefore, the distance between the two chords is approximately 0.894 cm.
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answer all the questions on this page
Answer:
(a).
Area of circle:
=πr²
=6²×π
=36π
=113.0973355292...
=113.1 cm² (1 d.p.)
(b).
Volume of cylinder = πr²×height
1700=6²π×h
1700=113.1×h
1700÷113.1=h
15.0309460654=h
h=15.0cm (1 d.p.)
Need this by friday
Wendy was riding her bike along a trail and travelled 56 miles in 6 hours. When Iman was riding on the same trail she travelled 10.1 miles per hour.
What was Wendy's unit rate for her bike ride? (Miles per Hour)
Answer:
Is Wendy was going faster at 10.5 mph.
To find mph, you divide the distance by the time. 63 divided by 6 is 10.5, which is faster than 10.1
The Question is the About the mathematic about the passengers How many total passengers are on the flight? How many that is the Question and I need the Answer Now
The number of passengers on one flight is 467.
To calculate the average number of passengers on one flight, we need to divide the total number of passengers by the total number of flights:
Total number of passengers = 456,705
Total number of flights = 995
Maximum passenger capacity per flight = 467
Therefore, the average number of passengers on one flight can be calculated as follows:
Average number of passengers per flight = Total number of passengers / Total number of flights
= 456,705 / 995
= 459.18 (rounded to two decimal places)
However, since the maximum passenger capacity per flight is 467, the actual number of passengers on one flight cannot be higher than that. Therefore, the answer is:
Number of passengers on one flight = Maximum passenger capacity per flight = 467
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The given question is incomplete, the complete question is:
United Airlines flies 456705 passengers every week. A Flight has a maximum passenger capacity of 467 passengers. if it completes 995 flights in a week ,then how many passengers are there on one flight?
Find the missing variable and indicated
angle measure.
A
X =
51°
(3x)°
B
C
mzBDC =
This is the last one for me to be done with the assignment please answer fast:) will mark brainiest
Answer:x=a
Step-by-step explanation:
Susan borrowed $25,000 from a bank for one year at the rate of 10.5% per annum. Compute the amount she must pay to the bank to clear her loan amount; interest is compounded half yearly.
Susan must pay back $27,693.90 to clear her loan amount, including the interest.
How to find half-yearly compounded interest?If the interest is compounded half-yearly, then we need to use the formula:
[tex]A = P(1 + \dfrac{r}{n})^{(nt)}[/tex]
Where:
A is the amount to be paid back
P is the principal amount borrowed (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the time duration of the investment in years
Here, P = $25,000, r = 10.5% = 0.105, n = 2 (since interest has compounded half yearly), and t = 1 year.
Substituting these values into the formula, we get:
[tex]A = 25,000(1 + \dfrac{0.105}{2})^{(2\times1)}[/tex]
= $25,000(1.0525)²
= $27,693.90 (rounded to the nearest cent)
Therefore, she must pay back $27,693.90 to clear her loan amount, including the interest.
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What is the frequency of each interval?
81, 65, 2, 24, 25, 44, 97, 12, 38, 37
Question 5 options:
81 - 100
61 - 80
1 - 20
21 - 40
41 - 60
1.
1
2.
2
3.
3
4.
4
5.
5
1 - 20, 21 - 40, 41 - 60, 1.1, 2.2, 3.3, 4.4, 5.5
Which function has a zero with a multiplicity of 2 ? f(x)=-3x^(2)+12x+12 f(x)=3x^(2)+24x+48 f(x)=x^(2)+3x+9 f(x)=x^(2)-2x-1
The function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.
To determine which function has a zero with a multiplicity of 2, we need to find the discriminant of each quadratic function and check if it is equal to zero.
The discriminant is given by the formula: b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
Let's calculate the discriminant for each function:
1) f(x) = -3x² + 12x + 12
Discriminant: b² - 4ac = (12)² - 4(-3)(12) = 144 + 144 = 288 (not zero)
2) f(x) = 3x² + 24x + 48
Discriminant: b² - 4ac = (24)² - 4(3)(48) = 576 - 576 = 0 (zero)
3) f(x) = x² + 3x + 9
Discriminant: b² - 4ac = (3)² - 4(1)(9) = 9 - 36 = -27 (not zero)
4) f(x) = x² - 2x - 1
Discriminant: b² - 4ac = (-2)² - 4(1)(-1) = 4 + 4 = 8 (not zero)
Based on the calculations, the function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.
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The demand function for a certain brand of compact discs is given by
p = −3x2 − 4x + 69
where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.
(a) Compute the price, p, when x = 4.
Price, p = dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 4. Do not round your answer.
Rate of change of demand, x' = thousands of units per dollar
(c) Compute the elasticity of demand when x = 4. Do not round your answer.
