The Length of RS = 7 units.
Collinear points:The set of points that lie on a straight line is known as Collinear points. In other words, two or more points that lie on the same straight line are said to be Collinear points.
Here we have
RS = x + 5, ST = 14x − 15, and RT = 20
When R, S, and T are considered as 3 collinear points
=> RT = RS + ST
From the given data,
=> 20 = x + 5 + 14x − 15
=> 15x − 10 = 20
=> 15x = 30
=> x = 2
Length of RS = x + 5 = 2 + 5 = 7
Therefore,
The Length of RS = 7 units
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NO LINKS!!
Consider the function f(x) = log₂(x - 2) - 5
Answer:
a) See attachment.
b) Domain: (2, ∞)
Range: (-∞, ∞)
c) As x → 2⁺, f(x) → -∞
As x → ∞, f(x) → ∞
d) f(3) = -5
e) x = 4
Step-by-step explanation:
Given logarithmic function:
[tex]f(x)=\log_2(x-2)-5[/tex]
Part aAs the argument of a log function can only take positive arguments, the domain is restricted to x > 2 and there is a vertical asymptote at x = 2.
To find the x-intercept, set the function to zero and solve for x:
[tex]\implies \log_2(x-2)-5=0[/tex]
[tex]\implies \log_2(x-2)=5[/tex]
[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies x-2=2^5[/tex]
[tex]\implies x-2=32[/tex]
[tex]\implies x=34[/tex]
Therefore, the x-intercept is (34, 0).
Substitute x = 10 into the function to find a second point on the curve:
[tex]\implies f(10)=\log_2(10-2)-5[/tex]
[tex]\implies f(10)=\log_2(8)-5[/tex]
[tex]\implies f(10)=\log_2(2)^3-5[/tex]
[tex]\implies f(10)=3\log_2(2)-5[/tex]
[tex]\implies f(10)=3(1)-5[/tex]
[tex]\implies f(10)=3-5[/tex]
[tex]\implies f(10)=-2[/tex]
Therefore, a second point on the curve is (10, -2).
Part bAs the argument of a log function can only take positive arguments, the domain is restricted to x > 2:
Interval notation: (2, ∞)The range is unrestricted:
Interval notation: (-∞, ∞)Part cAs x approaches x = 2 from the positive side, the function approaches negative infinity:
As x → 2⁺, f(x) → -∞As x approaches positive infinity, the function approaches positive infinity.
As x → ∞, f(x) → ∞Part dFrom inspection of the graph:
f(3) = -5Check by evaluating f(3) algebraically:
[tex]\implies f(3)=\log_2(3-2)-5[/tex]
[tex]\implies f(3)=\log_2(1)-5[/tex]
[tex]\implies f(3)=0-5[/tex]
[tex]\implies f(3)=-5[/tex]
Part e
From inspection of the graph:
f(x) = -4 ⇒ x = 4Check by evaluating f(x) = -4 algebraically:
[tex]\implies \log_2(x-2)-5=-4[/tex]
[tex]\implies \log_2(x-2)=1[/tex]
[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies x-2=2^1[/tex]
[tex]\implies x-2=2[/tex]
[tex]\implies x=4[/tex]
Flexural strength is a measure of a material's ability to resist failure in bending. The accompanying data are on flexural strength of concrete (in MegaPascal, MPa, where 1 Pa (Pascal) = 1.45 cross 10^-4 psi):
(a) Construct a stem-and-leaf display of the data. (Enter numbers from smallest to largest separated by spaces. Enter NONE for stems with no values.)
What appears to be a representative strength value?
MPa
Do the observations appear to be highly concentrated about the representative value or rather spread out?
1) highly concentrated around the representative value
2) spread out with a large range
(b) Does the display appear to be reasonably symmetric about a representative value, or would you describe its shape in some other way?
1) reasonably symmetric positive skewness
2) negative skewness completely random distribution
(c) Do there appear to be any outlying strength values?
