The first three terms of the sequence Tₙ = n^2 - 2n - 6 are -7, -8, and 0, and the sequence is a quadratic sequence with a parabolic graph that opens upward.
To find the first three terms of the sequence Tₙ = n^2 - 2n - 6, we simply need to substitute the first three positive integers for n, which gives us:
T₁ = 1^2 - 2(1) - 6 = -7
T₂ = 2^2 - 2(2) - 6 = -8
T₃ = 3^2 - 2(3) - 6 = 0
Therefore, the first three terms of the sequence are -7, -8, and 0.
The sequence Tₙ is a quadratic sequence, which means that it has a second-order difference. In other words, the differences between the terms of the sequence form a linear sequence.
Specifically, the first differences are 2, 4, 6, 8, and so on, which form an arithmetic sequence with a common difference of 2. The second differences are all equal to 2, which confirms that the sequence is quadratic.
The graph of the sequence Tₙ is a parabola that opens upward, with a vertex at (1, -7). This means that the sequence starts with a negative term, then decreases until it reaches a minimum at n = 1, and then increases indefinitely.
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Factor the equation
25w^2-81
Answer:
(5w + 9)(5w - 9)
Step-by-step explanation:
hmmm, dunno man.
(5w + 9)(5w - 9)
The diagonal of rectangle is 26 and its perimeter is 68 what are its dimensions
If the diagonal of the rectangle is 26 and its perimeter is 68, its dimensions are either 24 by 10 or 10 by 24.
Let's denote the length and width of the rectangle as L and W, respectively.
We know that the diagonal of the rectangle is 26, which we can express using the Pythagorean theorem:
[tex]L^2[/tex]+ [tex]W^2[/tex]=[tex]26^2[/tex]
We also know that the perimeter of the rectangle is 68:
2L + 2W = 68
We can simplify the second equation by dividing both sides by 2:
L + W = 34
We can use this simplified equation to express one variable in terms of the other:
L = 34 - W
This is what we get when we enter it into the first equation:
[tex](34 - W)^2[/tex] + [tex]W^2[/tex] = [tex]26^2[/tex]
Expanding the square and simplifying, we get:
[tex]2W^2[/tex] - 68W + 420 = 0
Dividing both sides by 2, we get:
[tex]W^2[/tex]- 34W + 210 = 0
The quadratic formula can be used to find W:
W = (-b ± t[tex]\sqrt{(b^2 - 4ac)}[/tex] )/ 2a
where a = 1, b = -34, and c = 210. Plugging in these values, we get:
W = (34 ± [tex]\sqrt{34^2 - 4(1)(210))}[/tex]) / 2(1)
W = (34 ± [tex]\sqrt{196}[/tex]) / 2
W= (34 ± 14)/2
W = 17 ± 7
As the breadth cannot be negative, we can disregard the negative solution. Therefore, we have:
W = 24 or W = 10
If W = 24, then L = 10 (using L = 34 - W). If W = 10, then L = 24. Therefore, the dimensions of the rectangle are either 24 by 10 or 10 by 24.
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Solve triangle ABC given that A = 56°, B = 57°, and b = 9. 0. Round the length of the sides to the nearest hundredth.
The sides of the triangle to the nearest hundredth are: a ≈ 8.99, b ≈ 7.96
and c ≈ 9.44.
To solve the triangle, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides.
Using this law, we can find the length of side a as follows:
sin A / a = sin B / b
sin 56° / a = sin 57° / 9
a = 9 * sin 56° / sin 57°
a ≈ 8.99
Similarly, we can find the length of side c as follows:
sin C / c = sin B / b sin (180° - A - B) / c = sin 57° / 9
sin 67° / c = sin 57° / 9
c = 9 * sin 67° / sin 57°
c ≈ 9.44
Now, we can use the Law of Cosines to find the length of the remaining side, side b:
b 2 = a 2 + c 2 - 2ac cos B
b 2 = (8.99) 2 + (9.44) 2 - 2(8.99)(9.44) cos 57°
b ≈ 7.96
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Claire buys furniture to the value of R35 000. she pays a 15% cash deposit and signs a hire purchase agreement for the balance. the interest rate will be 13% per annum and she will pay off the loan over a period of 3 years. what is the value of the deposit?
