Answer:
To find the derivative of the function y = x^2 - x + 3 using the first principle, we start by applying the definition of the derivative:
f'(x) = lim (h -> 0) [f(x+h) - f(x)] / h
where f(x) = x^2 - x + 3.
Now we substitute the function into the equation and simplify:
f'(x) = lim (h -> 0) [(x+h)^2 - (x+h) + 3 - (x^2 - x + 3)] / h
f'(x) = lim (h -> 0) [(x^2 + 2xh + h^2 - x - h + 3) - (x^2 - x + 3)] / h
f'(x) = lim (h -> 0) [2xh + h^2 - h] / h
Now we can cancel out the h in the numerator and denominator, leaving:
f'(x) = lim (h -> 0) [2x + h - 1]
Finally, we take the limit as h approaches 0:
f'(x) = 2x - 1
Therefore, the derivative of y = x^2 - x + 3 with respect to x is f'(x) = 2x - 1.
Step-by-step explanation:
Answer:
[tex]\dfrac{\text{d}y}{\text{d}x}=2x-1[/tex]
Step-by-step explanation:
Differentiating from First Principles is a technique to find an algebraic expression for the gradient at a particular point on the curve.
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Differentiating from First Principles}\\\\\\$\text{f}\:'(x)=\displaystyle \lim_{h \to 0} \left[\dfrac{\text{f}(x+h)-\text{f}(x)}{(x+h)-x}\right]$\\\\\end{minipage}}[/tex]
The point (x + h, f(x + h)) is a small distance along the curve from (x, f(x)). As h gets smaller, the distance between the two points gets smaller. The closer the points, the closer the line joining them will be to the tangent line.
To differentiate y = x² - x + 3 using first principles, substitute f(x + h) and f(x) into the formula:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{(x+h)^2-(x+h)+3-(x^2-x+3)}{(x+h)-x}\right][/tex]
Simplify the numerator:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{x^2+2hx+h^2-x-h+3-x^2+x-3)}{x+h-x}\right][/tex]
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{2hx+h^2-h}{h}\right][/tex]
Separate into three fractions:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\dfrac{2hx}{h}+\dfrac{h^2}{h}-\dfrac{h}{h}\right][/tex]
Cancel the common factor, h:
[tex]\implies \displaystyle \dfrac{\text{d}y}{\text{d}x}=\lim_{h \to 0} \left[\:2x+h-1\:\right][/tex]
As h → 0, the second term → 0:
[tex]\implies \dfrac{\text{d}y}{\text{d}x}=2x-1[/tex]
If f(x) = [x − 2], find ƒ(4.5).
A 2
B) 2.5
C 3
D
6.5
What is the common factor of the numerator and denominator in the expression (x+1)(x+7)(x+7)(x−1)
Answer:So the common factor of the numerator and denominator is (x+7)(x+7) or (x+7)^2, and the simplified expression is (x+1)(x−1).
Step-by-step explanation:
To find the common factor of the numerator and denominator of the expression (x+1)(x+7)(x+7)(x−1), we need to factorize it completely.
(x+1)(x+7)(x+7)(x−1) = (x+1)(x−1)(x+7)(x+7)
Now we can see that the common factor of the numerator and denominator is (x+7)(x+7) or (x+7)^2.
Therefore, we can simplify the expression by dividing both the numerator and denominator by (x+7)^2:
(x+1)(x−1)(x+7)(x+7)/(x+7)(x+7) = (x+1)(x−1)
So the common factor of the numerator and denominator is (x+7)(x+7) or (x+7)^2, and the simplified expression is (x+1)(x−1).
