Find the exact value of cos 105⁰.
a. √√√2-√6
4
b.
√2+√6
4
C.
4
d. √2+√6
4
Answers
Answer:
[tex]\dfrac{\sqrt{2}-\sqrt{6} }{4} }[/tex]
Step-by-step explanation:
Find the exact value of cos(105°).
The method I am about to show you will allow you to complete this problem without a calculator. Although, memorizing the trigonometric identities and the unit circle is required.
We have,
[tex]\cos(105\°)[/tex]
Using the angle sum identity for cosine.
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Angle Sum Identity for Cosine}}\\\\\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\end{array}\right}[/tex]
Split the given angle, in degrees, into two angles. Preferably two angles we can recognize on the unit circle.
[tex]105\textdegree=45\textdegree+60\textdegree\\\\\\\therefore \cos(105\textdegree)=\cos(45\textdegree+60\textdegree)[/tex]
Now applying the identity.
[tex]\cos(45\textdegree+60\textdegree)\\\\\\\Longrightarrow \cos(45\textdegree+60\textdegree)=\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)[/tex]
Now utilizing the unit circle.
[tex]\boxed{\left\begin{array}{ccc}\text{\underline{From the Unit Circle:}}\\\\\cos(45\textdegree)=\dfrac{\sqrt{2} }{2}\\\\\cos(60\textdegree)=\dfrac{1}{2}\\\\\sin(45\textdegree)=\dfrac{\sqrt{2} }{2}\\\\\sin(60\textdegree)=\dfrac{\sqrt{3} }{2} \end{array}\right}[/tex]
[tex]\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)\\\\\\\Longrightarrow \Big(\dfrac{\sqrt{2} }{2}\Big)\Big(\dfrac{1 }{2}\Big)-\Big(\dfrac{\sqrt{2} }{2}\Big)(\dfrac{\sqrt{3} }{2}\Big)[/tex]
Now simplifying...
[tex]\Big(\dfrac{\sqrt{2} }{2}\Big)\Big(\dfrac{1 }{2}\Big)-\Big(\dfrac{\sqrt{2} }{2}\Big)(\dfrac{\sqrt{3} }{2}\Big)\\\\\\\Longrightarrow \Big(\dfrac{\sqrt{2} }{4} \Big)-\Big(\dfrac{\sqrt{6} }{4} \Big)\\\\\\\therefore \cos(105\textdegree)= \boxed{\boxed{\frac{\sqrt{2}-\sqrt{6} }{4} }}[/tex]
Related Questions
How would you describe the difference between the graphs of f (x) = 3x²
and g(x) = -2² ?
OA. g(x) is a reflection of f(x) over the line y = x.
B. g(x) is a reflection of f(x) over the line y = -1.
C. g(x) is a reflection of f(x) over the x-axis.
D. g(x) is a reflection of f(x) over the y-axis.
Answers
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C).
The given functions are f(x) = 3x² and g(x) = -2².
To understand the difference between their graphs, let's examine the characteristics of each function individually:
Function f(x) = 3x²:
The coefficient of x² is positive (3), indicating an upward-opening parabola.
The graph of f(x) will be symmetric with respect to the y-axis, as any change in x will result in the same y-value due to the squaring of x.
The vertex of the parabola will be at the origin (0, 0) since there are no additional terms affecting the position of the graph.
Function g(x) = -2²:
The coefficient of x² is negative (-2), indicating a downward-opening parabola.
The negative sign will reflect the graph of f(x) across the x-axis, resulting in a vertical flip.
The vertex of the parabola will also be at the origin (0, 0) due to the absence of additional terms.
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C). This means that g(x) is obtained by taking the graph of f(x) and flipping it vertically. The reflection occurs over the x-axis, causing the parabola to open downward instead of upward.
Therefore, the correct answer is option C: g(x) is a reflection of f(x) over the x-axis.
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Given the graphs of y = f(x) and y = g(x),
g(x) = f(x) +
expresses g(x) in terms of f(x)
Answers
The expression g(x) = f(x) + represents the relationship between the two functions expression for g(x) in terms of f(x).