Answer:
Total amount = R41 337.50
Step-by-step explanation:
The value of the deposit is 15% of the total purchase price:
Deposit = 15/100 x R35 000
Deposit = R5 250
To calculate the balance that Claire will have to pay off, we need to subtract the deposit from the total purchase price:
Balance = Total purchase price - Deposit
Balance = R35 000 - R5 250
Balance = R29 750
To calculate the interest that Claire will have to pay on the balance, we can use the formula:
Interest = Principal x Rate x Time
where:
Principal is the amount of the balance
Rate is the annual interest rate (13%)
Time is the duration of the loan in years (3)
So, the interest that Claire will have to pay is:
Interest = R29 750 x 0.13 x 3
Interest = R11 587.50
Therefore, the total amount that Claire will have to pay back over the 3-year period is:
Total amount = Balance + Interest
Total amount = R29 750 + R11 587.50
Total amount = R41 337.50
Note that this assumes that the interest is calculated annually and added to the balance at the end of each year. In reality, the interest may be calculated differently depending on the terms of the hire purchase agreement.
what is the multiplier for 1%, and 2% decay?
"The multiplier for decay is given by the formula:
multiplier = (1 - (decay rate/100))^n
where "decay rate" is the percentage rate of decay per time period (usually per year), and "n" is the number of time periods.
For a decay rate of 1%, the multiplier would be:
multiplier = (1 - (1/100))^n = 0.99^n
For a decay rate of 2%, the multiplier would be:
multiplier = (1 - (2/100))^n = 0.98^n
The value of the multiplier depends on the number of time periods "n". For example, if we want to find the value of an initial quantity after 3 years of 1% decay, we would use the multiplier of 0.99 raised to the power of 3:
multiplier = 0.99^3 = 0.9703
This means that the final quantity would be 97.03% of the initial quantity.
Similarly, if we want to find the value of an initial quantity after 5 years of 2% decay, we would use the multiplier of 0.98 raised to the power of 5:
multiplier = 0.98^5 = 0.9039
This means that the final quantity would be 90.39% of the initial quantity." (ChatGPT, 2023)
The relationship between the number of decibels β and the intensity of a sound I
in watts per square meter is
β=10log(I10−12)
(a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter.
(b) Determine the number of decibels of a sound with an intensity of 10−2
watt per square meter.
(c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain.
(a) A sound with an intensity of 1 watt per square meter has a decibel level of -120.
(b) A sound with an intensity of 10⁻² watt per square meter has a decibel level of -140.
(c) The number of decibels is not 100 times as great as the intensity, even though the intensity in part (a) is 100 times greater than the intensity in part (b).
What is the sound level?
Using the relationship of intensity of sound and number of decibels, we can calculated the required number of decibels as follows;
β =10 log(I x 10⁻¹²)
(a) Decibel level: plugging in I = 1 watt per square meter into the equation, we get:
β = 10 log(1 x 10⁻¹²) = 10 log(10⁻¹²) = -120 decibels
(b) Decibel level: plugging in I = 10⁻² watt per square meter into the equation, we get:
β = 10 log(10⁻² x 10⁻¹²) = 10 log(10⁻¹⁴) = -140 decibels
(c) Let's calculate the decibel level for the sound in part (a) using the given formula:
β₁ = 10 log(I₁ x 10⁻¹²) = 10 log(100 x I₂ x 10⁻¹²) = 10 log(I₂ x 10⁻¹⁴) + 20
where;
I₂ is the intensity of the sound in part (b).Using the result from part (b), we can substitute I₂ = 10⁻² watt per square meter and get:
β₁ = -140 + 20 = -120 decibels
Comparing this result to the decibel level for the sound in part (a) (which is also -120 decibels), we see that they are the same.
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A discount store is selling 5 small tables with 8 chairs for $115. Three tables with 5 chairs cost $70.
Which system of linear equations could be used to find the cost of each table (x) and the cost of each chair (y)?
Determine the cost of each table (x) and the cost of each chair (y).
answer both question please and thank you
Using linear equations 5x + 8y = 115 and 3x + 5y = 70, the cost of each table is $15 and the cost of each chair is $5.
What is a linear equation, exactly?