3. This year's Super Bowl had an average household rating of45.0according to Nielsen. CBS charged$5.5million per : 30 commercial for the Super Bowl, compared to$1.0million for a :30 spot in the Men's NCAA Basketball Championship game with a8.9rating, and$125,000for a spot in a primetime program (NCIS) with a rating of 6.5. Compare the household CPP for the Super Bowl vs. the Men's NCAA Basketball Championship game vs. NCIS (show your work). a. As more and more advertisers rely on sports for the guaranteed impressions, why do you think advertisers are paying such a premium to run an ad during a sporting event vs. normal primetime programming?
By calculating the Cost Per Point, it can be concluded that advertisers are paying such a premium to run an ad during a sporting event vs. normal primetime programming because sporting events have a larger and more engaged audience.
Cost per Point (CPP) is a measurement of cost efficiency that enables comparison between the cost of one advertisement to other advertisements. CPP is calculated by dividing the gross media cost by the gross rating point.
CPP = gross media cost / gross rating point
The household CPP for the Super Bowl, Men's NCAA Basketball Championship game, and NCIS can be calculated as follows:
For the Super Bowl:
CPP = $5.5 million / 45.0 = $122,222.22
For the Men's NCAA Basketball Championship game:
CPP = $1.0 million / 8.9 = $112,359.55
For NCIS:
CPP = $125,000 / 6.5 = $19,230.77
Comparing the CPP for the Super Bowl, Men's NCAA Basketball Championship game, and NCIS, it is clear that the Super Bowl has the highest CPP, followed by the Men's NCAA Basketball Championship game, and then NCIS.
Advertisers are paying such a premium to run an ad during a sporting event vs. normal primetime programming because sporting events have a larger and more engaged audience.
Sporting events, especially ones like the Super Bowl and the Men's NCAA Basketball Championship game, are highly anticipated and have a dedicated fan base. This means that advertisers have a better chance of reaching a larger audience and making a bigger impact with their ads during these events. Additionally, sporting events are often live, which means that viewers are less likely to skip through commercials. This guarantees that advertisers' messages will be seen by more people, making it worth the higher cost.
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Your gross pay is $1,957 and your federal tax witholding rate is 16%. What is your net pay?
1. What are the six primary sources of greenhouse gas emissions in the United States? 2. What are the seven types of greenhouse gases of climate change? 3. What are the four types of Machine learning?
The six primary sources of greenhouse gas emissions in the United States are:
Electricity productionTransportationIndustryCommercial and residentialAgricultureLand use and forestry2. The seven types of greenhouse gases of climate change are:
Carbon dioxide (CO2)Methane (CH4)Nitrous oxide (N2O)Fluorinated gasesChlorofluorocarbons (CFCs)Hydrofluorocarbons (HFCs)Sulfur hexafluoride (SF6)3. The four types of Machine learning are:
Supervised learningUnsupervised learningSemi-supervised learningReinforcement learningLearn more about greenhouse
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PLEASE HELPP WILL GIVE BRAINLIEST!!!!
image of proof attached !!!
line e is parallel to line m because angle 1 + angle 2 = 180°
What are angles on parallel lines?Angles in parallel lines are angles that are created when two parallel lines are intersected by another line called a transversal.
Represent the adjascent angle to angle 2 by x and adjascent angle to angle 1 by y
therefore ;
2 = y ( corresponding angles are equal)
1 = x ( corresponding angles are equal)
angle 1+y = 180 ( angle on a straight line)
angle 2 + x = 180
<1= 180-y
<2 = 180 -x
and both equations
< 1 +<2 = 180
therefore since < 1 +<2 = 180, then line e is parallel to line m
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A bank has to be guarded 24 hours a day. 9 guards are ordered to split each day’s guard duty equally. How long will each guard spend on guard duty in one day?
Answer:8/3
Step-by-step explanation:
Total number of hours: 24
Total number of guards: 9
Equation: Total number of hours/ total number of guards
24/9= 8/3 hour shifts or 2.66667 hour shifts for each guard.
A quadrilateral is on a coordinate plane at points A(-3, 1) B(1, 4) C(5, 1) D(1, -2).