Yes
No
(d) What proportion of strength observations in this sample exceed 10 MPa? (Round your answer to two decimal places.)
The proportion of strength observations in this sample that exceed 10 MPa can be calculated by taking the total number of observations that exceed 10 MPa divided by the total number of observations.
To do this, we count the number of observations in the stem-and-leaf display greater than 10 MPa. We see that there are 8 observations greater than 10 MPa. We then divide 8 by the total number of observations, which is 24. This yields a proportion of 0.33 or 33%. This means that 33% of the strength observations in this sample exceed 10 MPa.There are six observations in the stem-and-leaf display that are greater than 10 MPa: 11, 11, 11, 11, 12, and 13. Therefore, the number of observations greater than 10 MPa is 6. The total number of observations is 11, as there are 11 stems with leaves. Therefore, the proportion of strength observations in the sample that exceed 10 MPa is (6/11) x 100 = 54.55%. This means that 54.55% of the observations in the sample exceed 10 MPa.
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x/3 = 9 please help whats supposed to go on top
The value of x in the equation x/3 = 9 is 27
What is linear equation?A linear equation is an algebraic equation where each term is either a constant or a variable raised to the first power. In other words, none of the exponents can be greater than 1. For example, x² is a variable raised to the second power, but x is a variable raised to the first power.
Example of linear equation with two variables is 2x + y = 5, where x and y are the variables . 5x + 2 = 6 is an example of linear equation with one variable
in the expression x/3 = 9
We multiply both sides by 3
therefore x/3 × 3 = 9 × 3
x = 27
therefore the value of x in the equation x/3 = 9 is 27
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show that -0.75 ( -4f + 12 ) and ( 5f + 9 ) - ( 2f + 18 ) are equivalent expressions.
The Linear terms -0.75(-4f + 12) and (5f + 9) - (2f + 18) are actually an equivalent expressions.
What is equivalent expressions?Equivalent algebraic expressions are ones that, despite their apparent differences, actually have the same value. As a result of their similarity, they will produce the same outcomes regardless of the values we choose to use as their variables.
If -0.75 ( -4f + 12 ) and ( 5f + 9 ) - ( 2f + 18 ) are equivalent expressions then
-0.75(-4f + 12) = (5f + 9) - (2f + 18)
3f − 9 = (5f + 9) - (2f + 18)
3f − 9 = 5f + 9 -2f - 18
3f − 9 = 5f - 2f + 9 - 18
3f − 9 = 3f − 9
Thus, The Linear terms -0.75(-4f + 12) and (5f + 9) - (2f + 18) are actually an equivalent expressions.
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Suppose a college bookstore buys a textbook from a publishing company and then marks up the price they paid for the book 20% and sells it to a student at the marked-up price. If the student pays 125$ for the textbook, what did the bookstore pay for it? Round your answer to the nearest cent.
The amount of money that the bookstore paid for the textbook is $104.17.
How much did the bookstore pay?When the price of an item is marked-up, it means that the price of the item is increased by a percentage. After a markup, the price of the item increases.
In order to determine the amount the bookstore paid for the bookstore, divide the price the student paid for the textbook by the percentage discount.
Price before the mark up = marked up price / (1 + percentage discount)
= $125 / ( 1 + 0.2)
= $125 / 1.2
= $104.17
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Mr. Kimball receives a $3000 annual salary
increase on the anniversary of his hiring if he receives
a satisfactory performance review. His starting salary
was $41,250. Write an equation to show k, Mr.
Kimball's salary after t years at this company if his
performance reviews are always satisfactory.
Answer:
OK so he gets 3000 a year
so 3000t is what we need. thats 3000 times the number of years.
His starting salary is 41250, so we would add that on.
K = 3000t + 41250.
Step-by-step explanation:
A middle school took all of its 6th grade students on a field trip to see a symphony at a theater that has 4100 seats. The students filled 2296 of the seats in the theater. What percentage of the seats in the theater were filled by the 6th graders on the trip?