To express the function g(x) in terms of f(x), we need to understand the relationship between the two functions.
The given expression g(x) = f(x) + indicates that the function g(x) is obtained by adding a certain value or expression to the function f(x). expression for g(x) in terms of f(x).
In general, if we have the function g(x) = f(x) + c, where c is a constant value, then g(x) can be expressed in terms of f(x) as:
g(x) = f(x) + c
In this case, g(x) is obtained by adding the constant value c to the corresponding values of f(x).
It's important to note that without additional information about the specific relationship between f(x) and g(x), such as a functional equation or given values, we cannot provide a more precise expression for g(x) in terms of f(x).
Therefore, the expression g(x) = f(x) + represents the relationship between the two functions expression for g(x) in terms of f(x).
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Answer: 3
Step-by-step explanation: just 3
Edge 2020
Write the equation of the trigonometric graph.
Answers
Answer:
[tex]y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}[/tex]
Step-by-step explanation:
The graph of the solid black line is the cosine parent function, y = cos(x).
The standard form of a cosine function is:
[tex]\boxed{y = A \cos(B(x + C)) + D}[/tex]
where:
A is the amplitude (height from the mid-line to the peak).2π/B is the period (horizontal distance between consecutive peaks).C is the phase shift (horizontal shift - positive is to the left).D is the vertical shift (the mid-line is y = D).From inspection of the graph, the x-values of the turning points (peaks and troughs) of the parent function and the new function are the same. Therefore, the period of both functions is the same, and there has been no horizontal shift. So, B = 1 and C = 0.
The mid-line of the new function is y = 3. Therefore, D = 3.
The y-value of the peaks is y = 5. The amplitude is the distance from the mid-line to the peak. Therefore, A = 2.
Substituting these values into the standard formula we get:
[tex]y = 2 \cos(1(x + 0)) + 3[/tex]
[tex]y=2 \cos (1(x))+3[/tex]
[tex]y= 2 \cos(x) + 3[/tex]
Therefore, the equation of the trigonometric graph is:
[tex]y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}[/tex]
What is the z score for Brazil?
Answers
The z-score for Brazil is given as follows:
Z = 0.87.
What is the z-score formula?The z-score formula is defined as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
In which:
X is the measure.[tex]\mu[/tex] is the population mean.[tex]\sigma[/tex] is the population standard deviation.The parameters for this problem are given as follows:
[tex]X = 6.24, \mu = 4.8, \sigma = 1.66[/tex]
Hence the z-score for Brazil is given as follows:
Z = (6.24 - 4.8)/1.66
Z = 0.87.
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Find the equations of the asymptotes of the hyperbola defined by the equation shown below. If necessary, round to the nearest tenth. 100pts
Answers
The equations of the asymptotes of the hyperbola are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
To find the equations of the asymptotes of the hyperbola defined by the equation:
[tex]-25x^2 + 81y^2 + 100x + 1134y + 1844 = 0[/tex]
We can rewrite the equation in the standard form by isolating the x and y terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y + 1844 = 0[/tex]
Rearranging the terms:
[tex]-25x^2 + 100x + 81y^2 + 1134y = -1844[/tex]
Next, let's complete the square for both the x and y terms:
[tex]-25(x^2 - 4x) + 81(y^2 + 14y) = -1844\\-25(x^2 - 4x + 4 - 4) + 81(y^2 + 14y + 49 - 49) = -1844\\-25((x - 2)^2 - 4) + 81((y + 7)^2 - 49) = -1844[/tex]
Expanding and simplify
[tex]-25(x - 2)^2 + 100 - 81(y + 7)^2 + 3969 = -1844\\-25(x - 2)^2 - 81(y + 7)^2 = -1844 - 100 - 3969\\-25(x - 2)^2 - 81(y + 7)^2 = -4913[/tex]
Dividing both sides by -4913:
[tex](x - 2)^2/(-4913/25) - (y + 7)^2/(-4913/81) = 1[/tex]
Comparing this equation to the standard form of a hyperbola:
[tex](x - h)^2/a^2 - (y - k)^2/b^2 = 1[/tex]
We can determine that the center of the hyperbola is (h, k) = (2, -7). The value of [tex]a^2[/tex] is (-4913/25), and the value of [tex]b^2[/tex] is (-4913/81).