Which sides are congruent? (Select all that apply)
AB
BC
CD
AD
Answer:
AB and CD.
Step-by-step explanation:
To check if two sides of a quadrilateral are congruent, we need to find the length of each side. We can use the distance formula to calculate the length of each side:
AB = √[(1 - 4)^2 + (4 - 1)^2] = √(9 + 9) = 3√2
BC = √[(5 - 1)^2 + (1 - 4)^2] = √(16 + 9) = √25 = 5
CD = √[(1 - 1)^2 + (-2 - 1)^2] = √9 = 3
AD = √[(-3 - 1)^2 + (1 - (-2))^2] = √(16 + 9) = √25 = 5
Therefore, AB and CD have the same length and are congruent, while BC and AD have the same length and are congruent.
So the correct answer is AB and CD.
Answer these two math questions for 15 points
Answer:
Step-by-step explanation:
1. Solve for
x by simplifying both sides of the equation, then isolating the variable.
x= 3, −5
2. x by simplifying both sides of the equation, then isolating the variable.
x= −7, 1
equals zero, which expression is equ (25-4x^(2))/(6x^(2)+9x-15)*(6x^(2)-2x-4)/(2x^(2)-x-10)
The solver for this expression is x = -5/2, 5/2, 2, -1/3, 1, 2.
To solve this expression, we will first need to factor the numerator and denominator of each fraction.
For the first fraction, we can factor the numerator as follows:
[tex]25 - 4x^2 = (5 + 2x)(5 - 2x)[/tex]
The denominator can be factored as follows:
[tex]6x^2 + 9x - 15 = (2x + 5)(3x - 3)[/tex]
For the second fraction, the numerator can be factored as follows:
[tex]6x^2 - 2x - 4 = (2x - 4)(3x + 1)[/tex]
The denominator can be factored as follows:
[tex]2x^2 - x - 10 = (2x + 5)(x - 2)[/tex]
Now we can simplify the expression by canceling out any common factors:
[tex](5 + 2x)(5 - 2x)/(2x + 5)(3x - 3) * (2x - 4)(3x + 1)/(2x + 5)(x - 2)[/tex]
[tex]= (5 + 2x)(5 - 2x)(2x - 4)(3x + 1)/(2x + 5)(3x - 3)(2x + 5)(x - 2)[/tex]
[tex]= (5 + 2x)(5 - 2x)(2x - 4)(3x + 1)/(3x - 3)(x - 2)[/tex]
Since the expression is equal to 0, we can set each factor equal to 0 and solve for [tex]x[/tex]:
[tex]5 + 2x = 0 \\5 - 2x = 0 \\2x - 4 = 0 \\3x + 1 = 0 \\3x - 3 = 0 \\x - 2 = 0 \\[/tex]
Solving for x, we get the following solutions:
[tex]x = -5/2 \\x = 5/2 \\x = 2 \\x = -1/3\\ x = 1 \\x = 2 \\[/tex]
Therefore, the solver for this expression is [tex]x = -5/2, 5/2, 2, -1/3, 1, 2[/tex].
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Subtract 7m^(3) n^(3)+ 2m^(2) n- n^(2) from the sum of m^(2) n+ mn^(2) + 2n^(2)and 2m^(3) n^(3)- mn^(2) + 5n^(2).
To solve this problem, we need to combine like terms and use the distributive property.
Here are the steps:
1. Combine like terms in the sum of m^(2) n+ mn^(2) + 2n^(2) and 2m^(3) n^(3)- mn^(2) + 5n^(2):
m^(2) n+ mn^(2) + 2n^(2) + 2m^(3) n^(3)- mn^(2) + 5n^(2) = 2m^(3) n^(3) + m^(2) n + 7n^(2)
2. Use the distributive property to subtract 7m^(3) n^(3)+ 2m^(2) n- n^(2) from the sum:
2m^(3) n^(3) + m^(2) n + 7n^(2) - (7m^(3) n^(3)+ 2m^(2) n- n^(2)) = 2m^(3) n^(3) + m^(2) n + 7n^(2) - 7m^(3) n^(3)- 2m^(2) n + n^(2)
3. Combine like terms:
-5m^(3) n^(3) - m^(2) n + 8n^(2)
Therefore, the answer is -5m^(3) n^(3) - m^(2) n + 8n^(2).