The elasticity of Demand =
2) The yearly demand function for Penn State Bakery cookie trays is given by
x2 + 3p2 + 8x + 12p = 216
where p is the wholesale unit price in dollars and x is the quantity demanded each year, measured in units of a thousand.
(a) Compute the price, p, when x = 10.
Price, p =
dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 10. Do not round your answer.
Rate of change of demand, x' = thousands of units per dollar
(c) Compute the elasticity of demand when x = 10. Do not round your answer.
Elasticity of Demand =
3)
The weekly demand equation is given by
p + x + 3xp = 53,
where x is the number of thousands of units demanded weekly and p is in dollars. If the price p is decreasing at a rate of 70 cents per week when the level of demand is 5000 units, then demand is
decreasing
increasing
at units per week.
4)
A company is decreasing production of math-brain protein bars at a rate of 100 cases per day. All cases produced can be sold. The daily demand function is given by
p(x) = 20 −
x
200
,
where x is the number of cases produced and sold, and p is in dollars.
If the daily production is 800 cases, then revenue is
increasing decreasing
at a rate of dollars per day.
5)
The wholesale price p of e-tablet writing styluses in dollars is related to the supply x in thousands of units by
400p2 − x2 = 14375,
If 5,000 styluses are available at the beginning of a week, and the price is falling at 30 cents per week, then supply is
falling rising
at a rate of styluses per week.
1. a) The price is 5 dollars. b) The rate of change of demand with respect to price, p, when x = 4 is 5/6 thousands of units per dollar. c) The elasticity of demand when x = 4 is 25/24.
2. a) The price is 2*sqrt(3) dollars. b) The rate of change of demand with respect to price, p, when x = 10 is - (3 * sqrt(3))/(7) thousands of units per dollar. c) The elasticity of demand when x = 10 is -6/7.
3. Demand is increasing at -23.30 thousands of units per week.
4. If the daily production is 800 cases, then revenue is decreasing at a rate of -1200 dollars per day.
5. Supply is rising at a rate of 150.90 thousands of styluses per week.
1)
(a) Compute the price, p, when x = 4.
Price, p = −3(4)^2 − 4(4) + 69 = -48 -16 + 69 = 5 dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 4.
Rate of change of demand, x' = - (2 * -3 * x + 4)/(-3 * 2 * x) = -(-24 + 4)/(-24) = 20/24 = 5/6 thousands of units per dollar
(c) Compute the elasticity of demand when x = 4.
The elasticity of Demand = (x'/x) * (p) = (5/6)/(4) * (5) = 25/24
2)
(a) Compute the price, p, when x = 10.
Price, p = sqrt((216 - 10^2 - 8*10)/3) = sqrt((216 - 100 - 80)/3) = sqrt(12) = 2*sqrt(3) dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 10.
Rate of change of demand, x' = - (2 * 3 * p)/(2 * x + 8) = - (6 * 2 * sqrt(3))/(20 + 8) = - (12 * sqrt(3))/(28) = - (3 * sqrt(3))/(7) thousands of units per dollar
(c) Compute the elasticity of demand when x = 10.
Elasticity of Demand = (x'/x) * (p) = (- (3 * sqrt(3))/(7))/(10) * (2*sqrt(3)) = -6/7
3)
The weekly demand equation is given by p + x + 3xp = 53,
where x is the number of thousands of units demanded weekly and p is in dollars. If the price p is decreasing at a rate of 70 cents per week when the level of demand is 5000 units, then demand is increasing at a rate of (53 - 5000 - 70)/(3*70 + 1) = -4917/211 = -23.30 thousands of units per week.
4)
A company is decreasing production of math-brain protein bars at a rate of 100 cases per day. All cases produced can be sold. The daily demand function is given by p(x) = 20 − x/200,
where x is the number of cases produced and sold, and p is in dollars.
If the daily production is 800 cases, then revenue is decreasing at a rate of (20 - 800/200) * (-100) + (800) * (-1/200) * (-100) = (20 - 4) * (-100) + (800) * (1/2) = -1600 + 400 = -1200 dollars per day.
5)
The wholesale price p of e-tablet writing styluses in dollars is related to the supply x in thousands of units by 400p^2 − x^2 = 14375,
If 5,000 styluses are available at the beginning of a week, and the price is falling at 30 cents per week, then supply is rising at a rate of (2 * 400 * p * (-0.30) - 2 * x * x')/(2 * -1 * x) = (800 * p * (-0.30) - 0)/(2 * -5000) = (800 * sqrt(14375 + 5000^2)/400 * (-0.30) - 0)/(2 * -5000) = (800 * sqrt(14375 + 25000000)/400 * (-0.30))/(2 * -5000) = (2 * sqrt(14375 + 25000000) * (-0.30))/(-5000) = 0.012 * sqrt(14375 + 25000000) = 150.90 thousands of styluses per week.
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