A linear equation is a mathematical expression that describes a straight line relationship between two variables. It can be written in the form y = mx + b, where x and y are variables, m is the slope of the line, and b is the y-intercept.
Now,
Let x = cost of each table and y= cost of each chair. Then, we can set up the following system of linear equations based on the given information:
5x + 8y = 115 (equation 1)
3x + 5y = 70 (equation 2)
Equation 1 represents the cost of 5 small tables with 8 chairs, and equation 2 represents the cost of 3 tables with 5 chairs.
Now,
Solve equation 2 for x:
3x + 5y = 70
3x = 70 - 5y
x = (70 - 5y)/3
Substitute x into equation 1:
5x + 8y = 115
5[(70 - 5y)/3] + 8y = 115
Simplify and solve for y:
350/3 - 25/3 y + 8y = 115
23/3 y = 115 - 350/3
y = 5
Substitute y = 5 into equation 2 and solve for x:
3x + 5y = 70
3x + 5(5) = 70
3x = 45
x = 15
Therefore,
the cost of each table is $15 and the cost of each chair is $5.
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Give each Polynomial a name for its Highest Degree & Number of Terms.
[tex]x^{2} +2x-3[/tex]
[tex]-5x-1[/tex]
[tex]-2v^{3} +7[/tex]
[tex]3x^{4} +2v^{3}-8[/tex]
[tex]9w^{5}[/tex]
Triangle a′b′c′ is a dilation of triangle abc about point p with a scale factor of 12. is the dilation a reduction or an enlargement? responses reduction reduction enlargement
The dilation of triangle ABC about point P with a scale factor of 12 is an enlargement.
In a dilation, the size of a figure is changed by multiplying its dimensions by a scale factor. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is between 0 and 1, the figure is reduced in size. In this case, the scale factor is 12, which is greater than 1. Therefore, triangle ABC is enlarged to form triangle A'B'C' by a factor of 12. Therefore, the dilation is an enlargement.
In the context of triangles, an enlargement means that all sides and angles of the original triangle are increased by the same factor to create a larger triangle. In this case, the sides and angles of triangle ABC are multiplied by a factor of 12 to create triangle A'B'C'. As a result, the corresponding sides and angles of the two triangles are proportional, but the enlarged triangle is 12 times larger than the original triangle.
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What the rate if the principal is 5,000, the time is 10 years, and the interest is 2,950?
Answer: 5.9%
Hope this helps!
Part E Graph the average profit function you created in part C by going to your math tools and opening the Graph tool. Set the scale of the graph to go from -20 to 20 on the x-axis and from -100,000 to 100,000 on the y-axis. Select the graph generated, copy it, and paste an image of your graph in the space provided
Answer:
huh? where is the pic of graph
Step-by-step explanation:
no pic was shown
What is 68+1=
What is 69+0=
Answer:
69 and 69
Step-by-step explanation:
68 + 1 = 69
69 + 0 = 69
look how much people i hav helped
3t-18=4(-3- 3/4t
solve for t
Answer:
t = 1
Step-by-step explanation:
I will assume that there is a parenthesis missing: the equation should read 3t-18=4(-3- 3/4t) and not 3t-18=4(-3- 3/4t
3t-18=4(-3- 3/4t)
3t-18= -12- 12/4t [Remove parentheses by multiplying each term by 4]
3t-6= - 3/t [Simplify]
3t^2-6t = -3 [Multiply both sides by t]
3t^2-6t +3 = 0
3(t^2-2t+1)=0
3(t-1)*(t-1) = 0
Two solutions, both the same value:
(t-1) = 0
t = 1
===
Does t=1 result in 0 for the equation?
3t-18=4(-3- 3/4t)
3(1)-18=4(-3- 3/4*(1))
3-18 = 4(-3-3/4)
-15 = 4(-3.75)
-15 = -15 YES
If you were to publish this chart in a news article, what would your headline be and why?
The headline should be informative and accurately reflect the data presented in the chart.
What is chart?A chart is a visual representation of data or information that can be used to help people understand complex information quickly and easily. Charts can take many different forms, such as bar charts, line charts, scatter plots, pie charts, and more. They are often used in business, science, and journalism to present data in a way that is clear and easy to interpret. By using charts, people can quickly see patterns and trends in data that might be difficult to discern from a simple list of numbers or words.