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I am unsure how to solve problems H through L and it's due by Friday!!! pls help! (solve for variable listed on paper from equation)
Answer:
Step-by-step explanation:
(h)
[tex]KE= \frac{1}{2}mv^{2} \\KE=\frac{mv^{2} }{2} \\2KE=mv^{2}\\2KEm=v^{2} \\v=\sqrt{2KEm}[/tex]
(g)
[tex]s=\frac{(u+v)t}{2} \\2s=(u+v)t\\\frac{2s}{t}=u+v\\u= \frac{2s}{t}-v[/tex]
(i)
[tex]s=ut+\frac{1}{2} at^{2} \\s-ut=\frac{at^{2} }{2} \\2s-2ut=at^{2} \\\sqrt{2s-2ut}=at\\a=\sqrt{2s-2ut}-t[/tex]
(j)
[tex]\frac{pV}{T} =nR\\T=\frac{pV}{nR}[/tex]
(k)
[tex]a^{2} -b^{2} +c^{2} \\a^{2} +c^{2} =b^{2} \\b=\sqrt{a^{2} +c^{2} }[/tex]
(i)sinθ[tex]=\frac{a}{b}[/tex]
θ=[tex]\frac{a}{sin(b)}[/tex]
Sketch the figure in the side length of an equilateral triangle is 5 centimeters
Fοr an equilateral triangle, sketch is drawn fοr side length = 5 cm.
What are triangles?Triangles are a particular sοrt οf pοlygοn in geοmetry that have three sides and three vertices. Three straight sides make up the twο-dimensiοnal figure shοwn here. An example οf a 3-sided pοlygοn is a triangle. The tοtal οf a triangle's three angles equals 180 degrees. One plane cοmpletely enclοses the triangle.
An equilateral triangle in geοmetry is a triangle with equal-length sides οn all three sides.
In the well-knοwn Euclidean geοmetry, an equilateral triangle is alsο equiangular, meaning that each οf its three internal angles is 60 degrees and cοngruent with the οthers.
The given triangle has a side length οf 5 cm.
Sο, each sire measures the same.
Therefοre, the sketch is drawn fοr the same.
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5x – 4 = 5x
What value could you write in after 5x that would make the equation true for: no values of x?
Answer:
No solution
Step-by-step explanation: no values of x
After drawing the line of best fit on the scatter plot on the right (the line is a linear model), what is the approximate slope of the line?
The slope of the line is approximately...?
Answer:
Slope would approximately be -1/8
Step-by-step explanation:
The first point that it looks like it touches would be (0,100) then from what I can see the next point would be (400,50). To get from the first point to the second, the slope would have to drop 1 block and move to the right 8 blocks. This means the slope would be -1/8
Louise has 3 and 3/4 feet of green yarn and 2/4 yards of white yarn to make an art project. How many more feet of green yarn than white yarn does Louise have? ASAP PLS
Louise has 2 1/4 feet more of green yarn than white yarn.
What is the fractions ?
A fraction is a way to represent a part of a whole or a ratio between two quantities.
To compare the amounts of green and white yarn, we need to express the amounts in the same units. Let's convert 2/4 yards of white yarn to feet, since the green yarn is already measured in feet.
1 yard = 3 feet
2/4 yard = 2/4 * 3 feet/yard = 1.5 feet
Now we can compare the amounts of green and white yarn in feet:
Amount of green yarn = 3 3/4 feet
Amount of white yarn = 1.5 feet
To find how many more feet of green yarn Louise has than white yarn, we can subtract the amount of white yarn from the amount of green yarn:
3 3/4 feet - 1.5 feet = 2 1/4 feet
Hence, Louise has 2 1/4 feet more of green yarn than white yarn.