Answer:
56%
Step-by-step explanation:
4100 seats.
filled 2296
% = 2296/4100 = 56%
Answer:
56
Step-by-step explanation:
it is Right
What would the cash discount statement "3/10, 2/20, net/30" mean on an
invoice?
Cash discounts describe a reward that a seller provides to a buyer in exchange for paying a bill in full before the due date.
Is cash discount recorded in income statement?In contrast to trade discounts, which are recorded in buy and sales books, cash discounts are an expense to a corporation that appear in the profit and loss account.Using the price and the rate, the formula determines the cash discount. Cash Discount is calculated as Purchase Price x Discount Rate. For instance, if a product costs $200 and a 10% discount is applied, the cash discount would be $20, translating to a $20 savings for the customer.The discount that the seller of a product provides to the buyer at the time of payment for the purchase is referred to as a cash discount. This discount is offered at the amount of the invoice.To learn more about discounts refer to:
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With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is okay. The ABC Electronics Company has just manufactured 1500 write-rewrite CDs, and 160 are defective. If 6 of these CDs are randomly selected for testing, what is the probability that the entire batch will be accepted?
To find the probability that the entire batch will be accepted, we need to find the probability that all 6 CDs selected for testing are not defective. This is known as the probability of a "success" in the binomial distribution, which is (1 - population proportion of defective CDs) raised to the power of the number of CDs selected for testing.
The population proportion of defective CDs is 160/1500 = 0.107. So the probability of a CD being not defective is 1- 0.107 = 0.893. Therefore, the probability of all 6 CDs selected for testing are not defective is (0.893)^6 = 0.827 which is about 83%
Therefore, the probability that the entire batch of 1500 CDs will be accepted if 6 CDs are randomly selected for testing is 0.827 or about 83%.
It's worth noting that this method is based on some assumptions and the actual probability might be different.
Haley solves the inequality - 13 ≥r+7 and
graphs the solution on a number line with a solid circle at -20 and an
arrow pointing left. Is she correct? Support your answer, and give
the correct description if she is incorrect.
Haley is right because the solution is r ≤ -20
What is an equation?An equation is an expression that shows the relationship between numbers and variables.
Inequalities is used for the non equal comparison of these numbers and variables.
Given the inequality:
-13 ≥ r + 7
Add 13 to both sides:
-13 + 13 ≥ r + 7 + 13
0 ≥ r + 20
subtract r from both sides:
0 - r ≥ r + 20 - r
-r ≥ 20
divide both sides by -1:
r ≤ -20
The solution is a graph on a number line with a solid circle at -20 and an arrow pointing left
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Andrew is preheating his oven before using it to bake. The initial temperature of the oven is 65° and the temperature will increase at a rate of 10° per minute after being turned on. What is the temperature of the oven 15 minutes after being turned on? What is the temperature of the oven t minutes after being turned on?
Answer:
see belwo
Step-by-step explanation:
65
increase 10/min
in 15min it'll increase 150
so 65+150 =215
over t mins it'll be 65+10*t
Answer: 215°
t = minutes after being turned on
If the oven temperature increases by 10 per minute, and the initial temperature is 65, this can be represented as:
65 + 10t
After 15 minutes:
65 + 10(15)
65 + 150
215°
If x = temperature of the oven, the temperature of the oven t minutes after being turned on is:
65 + 10t = x
A financial analyst following Singh, Inc. has created the following probability distribution to describe Singh's expected return over the next year. Singh's fortunes rely on the success of a new product. The probability distribution shown reflects the analyst's expectation of the stock's return depending on how well the new product sells. What is the expected return on the stock? 7.50% 2.00% 22.50% 19.00% 14.00% What is the expected standard deviation of stock returns? 30.51% 26.57% 36.82% 24.23% 31.21%
The expected return on the stock is 22.50% and the expected standard deviation of stock returns is 26.57%
The standard deviation is a measure of the spread of a set of data around its mean (average) value. It is a statistical measure of the dispersion of data. The standard deviation is the square root of the variance, which is the average of the squared differences between each data point and the mean.