The equations of the asymptotes can be found using the formula:
y - k = ±(b/a)(x - h)
Substituting the values we found:
y + 7 = ±(√(-4913/81) / √(-4913/25))(x - 2)
Simplifying:
y + 7 = ±(√(4913) / √(81)) × √(25/4913) × (x - 2)
y + 7 = ±(√(4913) / 9) × √(25/4913) × (x - 2)
Rationalizing the denominators and simplifying:
y + 7 = ±(5/9) ×(x - 2)
Finally, rearranging the equation to isolate y:
y = ±(5/9)x - 10/9 - 7
Simplifying further:
y = ±(5/9)x - 79/9
In light of this, the equations for the hyperbola's asymptotes are y = (5/9)x - 79/9 and y = -(5/9)x + 79/9.
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Answer:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
Step-by-step explanation:
First, rewrite the given equation in the standard form of a hyperbola by completing the square.
Given equation:
[tex]-25x^2+81y^2+100x+1134y+1844=0[/tex]
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side:
[tex]-25x^2+100x+81y^2+1134y=-1844[/tex]
Factor out the coefficient of the x² term and the coefficient of the y² term:
[tex]-25(x^2-4x)+81(y^2+14y)=-1844[/tex]
Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:
[tex]-25(x^2-4x+4)+81(y^2+14y+49)=-1844-25(4)+81(49)[/tex]
Factor the two perfect trinomials on the left side and simplify the right side:
[tex]-25(x-2)^2+81(y+7)^2=2025[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\dfrac{-25(x-2)^2}{2025}+\dfrac{81(y+7)^2}{2025}=\dfrac{2025}{2025}[/tex]
[tex]\dfrac{-(x-2)^2}{81}+\dfrac{(y+7)^2}{25}=1[/tex]
[tex]\dfrac{(y+7)^2}{25}-\dfrac{(x-2)^2}{81}=1[/tex]
As the y²-term is positive, the hyperbola is vertical (opening up and down).
The standard equation of a vertical hyperbola is:
[tex]\boxed{\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1}[/tex]
Therefore, comparing this with the rewritten equation:
h = 2k = -7a² = 25 ⇒ a = 5b² = 81 ⇒ b = 9The formula for the equations of the asymptotes of a vertical hyperbola is:
[tex]\boxed{y=\pm \dfrac{a}{b}(x-h)+k}[/tex]
Substitute the values of h, k, a and b into the formula:
[tex]y=\pm \dfrac{5}{9}(x-2)-7[/tex]
Therefore, the equations for the asymptotes are:
[tex]\boxed{y=\dfrac{5}{9}x-\dfrac{73}{9}}\;\; \textsf{and} \;\;\boxed{ y=-\dfrac{5}{9}x-\dfrac{53}{9}}[/tex]
omari's monthly taxable income is ksh 24200. calculate the tax charged on omari's monthly earning
Answers
The tax charged on Omari's monthly earning of Ksh 24,200 is Ksh 3,340.
To calculate the tax charged on Omari's monthly earning, we need to consider the tax brackets and rates applicable in the specific tax system or country. Since you haven't specified a particular tax system, I will provide a general explanation.
Assuming we have a simplified progressive tax system with three tax brackets:
For the first tax bracket, let's say income up to Ksh 10,000 is taxed at a rate of 10%.
For the second tax bracket, income between Ksh 10,001 and Ksh 20,000 is taxed at a rate of 15%.
For the third tax bracket, income above Ksh 20,000 is taxed at a rate of 20%.
To calculate the tax charged on Omari's monthly earning of Ksh 24,200, we can divide it into the respective tax brackets:
Ksh 10,000 falls in the first tax bracket. So, the tax for this portion is 10% of Ksh 10,000, which is Ksh 1,000.