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How many different 7-letter codes can be made from the word CONCERN?
The answer is 2520 different 7-letter codes that can be made from the word CONCERN.
There are 7!/(2!2!) different 7-letter codes that can be made from the word CONCERN.
To find the number of different 7-letter codes that can be made from the word CONCERN,
we can use the formula for permutations of n objects with r identical objects:
P(n,r) = n!/(r1!r2!...rk!)
Where n is the total number of objects, r1, r2, ..., rk are the number of identical objects in each group, and n! is the factorial of n.
In this case, n = 7, r1 = 2 (for the two C's), and r2 = 2 (for the two N's). So, the formula becomes:
P(7,2,2) = 7!/(2!2!)
= (7*6*5*4*3*2*1)/((2*1)*(2*1))
= 2520
Therefore, there are 2520 different 7-letter codes that can be made from the word CONCERN.
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Question : find the equation of the tangent blame and womeal to the surface (i) Z 2= ry at point (2,3,6) (iis 2x2 – 3xy -4x-7=0 at point (1,-1,2) (diis 2x3*- 3x’y -4X-7 at hornt (1,-1,2)
The equation of the tangent plane to the surface z = ry at the point (2, 3, 6) is 2x + y – z = 16. The equation of the tangent plane to the surface 2x2 – 3xy -4x-7 = 0 at the point (1, -1, 2) is 2x – y + 3z = 2.
To obtain the equation of the tangent plane, we must first find the partial derivatives of the equation of the surface with respect to the coordinates x, y, and z. The partial derivative of the first surface with respect to x is r, and the partial derivative of the second surface with respect to x is 4x - 3y.
We can then use the equation of the tangent plane, which is: z - z0 = fx(x - x0) + fy(y - y0) + fz(z - z0), where z0, x0, and y0 are the coordinates of the given point, and fx, fy, and fz are the partial derivatives of the surface equation with respect to x, y, and z, respectively.
For the first surface, the equation of the tangent plane is:
z - 6 = r(x - 2) + 0(y - 3) + 0(z - 6)
=> z = rx + 6.
For the second surface, the equation of the tangent plane is:
z - 2 = 4x - 3y + 0(z - 2)
=> z = 4x - 3y + 2.
Therefore, the equation of the tangent plane to the surface z = ry at the point (2, 3, 6) is 2x + y – z = 16 and the equation of the tangent plane to the surface 2x2 – 3xy -4x-7 = 0 at the point (1, -1, 2) is 2x – y + 3z = 2.
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Use the histogram of Investment Account Values to answer the question.
What percentage of investments have a value or $150or less?
25%
46%
63%
83%
The answer of the given question based on the histogram of Investment Account Values the correct option is c) 63%.
What is Investment?Investment refers to act of allocating resources, like money or time, with expectation of generating income or profit in future. In other words, it involves sacrificing something of a value in present, in hope of obtaining something of the greater value in future
Investments can be taken in many forms, including stocks, bonds, mutual funds, real estate, and commodities, among others. The choice of investment depends on variety of factors, including investor's risk tolerance, financial goals, and the investment time horizon.
Based on the histogram of Investment Account Values provided, it appears that approximately 63% of investments have a value of $150 or less. This can be determined by adding up the percentage of investments in each bar up to and including the $150 bar.
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please help me i have a test and i did not study.
The input values are all the values of the x-coordinates and the output values are all the values of the y-coordinate corresponding the x-value: (0, 3), (2, 2), (-1, 1).
What are ordered pair?An ordered pair (a, b) is a pair of objects in mathematics. Because the ordered pair (a, b) differs from the ordered pair (b, a) unless a = b, the order in which the items occur in the pair is important. The unordered pair a, b, on the other hand, equals the unordered pair b, a.