The expected return on the stock is the weighted average of the possible returns, with the weights being the probabilities of those returns occurring. To calculate the expected return, we multiply each return by its corresponding probability and add the products together.
Expected return = (7.50% * 0.2) + (2.00% * 0.1) + (22.50% * 0.3) + (19.00% * 0.1) + (14.00% * 0.3) = 7.50% + 0.2% + 6.75% + 1.90% + 4.20% = 22.50%
The expected standard deviation of stock returns is a measure of the volatility or risk of the stock. It is calculated by taking the square root of the variance, which is the average of the squared differences between each return and the expected return.
Expected standard deviation = √(((7.50% - 4.20%)^2 * 0.2) + ((2.00% - 4.20%)^2 * 0.1) + ((22.50% - 4.20%)^2 * 0.3) + ((19.00% - 4.20%)^2 * 0.1) + ((14.00% - 4.20%)^2 * 0.3)) = √(26.57%)
So, the expected return on the stock is 4.20% and the expected standard deviation of stock returns is 26.57%.
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What number needs to be added to 4 and 9 so that the ratio of the first number to the second becomes 2 : 3?
Answer:
6
Step-by-step explanation:
4+6=10
9+6=15
the ratio of 10 and 15= 2:3
Amelia grabbed 50% of the M&M's from a bowl. Then, Jess took 50% of what was left. Dominick takes 50% of the remaining M&M's. There were 3 left. How many M&M's were in the bowl before Amelia arrived.
The number of M&M's that were in the bowl before Amelia arrived is; 12
How to solve Percentage problems?Let the original amount in the bowl be x. Thus;
If Amelia grabbed 50%, then amount left = (100% - 50%)x = 50%x = 0.5x
Now, we are told that domicick took 50% of what was keft after amelia. Thus; New amount left = 100%x - (0.5x + 0.25x) = 0.25x
Now, we are told that there were finally 3 left after all the deductions. Thus;
0.25x = 3
x = 3/0.25
x = 12
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Q"
R
X
S
15
12
9
N
Given AQRS~AXYZ, what is the value of tan(Q)?
OOO
m/n M/T +/n +/m
²
Answer:
3/4
Step-by-step explanation:
since they are similar, they have the exact same angles.
angle Q = angle X
tan(angle) = sin(angle)/cos(angle)
that is the same ratio here as
9/12 = 3/4
Need help with #20 please
The length of the side DE is 9.
What is Parallelogram?Parallelogram is a type of polygon with four sides, four angles and four vertices. It is a type of Quadrilateral.
Opposite sides are parallel, and the opposite sides and opposite angles are equal.
Given is a figure of a quadrilateral DEFG.
We have from the figure, two triangles, ΔDGF and ΔDEF.
Diagonal of the quadrilateral DF is a common side for both triangles.
Given that DG = EF
Also ∠FDG = ∠EFD
So we get the two triangles are congruent by SAS theorem.
So the given figure is parallelogram.
So Length of DE = Length of GF = 4x + 1.
We have EF = DG
6x + 4 = 4x + 8
2x = 4
x = 2
Length of DE = 4x + 1 = 9
Hence the length of the side DE of the parallelogram is 9.
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Fix the inequality below to correctly show the domain of
the function.
The domain of the graph will be [-1.50, 1). Then the domain in the form of inequality will be - 1.50 ≤ x < 1.
How to draw the graph of the function?The collection includes all locations on the surface of the shape (x, f(x)) that make up a function of f's graph. We may alternatively say that the graph of f is the curve of y = f. (x). As a result, the diagram of an equation is a particular instance of the graphs of functions.
The domain of the graph is given as,
- 1.50 ≤ x < 1
The domain of the graph will be [-1.50, 1). Then the domain in the form of inequality will be - 1.50 ≤ x < 1.
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The complete question is attached below.
Which of the following sets is closed under division?A. Natural numbersB. Non zero intergersC. Irrational numbersD. Non zero rational numbers
A set is considered closed under a particular operation if the result of that operation applied to any two elements of the set is also an element of the set.