Ksh 20,000 - Ksh 10,000 = Ksh 10,000 falls in the second tax bracket. The tax for this portion is 15% of Ksh 10,000, which is Ksh 1,500.
The remaining amount, Ksh 24,200 - Ksh 20,000 = Ksh 4,200, falls in the third tax bracket. The tax for this portion is 20% of Ksh 4,200, which is Ksh 840.
Now, we can sum up the taxes for each bracket:
Total Tax = Tax in the first bracket + Tax in the second bracket + Tax in the third bracket
Total Tax = Ksh 1,000 + Ksh 1,500 + Ksh 840
Total Tax = Ksh 3,340
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A rectangular pyramid is sliced. The slice passes through line segment AB and is parallel to the base.
Which two-dimensional figure represents the cross section?
A. A rectangle the same size as the base
B. A rectangle that is smaller than the base
C. A quadrilateral that is not a rectangle
D. A triangle with a height the same as the pyramid
Answers
Answer:
Step-by-step explanation:
The correct answer is A. A rectangle the same size as the base.
When a rectangular pyramid is sliced parallel to the base, the resulting cross-section is a rectangle that is the same size as the base. The parallel slicing ensures that the cross-section maintains the same dimensions as the base of the pyramid. Therefore, option A, a rectangle the same size as the base, represents the cross-section.
Devaughn's age is three times Sydney's age. The sum of their ages is 80 . What is Sydney's age?
Answers
[tex]\qquad\displaystyle \rm \dashrightarrow \: let \: \: Sydney's \: \: age \: \: be \: \: 'y'[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: Devaughn's \: \: age \: \: will \: \: be \: \: 3y[/tex]
Sum up ;
[tex]\qquad\displaystyle \tt \dashrightarrow \: 3y + y = 80[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: 4y = 80[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 80 \div 4[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 20[/tex]
So, Sydney's age is 20 years, n that of Devaughn is 20 × 3 = 60 years
Answer:
Sydney= 20, Devaughn= 60
Step-by-step explanation:
Let Sydney's age be 'x'
Devaughn's age = 3 times x = 3x
We Know That
The sum of their ages is 80.
So,
3x + x = 80
4x = 80
If we shift the 4 to the 80 side
x = 80/4
x = 20
So, Sydney's age is 20
Therefore, Devaughn's age =
3x = 3 times x
= 3 times 20
= 60
what best describes the relationship between the computed mean of 52.4 and the actual mean of 52.7
Answers
The computed mean of 52.4 and the actual mean of 52.7 suggest a close relationship in terms of central tendency.
A computed mean is a statistical measure calculated by summing up a set of values and dividing by the number of observations. In this case, the computed mean of 52.4 implies that when the values are averaged, the result is 52.4.
The actual mean of 52.7 refers to the true average of the population or data set being analyzed. Since it is higher than the computed mean, it indicates that the sample used for computation might have slightly underestimated the true population mean.
However, the difference between the computed mean and the actual mean is relatively small, with only a 0.3 unit discrepancy.
Given the proximity of these two values, it suggests that the computed mean is a reasonably accurate estimate of the actual mean.
However, it's important to note that without additional information, such as the sample size or the variability of the data, it is difficult to draw definitive conclusions about the relationship between the computed mean and the actual mean.
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A village P is 12 km from village Q. It takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6 km/h from P to Q and (6 + x) km/h back to P, find the value of x.
Answers
Answer:
Hope this helps and have a nice day
Step-by-step explanation:
To find the value of x, we can use the formula:
Time = Distance / Speed
Let's calculate the time taken to travel from Q to P and back to Q.