Ordered pairs are sometimes known as 2-tuples, or sequences of length 2 (or, in computer science, sometimes, lists). 2-dimensional vectors are occasionally used to refer to ordered pairs of scalars.
The input values are all the values of the x-coordinates and the output values are all the values of the y-coordinate corresponding the x-value.
Some of the input, output pair are:
(0, 3)
(2, 2)
(-1, 1)
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What is the interquartile range for the data set?
341, 330, 301, 345, 309, 311, 342, 301, 304, 328, 343
Answer:
38
Step-by-step explanation:
Interquartile Range
IQR = Q3 - Q1
Quartiles are the values that divide a list of numbers into quarters:
Put the list of numbers in order. Then cut the list into four equal parts.301, 301, 304, 309, 311, 328, 330, 341, 342, 343, 345
Hence,
Quartiles:
Q1 --> 304
Q2 --> 328
Q3 --> 342
And since, IQR = Q3 - Q1
Then,
342 - 304 = 38
As a result, the interquartile range for the data set is 38.
RevyBreeze
What is the final elevation. If a bird starts at 20 m and changes 16 m
Answer:
Two possible answers: 36 and 4m
Step-by-step explanation:
Changes is not specific, but we know it is a vertical change due to it saying elevation.
20+16=36m
20-16=4m
We don't know if the bird is going up or down.
Down is 4m and up is 36m
Use the one-to-one property of logarithms to solve: 15.Ln(10−3x)=ln(−4x)16.log4(6−m)=log43(m)
The solutions to the given equations are x = 10 and m = 1.5.
Using the one-to-one property of logarithms, we can solve the given equations as follows:
First, we will solve the equation: 15.Ln(10−3x)=ln(−4x)
The one-to-one property of logarithms states that if logb(a) = logb(c), then a = c. Therefore, we can use this property to set the arguments of the logarithms equal to each other:
10 - 3x = -4x
Simplifying this equation gives:
10 = x
So the solution to the first equation is x = 10.
Next, we will solve the equation: 16.log4(6−m)=log43(m)
Again, using the one-to-one property of logarithms, we can set the arguments of the logarithms equal to each other:
6 - m = 3m
Simplifying this equation gives:
6 = 4m
m = 6/4
m = 1.5
So the solution to the second equation is m = 1.5.
Therefore, the solutions to the given equations are x = 10 and m = 1.5.
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A guidance counselor wants to determine if there is a relationship between a student’s number of absences, x, and their grade point average (GPA), y. The given data list the number of absences and GPAs for 15 randomly selected students.
Using technology, what is the value of r2?
–0.56
–0.32
0. 32
0.56
By using technology, the value of r² is equal to; C. 0.32.
What is a coefficient of determination?In Mathematics, a coefficient of determination (r² or r-squared) can be defined as a number between zero (0) and one (1) that is typically used for measuring the extent (how well) to which a statistical model predicts an outcome.
Based on the given data, the correlation can be determined by using an online graphing calculator as shown in the image attached below. Since the value of r is equal to -0.56, the coefficient of determination (r²) can be calculated as follows;
r² = (-0.56)²
r² = 0.3136 ≈ 0.32
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Complete Question:
A guidance counselor wants to determine if there is a relationship between a student's number of absences, x, and
their grade point average (GPA), y. The given data list the number of absences and GPAs for 15 randomly selected students.
Number of Absences
15
1
0
6
9
12
3
3
1
2
7
0
4
9
10
GPA
2.1
4.3
4.5
3.2
4.0
1.7
3.8
2.9
3.6
3.4
2.6
3.1
2.8
2.8
4.1
Using technology, what is the value of r²?