Here,
However, a good headline for a chart should accurately and concisely summarize the main findings or trends depicted in the chart. It should also capture the reader's attention and make them want to read more. Depending on the chart, some possible headlines could be:
"New study finds rising trend in global temperatures over past decade"
"Stock market reaches all-time high, driven by tech sector growth"
"Poll shows majority of Americans support stricter gun control laws"
"Obesity rates continue to climb in US, particularly in southern states"
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I’m confused. Please help ASAP. How would I find this out?!
Answer:
i think 150 is the answer
Step-by-step explanation:
150 is a 3 digit numbers (as written in question (a)) and is a multiple of 5 and 3
150/5 = 30
150/3 = 50
Kayden is trying to find the height of a radio antenna on the roof of a local building.
He stands at a horizontal distance of 16 meters from the building. The angle of
elevation from his eyes to the roof (point A) is 41°, and the angle of elevation from
his eyes to the top of the antenna (point B) is 49°. If his eyes are 1.5 meters from the
ground, find the height of the antenna (the distance from point A to point B).
Round your answer to the nearest meter if necessary.
By answering the above question, we may state that As a result, the equation antenna's height is roughly 25 metres.
What is equation?A math equation is a mechanism for connecting two claims by using the equals sign (=) to indicate equivalence. A mathematical statement that establishes the equivalence of two mathematical expressions is known as an equation in algebra. The equal symbol, for example, splits the numbers 3x + 5 and 14. A mathematical formula can be used to describe the relationship between two phrases written on opposite sides of a letter. The logo and programme are frequently the same. 2x - 4 Equals 2, for example.
draw a diagram of the situation
B
|
| h
|
A------------C---------
| θ2| θ1
| |
| | 16 m
| |
D E
We need to calculate the height h of point B above point A. To begin, we may use the tangent function to get the heights of points C and E:
[tex]hC = 16 * tan(θ1)\\hE = 16 * tan(θ2)\\DE = hE - hC\\hB = hC + DE * tan(θ2)\\h = hB + 1.5\\θ1 = 41\\θ2 = 49\\DE = 16 * (tan(θ2) - tan(θ1)) ≈ 9.7587 m\\hC = 16 * tan(θ1) ≈ 12.2272 m\\hB = hC + DE * tan(θ2) ≈ 23.7333 m\\h ≈ hB + 1.5 ≈ 25.2333 m\\[/tex]
As a result, the antenna's height is roughly 25 metres.
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The owner of a bike shop would like to analyze the sales data to determine if the business is growing, declining, or remaining flat. The owner has the following data:
Sales Revenue Last Year =$125,000
Sales Revenue Current Year = $150, 000
What is the Sales Growth?
Do not forget to type the % symbol after multiplying your result by 100 to convert to percentage!
Hint: Your answer will NOT have a decimal point.
Example: 75%
Answer: To calculate the sales growth, we need to find the difference between the current year's revenue and the last year's revenue, divide it by the last year's revenue, and then multiply by 100 to convert it to a percentage.
Sales Growth = ((Sales Revenue Current Year - Sales Revenue Last Year) / Sales Revenue Last Year) x 100%
Sales Growth = (($150,000 - $125,000) / $125,000) x 100%
Sales Growth = ($25,000 / $125,000) x 100%
Sales Growth = 0.20 x 100%
Sales Growth = 20%
Therefore, the sales growth is 20%.
Brainliest is appreciated. (:
Step-by-step explanation:
Chin paid $16. 24 for 7 bags of flour that were on sale for 10% off. Each bag contained 2 pounds of flour. What was the regular price for 1 pound of flour
In the following question, if Chin paid $16. 24 for 7 bags of flour that were on sale for 10% off. Each bag contained 2 pounds of flour. The regular price for 1 pound of flour is $1.29.
Chin paid $16.24 for 7 bags of flour that were on sale for 10% off.
Let's first find the total cost of the 7 bags of flour before the discount:
Total cost before discount = $16.24 ÷ 0.9 = $18.04
We know that each bag contained 2 pounds of flour, so the total amount of flour Chin bought was:
7 bags x 2 pounds/bag = 14 pounds
So the cost of 1 pound of flour is:
$18.04 ÷ 14 pounds = $1.29 per pound
Therefore, the regular price for 1 pound of flour would be $1.29.