Division is an operation on two elements, where one element is divided by the other.
A. Natural numbers are not closed under division since when we divide two natural numbers, the result may not be a natural number. For example, when we divide 4 by 2, the result is 2, which is a natural number. However, when we divide 6 by 3, the result is 2.00, which is not a natural number.
B. Non zero intergers are not closed under division since when we divide two non zero integers, the result may not be a non zero integer. For example, when we divide 8 by 2, the result is 4, which is a non zero integer. However, when we divide 6 by 3, the result is 2.00, which is not a non zero integer.
C. Irrational numbers are not closed under division since when we divide two irrational numbers, the result may not be an irrational number. For example, when we divide √2 by √2, the result is 1, which is an irrational number. However, when we divide √3 by √2, the result is 1.73, which is not an irrational number.
D. Non zero rational numbers are closed under division since when we divide two non zero rational numbers, the result will always be a non zero rational number. For example, when we divide 6 by 3, the result is 2, which is a non zero rational number. Similarly, when we divide 8 by 4, the result is 2, which is also a non zero rational number.
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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 8x^3, y = 0, x = 1; about x = 2
Answer is 24pi/5
I just don't know how to get it.
Volume of the solid obtained by rotating the region bounded by the given curves about the specified line is [tex]\frac{24\pi }{5}[/tex]
Now, According to the question:
The shell method formula sums the volume of a cylindrical shell over the radius of the cylinder, and the volume of a cylinder is given by V=2πrh V = 2 π r h where r is the radius and h is the height.
Take a vertical cross section of the region. The width of the cross section is dx, and the height is [tex]8x^3.[/tex]
Rotate the cross section about the line x = 2. The height of the shell is [tex]8x^3.[/tex] the radius is 2 - x, and the thickness is dx.
Volume of shell = 2π(2-x)([tex]8x^3[/tex])dx = 2π([tex]16x^3 - 8x^4[/tex])dx
Volume of solid = [tex]2 \pi \int\limits^1_0 [16x3 - 8x4]dx[/tex]
[tex]= 2\pi (4x^4 - (8/5)x^5)^1_0[/tex]
= 24π/5
Hence, Volume of the solid obtained by rotating the region bounded by the given curves about the specified line is [tex]\frac{24\pi }{5}[/tex].
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What is 2+2
PLEASE HELP!!!!
Answer:
4
Step-by-step explanation:
2+2=4
propose five combinations of aqueous ionic reagents that likely would form a precipitate when they are mixed. Write the balanced full molecular equation and the balanced net ionic equation for each of your choices.
Equations will be AgNO3(aq)+HCl(aq)⟶AgCl(s)+HNO3(aq)
AgNO3+HCl⟶AgCl+HNO3 for acid
Ag:+1⟶+1
NO3:−1⟶−1
H:+1⟶+1
Cl:−1⟶−1
According to the solubility table, nitrates are always soluble, so the strong ionic bond between silver ions and nitrate ions are broken by water molecules because of ion-dipole attraction. Acids ionize in water to give mobile ions, so hydrogen chloride in aqueous solution gives out hydrogen ions (and form hydronium ions) and chloride ions.
There are five types of particles in the solution :
H2O molecules, Ag+ ions, NO3− ions, H+ ions, and Cl− ions. (Slight ionization of water is neglected in this case.) They freely bump into each other as they are mobile. However, this freely moving condition is inhibited by the interaction between Ag+ ions and Cl− ions.
According to the solubility table, AgCl is insoluble in water. When Ag+ ions and Cl− ions bump into each other, they strongly attract each other, in which the strong ionic force cannot be separated by the ion-dipole force between them and H2O molecules. As a result, AgCl evolves as a white solid.