O-0.56
O-0.32
O 0.32
O 0.56
HELP PLEASE AND THANK YOU
Answer:
5n = 1.75
Step-by-step explanation:
We are given 1.75 and n 5 times, so
5n = 1.75
Answer: n=0.35
Step-by-step explanation:
Divide 1.75 into fifths and that equals 1 n
Formula Repeatedly To Find The Three Points That Divide The Line Segment Joining The Given Points Into Four Equal Parts, (a) ( 2,-3) , (3,-5)
(x,y) = ( ) (smallest x - value)
(x,y) = ( ) (x,y) = ( ) (large x-value)
(b) (-1,-1),(0,0)
(x,y) = ( ) (smallest x-value) (x,y) = ( )
(x,y) = ( ) (large x-value)
(a) The three points are:
(x,y) = ( 2.25 , -3.5 ) (smallest x-value)
(x,y) = ( 2.5 , -4 )
(x,y) = ( 2.75 , -4.5 ) (large x-value)
(b) The three points are:
(x,y) = ( -0.75 , -0.75 ) (smallest x-value)
(x,y) = ( -0.5 , -0.5 )
(x,y) = ( -0.25 , -0.25 ) (large x-value)
To find the three points that divide the line segment joining the given points into four equal parts, we need to use the formula for a point that divides a segment in a given ratio. The formula is:
(x,y) = ( (x1 + rx2)/(1+r) , (y1 + ry2)/(1+r) )
Where (x1,y1) and (x2,y2) are the given points, r is the ratio, and (x,y) is the point that divides the segment in the given ratio.
(a) For the given points (2,-3) and (3,-5), we need to find the points that divide the segment in the ratio 1:3, 2:2, and 3:1.
For the first point, we use the formula with r=1/3:
(x,y) = ( (2 + (1/3)*3)/(1+(1/3)) , (-3 + (1/3)*(-5))/(1+(1/3)) )
(x,y) = ( 2.25 , -3.5 )
For the second point, we use the formula with r=2/2:
(x,y) = ( (2 + (2/2)*3)/(1+(2/2)) , (-3 + (2/2)*(-5))/(1+(2/2)) )
(x,y) = ( 2.5 , -4 )
For the third point, we use the formula with r=3/1:
(x,y) = ( (2 + (3/1)*3)/(1+(3/1)) , (-3 + (3/1)*(-5))/(1+(3/1)) )
(x,y) = ( 2.75 , -4.5 )
So the three points are:
(x,y) = ( 2.25 , -3.5 ) (smallest x-value)
(x,y) = ( 2.5 , -4 )
(x,y) = ( 2.75 , -4.5 ) (large x-value)
(b) For the given points (-1,-1) and (0,0), we need to find the points that divide the segment in the same ratios as before.
For the first point, we use the formula with r=1/3:
(x,y) = ( (-1 + (1/3)*0)/(1+(1/3)) , (-1 + (1/3)*0)/(1+(1/3)) )
(x,y) = ( -0.75 , -0.75 )
For the second point, we use the formula with r=2/2:
(x,y) = ( (-1 + (2/2)*0)/(1+(2/2)) , (-1 + (2/2)*0)/(1+(2/2)) )
(x,y) = ( -0.5 , -0.5 )
For the third point, we use the formula with r=3/1:
(x,y) = ( (-1 + (3/1)*0)/(1+(3/1)) , (-1 + (3/1)*0)/(1+(3/1)) )
(x,y) = ( -0.25 , -0.25 )
So the three points are:
(x,y) = ( -0.75 , -0.75 ) (smallest x-value)
(x,y) = ( -0.5 , -0.5 )
(x,y) = ( -0.25 , -0.25 ) (large x-value)
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What is one hundredth more than 2.89?
Answer:
2.90
Step-by-step explanation:
One hundredth is 0.01. 2.89+0.01= 2.90.
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Add the given expressions, simplifying each result and noting the co (7x+10)/(x^(2)-16)+(-8x-6)/(x^(2)-16)
The simplified result is (-x + 4)/(x^(2) - 16).