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NEED AN ANSWER NOW!!
5 c = ____ pt ____c
5 c 4 fl oz - 4 c 3 fl oz
7 gal 2 qt + 3 gal 1 qt
6 qt 1 pt - 2 qt 1 pt
ESTIMATE the number of pints in 445 ounces
The estimate of 445 pints in ounces is 7120 ounces
How to estimate the number of pintsThe given measurement from the question can be represented as
Measure = 445 pints
By the standard metrics of conversion, we have
1 pint = 16 ounces
Using the above conversion rules as a guide, we have the following:
Measure = 445 * 16 ounces
Evaluate the product
So, we have the following representation
Measure = 7120 ounces
The above represents the required estimate
Hence, the estimate is 7120 ounces
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Which inequality is equivalent to −4≥28?
Answer:
-1 ≥ 7
Step-by-step explanation:
-1 ≥ 7 = -4 ≥ 28
-4 ≥ 28
We simplify both sides by 4 and get -1 ≥ 7
Suppose a city with 400,000 population has been growing at a rate of 4% per year. If this rate continues, find the population of this city in 17 years
The population of a city with 400,000 population growing at a rate of 4% per year can be calculated using the following formula:
P(t) = P0 (1 + r)t
Where P0 is the initial population, r is the growth rate, and t is the number of years the growth is applied.
In this example, the initial population is 400,000, the growth rate is 0.04, and t is 17 years. Applying this formula, we get:
P(17) = 400,000 (1 + 0.04)17
P(17) = 615,288
Therefore, the population of this city in 17 years will be 615,288 if the growth rate of 4% per year continues.
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Pls help me I don’t know the answer and don’t understand it
If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.
What is area of rectangle?
Area of rectangle is the region covered by the rectangle in a two-dimensional plane. A rectangle is a type of quadrilateral, a 2d shape that has four sides and four vertices.
We know the perimeter of rectangle is 20 meters. So according to formula of Perimeter of rectangle = 2 (L + B) we can conclude that sum of length and width will be 10 meters.
We are given length as 1, 3, 5, 7, and 9.
So width of the rectangle will be 9, 7, 5, 3 and 1.
Area of Rectangle = L x B
So area of rectangle will be 9, 21, 25, 21, and 9.
If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.
Values are changing in a linear way and not in an exponential way.
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for the following two sequences (i) give the first 5 terms of the sequence. (ii) determine if the sequence converges or diverges. if the sequence converges, determine the limit of the sequence
The sequences can converge to 0 and also diverge towards infinity.
The two sequences are not mentioned in the question. Please provide the two sequences so that I can give you the first 5 terms of each sequence and determine if each sequence converges or diverges.
To find the first 5 terms of a sequence, you need to use the formula given for the sequence and plug in the values for the first 5 terms. For example, if the sequence is given by a_n = 2n + 1, the first 5 terms would be a_1 = 2(1) + 1 = 3, a_2 = 2(2) + 1 = 5, a_3 = 2(3) + 1 = 7, a_4 = 2(4) + 1 = 9, and a_5 = 2(5) + 1 = 11.
To determine if a sequence converges or diverges, you need to find the limit of the sequence as n approaches infinity. If the limit exists, then the sequence converges to that limit. If the limit does not exist, then the sequence diverges.
For example, the sequence a_n = 1/n converges to 0 as n approaches infinity, while the sequence a_n = n^2 diverges as n approaches infinity.
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i need help with this one
The missing values are given as shown in the table
What is a cylinder?A cylinder is a three-dimensional solid figure which has two identical circular bases joined by a curvilinear surface at a particular distance from the center which is the height of the cylinder.
We should understand that the area of a cylinder is given by
Area = 2λr(r+h)
and the volume of the cylinder to be;
volume - λr²h
by substitution we get the values as
diameter radius base area height `cylinder volume
units units square units cubic units
units
4 2 150.9 10 62.9
6 3 151.0 7 63π
8 4 25π 0.875 80π
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Someone PLEASE help me ASAP!! It’s due today!! Show work please! I will mark brainliest if it’s done correctly
The correct answer is experimental 22.5% theoretical 18.2%.