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The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
Answer:Option B , C and E are correct
Step-by-step explanation:
y^2-5y=750
750-y(y-5)=0
(y+25)(y-30)=0
Need help with #6 please help
Answer:
20 m
Step-by-step explanation:
Eric works at the deil on the weekends to earn extra money he makes 10 per hour making sandwiches and 14 per hour delivering orders eric puts half of his total earning in a savings account for college he wants to know how much he saves each week help i need answers fast
Answer:
you do it step by step so it can be expressed
a group of librarians is interested in the numbers of books and other media that patrons check out from their library. they examine the checkout records of 150 150150 randomly selected adult patrons.
For the given situation of number of visitors checkout out the library and 150 are randomly selected the population and the sample is represented by :
Option B. The population is all the given adult visitors of the library ; and the sample is the 150 selected visitors.
Total population is represented by all the adult visitors check out the library.
Number of adult visitors examine by the librarian = 150 randomly selected adult visitors.
Here 150 randomly selected adult visitors represents the sample of the whole population.
Therefore, the population and the sample for the given situation of visiting the library is given by :
Option B. The population is all the given adult visitors of the library ; and the sample is the 150 selected visitors.
The above question is incomplete, the complete question is :
A librarian is interested in the numbers of books that visitors check out from the library. She examines the checkout records of 150
randomly selected adult visitors.
Identify the population and sample for this situation.
A The population is all visitors of the library; the sample is the adult visitors
of the library.
B The population is all adult visitors of the library; the sample is the 150
visitors selected.
C The population is all visitors who check out at least 1 book from the
Library; the sample is the 150 visitors selected.
D The population is the 150 visitors selected; the sample is the 150 visitors
who check out at least 1 book from the library.
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division story for 40 divided by 4
I have 40 cupcakes in my house. That day, 3 friends came over. All of us have the same number of cupcakes each. How many cupcakes did we each eat?
The story of 40 divided by 4 reminding everyone about the importance of fairness and sharing.
There were 40 colorful candies waiting to be shared equally among 4 friends.
The candies were so delicious and enticing that the friends couldn't wait to indulge in their sweet treat.
The four friends, named Alex, Ben, Chris, and Dana, gathered around a table with their empty candy bags in front of them.
They were filled with excitement as they knew it was time to divide the candies fairly.
With the candies in the center of the table, they decided to take turns dividing them into smaller piles.
Alex went first and carefully divided the candies into 10 piles, making sure each pile had an equal number of candies.
This left 10 candies in each pile.
Then, it was Ben's turn. He took one pile of 10 candies and divided it into two equal piles, with 5 candies in each.
Ben did the same with the other nine piles, resulting in a total of 20 piles, each containing 5 candies.
Next, it was Chris's turn. He picked one pile with 5 candies and divided it into two equal piles, leaving 2 candies in each pile.
Chris repeated this process with the remaining 19 piles, resulting in a total of 40 piles, each containing 2 candies.
Finally, it was Dana's turn. Dana took one pile with 2 candies and divided it equally between two empty candy bags, leaving 1 candy in each bag. Dana repeated this process with the remaining 39 piles, resulting in a total of 80 candy bags, each containing 1 candy.
In the end, all four friends had received their fair share of candies. They joyfully took their candy bags and started savoring the delicious treats, grateful for their successful division.
And so, the story of 40 divided by 4 came to a sweet conclusion, reminding everyone about the importance of fairness and sharing.
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In a classroom, 100 plastic fish are in a tub. The tub is hidden from the student view. Some fish are green, and the rest are yellow. The students know that there are 100 fish, but they don’t know how many of each color there are. The students go fishing, each time, picking a random fish from the tub, recording its color, and throwing the fish back in the tub. At the end of the day, 65 fish have been chosen, 12, green and 53 yellow. What is the best estimate you can give for the number of green fish in the number of yellow fish in the tub? Describe how to calculate this best estimate, and explain why your method of calculation makes sense in a way that a seventh grader my understand. Is your best estimate necessarily accurate? Why or why not?
The best estimate for the number of fish is that there are 82 yellow fish and 18 green fish, which you can calculate using the rule of three. This estimate is expected to be accurate.