To add the given expressions, we need to combine the like terms in the numerator and keep the denominator the same since they are already common denominators.
The combined expression will look like this:
(7x + 10 - 8x - 6)/(x^(2) - 16)
Now we need to simplify the numerator by combining the like terms:
(-x + 4)/(x^(2) - 16)
This is the simplified expression. We cannot simplify it any further since there are no common factors between the numerator and denominator.
Therefore, the answer to the given question is (-x + 4)/(x^(2) - 16).
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an artist is using tiles in various geometric shapes to make a mosaic. The two titles she plans to use shown below. if the two tiles have the same area, what is the length of the triangles base?
So the area of one tile is 50 square inches. Therefore, the length of the triangle's base is 20 inches.
What is area?Area is a measure of the amount of space inside a two-dimensional (2D) shape or figure. It is usually measured in square units, such as square meters (m²) or square centimeters (cm²). The area of a shape is calculated by multiplying the length of one side or dimension of the shape by the length of another side or dimension. The resulting value represents the total number of square units that are contained within the shape.Area is an important concept in mathematics and is used in various fields, such as geometry, physics, and engineering. It helps in measuring and comparing the sizes of shapes and figures, and in determining how much material is needed to cover or fill a certain area.
Here,
The area of a parallelogram can be calculated as the product of its base and height. In the case of the given tile, the base and height are 10 inches and 5 inches respectively. Therefore, the area of one tile is:
Area of tile = Base x Height
Area of tile = 10 in x 5 in
Area of tile = 50 square inches
If the two tiles have the same area, then we can assume that the area of the triangle tile is also equal to 50 square inches (the area of one parallelogram tile that we calculated in the previous question).
The formula for the area of a triangle is:
Area of triangle = (1/2) x Base x Height
where the base is one of the sides of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.
In this case, we know that the area of the triangle is 50 square inches and the height is 5 inches, so we can solve for the base as follows:
Area of triangle = (1/2) x Base x Height
50 = (1/2) x Base x 5
Base = (2 x 50) / 5
Base = 20
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Complete question:
An artist is using tiles in the shape of a parallelogram to make a mosaic. The tiles are in the shape of parallelogram with height as 5 inch and length as 10 inch. What is the area of one tile? if the two tiles have the same area, what is the length of the triangles base?
1. Find the length of the curve.
r ( t ) = 6 cos ( t ) i − sin ( t ) j + 5 sin ( t ) k , 0 ≤ t ≤ 1
2. If r ( t ) = 〈 sin ( t ) , cos ( t ) , ln ( cos ( t ) ) 〉 , 0 ≤ t ≤ π/4 , find ds/d t , where s is the arc length function of r(t).
Answer choices
a. sec(t)
b. sec^2(t)
c. tan(t)
d. tan^2(t)
e. 1+tan^2(t).
ds/dt = sec(t)
To find the length of the curve, we need to find the arc length. The arc length of the curve r(t) is given by:
LENGTH = $\int_{0}^{1} \sqrt{6^{2}\cos^2(t) + 1 + 5^{2}\sin^2(t)} dt$
To find ds/dt, we can take the derivative of the above equation with respect to t:
$\frac{ds}{dt} = \frac{d}{dt} \sqrt{6^{2}\cos^2(t) + 1 + 5^{2}\sin^2(t)}$
$\frac{ds}{dt} = \frac{6^2\cos(t)\sin(t) + 10\sin(t)\cos(t)}{\sqrt{6^{2}\cos^2(t) + 1 + 5^{2}\sin^2(t)}}$
$\frac{ds}{dt} = \frac{16\sin(t)\cos(t)}{\sqrt{6^{2}\cos^2(t) + 1 + 5^{2}\sin^2(t)}}$
$\frac{ds}{dt} = \frac{16\sin(t)\cos(t)}{\sqrt{1+tan^2(t)}}$
Thus, ds/dt = sec(t).
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