What is theoretical probability?The theoretical probability is the likelihood of the event to happen without conducting an experiment,
The experimental probability is the result of the actual experiment, which in this case is the frequency of the letter M when chosen from the bag. Theoretical probability in this case is the number of M's in the bag divided by the total number of letters in the bag.
In this case, there are nine M's out of a total of 39 letters, giving a theoretical probability of 18.2%. Since the number of M's chosen in the experiment was nine, out of a total of 39 chosen, the experimental probability is 22.5%.
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The correct answer is experimental 22.5% theoretical 18.2%. To calculate the experimental probability of drawing the letter M, we must first identify the total number of items in the bag.
What is theoretical probability?The theoretical probability is the likelihood of the event to happen without conducting an experiment,
The total number of items in the bag is 9 + 12 + 7 + 2 + 4 + 1 + 3 + 2 = 40. Next, we divide the number of items with the letter M (9) by the total number of items to get the experimental probability of drawing the letter M.
Therefore, 9/40 = 0.225
= 22.5%.
To calculate the theoretical probability of drawing the letter M, we must first identify the total number of letters in the bag.
There are 8 letters in the bag (M, A, T, H, E, I, C, S).
Therefore, the theoretical probability of drawing the letter M is
8/40 = 0.2
≈18.2%
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Please help me with this question!
In the triangle, the values are p=10.3, [tex]\angle Q=19.5\textdegree[/tex] and [tex]\angle R = 112.5\textdegree[/tex].
What is trigοnοmetric functiοn?The functiοns οf an angle in a triangle are knοwn as trigοnοmetric functiοns, cοmmοnly referred tο as circular functiοns. In οther wοrds, these trig functiοns prοvide the relatiοnship between a triangle's angles and sides. There are five fundamental trigοnοmetric functiοns: sine, cοsine, tangent, cοtangent, secant, and cοsecant.
Here in the given triangle PQR, [tex]\angle P = 48\textdegree[/tex] , q=5 and r=13.
Now using law of cosine fοrmula,
=> [tex]p^2=q^2+r^2-2qr.cos\angle P[/tex]
=> [tex]p^2=5^2+13^2-2.5.13.cos 48\textdegree[/tex]
=> [tex]p^2=25+169-130cos48\textdegree[/tex]
=> [tex]p^2=107[/tex]
=> p = 10.3
Now using law of sine then,
=> [tex]\frac{p}{sinP}=\frac{q}{sinQ}[/tex]
=> [tex]sinQ=\frac{q sinP}{p}[/tex]
=> [tex]sin Q= \frac{5sin 48\textdegree}{10.3}[/tex]
=> sin Q = 0.344
=>[tex]\angle Q = 19.5\textdegree[/tex]
Now sum οf all angles in triangle is 180°. Then,
=>[tex]\angle P+\angle Q+\angle R = 180\textdegree[/tex]
=> [tex]\angle R = 180-48-19.5=112.5\textdegree[/tex]
Hence the answers are p=10.3, [tex]\angle Q=19.5\textdegree[/tex] and [tex]\angle R = 112.5\textdegree[/tex].
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Find the standard matrix of the linear transformation T: R2 to R2 that reflects each vector through the line x1=-x2.
The standard matrix is:
| 0 -1 |
| -1 0 |
The standard matrix of the linear transformation T: R2 to R2 that reflects each vector through the line x1=-x2 can be found by following these steps:
1. Start with the standard basis vectors for R2, which are (1,0) and (0,1).
2. Apply the transformation T to each of these vectors to find the new vectors. For the first vector, (1,0), we have x1=1 and x2=0. The reflection through the line x1=-x2 gives us a new vector with x1=-x2=-0=0 and x2=-x1=-1. So the new vector is (0,-1).
3. For the second vector, (0,1), we have x1=0 and x2=1. The reflection through the line x1=-x2 gives us a new vector with x1=-x2=-1 and x2=-x1=-0=0. So the new vector is (-1,0).
4. The standard matrix of the transformation T is the matrix whose columns are the new vectors we found. So the standard matrix is:
```
| 0 -1 |
| -1 0 |
```
This is the standard matrix of the linear transformation T: R2 to R2 that reflects each vector through the line x1=-x2.
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Juan ran 5 miles at the same pace and
walked for 10 minutes. Write an expression
that represents the total amount of time Juan
spent exercising.