What is the estimated number of fish of each color?This can be calculated using a rule of three as follows. Let's start with the yellow fish:
53 = 65
x = 100
53 x 100 / 65 = 82 yellow fish
Green fish:
12 = 65
x = 100
12 x 100 / 65= 18 green fish
Is this estimate accurate?This estimate is expected to be accurate; however, there might be small differences in real life such as 80 yellow fish and 20 green fish.
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1. Jamar has two bowls of fruit from which he can choose a
snack when he gets home from school. One bowl has 1 yellow,
1 red, and 2 green apples. The other bowl has 5 each of
boysenberries, strawberries, blueberries, and blackberries.
What is the probability that he gets a yellow apple and a
boysenberry when he reaches into the bowls?
Answer:
One bowl has 1 yellow, 1 red, and 2 green apples. The other bowl has 5 each of boysenberries, strawberries, blueberries, and blackberries.
Step-by-step explanation:
The deck of a bridge is suspended 285 feet above a river. If a pebble falls off the side of the bridge, the height, in feet, of the pebble above the water surface after t seconds is given by
y = 285 − 16t2.
(a). Find the average velocity (in ft/s) of the pebble for the time period beginning when t = 2 and lasting the following amount of time.
(I) 0.1 seconds = ___ ft/s (ii) 0.05 seconds = _____ft/s (iii) 0.01 seconds = _____ ft/s (b). Estimate the instantaneous velocity (in ft/s) of the pebble after 2 seconds. = ____ft/s
(a) (i) Average velocity = -160 ft/s, (ii) Average velocity = -320 ft/s, (iii) Average velocity = -1600 ft/s. (b) Instantaneous velocity = -320 ft/s.
(a) (i) Average velocity = change in height/change in time
= (285 - (285 - 16(2.1)^2))/0.1
= -160 ft/s
(ii) Average velocity
= (285 - (285 - 16(2.05)^2))/0.05
= -320 ft/s
(iii) Average velocity
= (285 - (285 - 16(2.01)^2))/0.01
= -1600 ft/s.
(b) Instantaneous velocity = -320 ft/s.
The average velocity of the pebble after t seconds is given by the equation y = 285 − 16t2. To find the average velocity for the time period beginning when t = 2, we can calculate the change in height of the pebble over the given time period and divide it by the change in time. For instance, for 0.1 seconds, we can calculate the average velocity by dividing the change in height (285 - (285 - 16(2.1)^2)) by the change in time (0.1). This gives us an average velocity of -160 ft/s. Similarly, for 0.05 and 0.01 seconds, the average velocity can be calculated to be -320 ft/s and -1600 ft/s respectively. To estimate the instantaneous velocity of the pebble after 2 seconds, we can assume that the change in time is very small and approximate the velocity to be the average velocity for 0.01 seconds. This gives us an instantaneous velocity of -320 ft/s.
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show that if a and b both have linearly independent column vectors, then the column vectors of c will also be linearly independent.
To show that if the column vectors of matrix A and B are linearly independent, then the column vectors of matrix C = A*B will also be linearly independent, we can use the following proof:
Assume that the column vectors of matrix A and B are linearly independent. Let's assume that the column vectors of matrix C = A*B are linearly dependent.
This means that there exists a non-trivial linear combination of the column vectors of C that results in the zero vector.
C = AB => c_1 = Ab_1, c_2 = Ab_2, ... c_n = Ab_n
Let's assume that a linear combination of c1, c2, ..., cn results in the zero vector.
a_1c_1 + a_2c_2 + ... + a_n*c_n = 0
Now, we can substitute the matrix multiplication for ci:
a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n = 0
Since matrix A is non-singular, we can left multiply by A^-1
A^-1*(a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n) = A^-1*0
This simplifies to
a_1b_1 + a_2b_2 + ... + a_n*b_n = 0
This is a non-trivial linear combination of the column vectors of matrix B that results in the zero vector, which contradicts the assumption that the column vectors of matrix B are linearly independent. Therefore, the column vectors of matrix C = A*B must also be linearly independent.
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