The expression for time will be, Time = 5/r + 10/w.
What exactly are expressions?
In mathematics, an expression is a combination of numbers, symbols, and operators that represent a value or a computation.
Expressions can be simple or complex, and they can contain variables, constants, and functions. They can be evaluated to produce a single value, or they can be simplified or transformed using mathematical rules and operations.
Now,
Let's assume that Juan ran at a speed of "r" miles per minute, and let's also convert the 10 minutes of walking to miles per minute by dividing by the walking speed "w".
Then, the total time Juan spent exercising can be expressed as:
Time = Time Running + Time Walking
Time Running = Distance / Speed = 5 / r
Time Walking = 10 / w
Therefore,
the expression for the total amount of time Juan spent exercising is:
Time = 5/r + 10/w
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Mr. Dieter wants to tile the family room in his basement. He has selected a pattern of
square tiles that measure 9 inches by 9 inches each. The shape of the floor to be tiled is
shown below. (3 points for each part)
a. What is the area of the family room in square feet?
b. How many of the 9 inch by 9 inch tiles will it take to cover the floor? (NOTE: You
will need to convert the area in Part a from square feet to square inches.)
c. If the tile is sold only in boxes of 12 tiles per box, how many boxes will Mr. Dieter
have to buy to tile the family room?
a. The area of the family room is 162 ft².
b. 24 tiles will be needed to cover the floor.
c. Mr. Dieter will have to buy 2 boxes to tile the family room.
The solution has been obtained by using the arithmetic operations.
What are arithmetic operations?
All real numbers are thought to be sufficiently described by the four basic operations, also referred to as "arithmetic operations." The mathematical operations quotient, product, sum, and difference come after division, multiplication, addition, and subtraction.
a. There are two different shapes of which one is rectangle and other is triangle.
So,
⇒ Area of rectangle = length x width
⇒ Area of rectangle = 16 x 8
⇒ Area of rectangle = 128 ft²
Also,
⇒ Area of first triangle = [tex]\frac{1}{2}[/tex] x base x height
⇒ Area of first triangle = [tex]\frac{1}{2}[/tex] x 4 x 3
⇒ Area of first triangle = 6 ft²
Similarly,
⇒ Area of first triangle = [tex]\frac{1}{2}[/tex] x base x height
⇒ Area of first triangle = [tex]\frac{1}{2}[/tex] x 8 x 7
⇒ Area of first triangle = 28 ft²
On adding the three areas, we get
⇒ Area of family room = 128 + 6 + 28
⇒ Area of family room = 162 ft²
b. We will convert the area into inches.
We know that 1 feet = 12 inches.
So, using the multiplication, we get
162 feet = 162 * 12 in²
162 feet = 1944 in²
Now,
⇒ Area of tile = 9 * 9
⇒ Area of tile = 81 in²
So,
⇒ Number of tiles = [tex]\frac{1944}{81}[/tex]
⇒ Number of tiles = 24
c. Since, tiles are sold in boxes of 12 so, using the division operation, we get
⇒ Number of boxes needed = [tex]\frac{24}{12}[/tex]
⇒ Number of boxes needed = 2
Hence, the required solution has been obtained.
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Find the inverse of the following function: f(x) = 2(x - 4)^3 + 7
The inverse of the function f(x) = 2(x - 4)³ + 7 is g(x) = ∛[(x - 7) / 2] + 4.
What is a function?A mathematical notion known as a function explains the connection between two sets of values known as the domain and the range. A function takes a value from the domain as an input, applies some logic to it, and returns a value in the range. Just one output value in the range corresponds to each input value in the domain. Functions can be described verbally or with mathematical symbols. They are employed throughout the mathematical discipline as well as in the sciences of physics, engineering, economics, and computer science.
The given function is f(x) = 2(x - 4)³ + 7.
Swap the variables, that is substitute x = y and vice versa as follows:
x = 2(y - 4)³ + 7
x - 7 = 2(y - 4)³
(x - 7) / 2 = (y - 4)³
Taking the cube root of both sides gives:
y - 4 = ∛[(x - 7) / 2]
y = ∛[(x - 7) / 2] + 4
Therefore, the inverse of f(x) = 2(x - 4)³ + 7 is g(x) = ∛[(x - 7) / 2] + 4.
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