From Q to P:
Distance = 12 km
Speed = 6 km/h
Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours
From P to Q:
Distance = 12 km
Speed = (6 + x) km/h
Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h
Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:
Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours
According to the problem, the total time is the sum of the time from Q to P and from P to Q:
Total time = Time taken from Q to P + Time taken from P to Q
Substituting the values:
10/3 hours = 2 hours + 12 km / (6 + x) km/h
Simplifying the equation:
10/3 = 2 + 12 / (6 + x)
Multiply both sides by (6 + x) to eliminate the denominator:
10(6 + x) = 2(6 + x) + 12
60 + 10x = 12 + 2x + 12
Collecting like terms:
8x = 24
Dividing both sides by 8:
x = 3
Therefore, the value of x is 3.
Answer:
x = 3
Step-by-step explanation:
speed = distance / time
time = distance / speed
Total time from P to Q to P:
T = 3h 20min
P to Q :
s = 6 km/h
d = 12 km
t = d/s
= 12/6
t = 2 h
time remaining t₁ = T - t
= 3h 20min - 2h
= 1 hr 20 min
= 60 + 20 min
= 80 min
t₁ = 80/60 hr
Q to P:
d₁ = 12km
t₁ = 80/60 hr
s₁ = d/t₁
[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]
= 9
s₁ = 9 km/h
From question, s₁ = (6 + x)km/h
⇒ 6 + x = 9
⇒ x = 3
At what points is the function y=sinx/3x continuous?
Answers
Answer: [tex](-\infty, 0) \cup (0, \infty)[/tex]
Step-by-step explanation:
The graph of [tex]\frac{\sin x}{x}[/tex] is continuous for all real [tex]x[/tex] except [tex]x=0[/tex], and multiplying this by [tex]1/3[/tex] does not change this.
PLEASE HELPP: 2.11.2 Project: Performance Task: The Parallax Problem (For San Francisco)
The Scenario: You’re looking for a sponsor to pay for you to participate in a sailboat race. Now that you’ve solved the parallax problem, use the same skills you used there to write a proposal that shows that you can win the race.
The Project: Use the information provided in the performance task to estimate your travel costs and to calculate your average speed and the speed of last year’s winner. Use the questions below to help you gather information to write your proposal
3. What is the distance between buoy A and B? (5 points)
4. What are the lengths of the other two triangle legs? (4 points: 2 points each)
Remember what you know about the shape of the Race Course.
5. What is the total length of the race course? (4 points: 3 for calculation, 1 for answer)
Part VIII: Calculate the winner’s speed. (10 points)
1. What was the winner’s speed during last year’s race? (5 points: 3 points for speed. 2 points for conversion to knots).
2. How does the winner’s speed compare with your average speed? How much faster or slower are you? (5 points)
Part IX: Write your proposal. (8 points)
Now it’s time to make your proposal to the sponsor. Your sponsor will have their logo on your boat, so they want to be sure it’s likely to do well. The sponsor also needs to know what the expenses and risks are, so they know how much their investment in you will cost.
1. Complete the table to summarize the results of your study. (4 points)
Category:
Race:
Risk Analysis:
Itemized Travel Cost
Safety hazards
Competitive Analysis:
My time and speed
Last year's winning time and speed
Reward Analysis:
My chances of winning
2. Write a summary paragraph explaining why the sponsor should accept your proposal. (4 points)
Answers
The proposal is as follows
Part III - The distance between buoys A and B is 12.8 kilometers.
Part IV - The length of the other two triangle legs are 10.2 kilometers and 8.4 kilometers.
Part V - The total length of the race course is 31.4 kilometers.
Part VIII - The winner's speed during last year's race was 10.8 knots.
See the proposal attached.
Why the sponsor should accept your proposalDear Sponsor,
I'm seeking sponsorship for the San Francisco sailboat race.
With a proven track record and the determination to win, your investment of $5,500 covers travel costs and potential hazards.
By associating your brand with a winning sailor, you'll gain significant exposure to thousands of spectators. Join me in this thrilling race for success.
Sincerely,
[Your Name]
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10 donuts cost $2.99 how much 1 cost?
Answers
If FE =14 find the length of BC
Please give a very in-depth explanation and I will mark Brainliest!!
Answers
HI Your answer is 42
I have calculated it you can trust me
Well you have marked right in the pic
PLEASE MARK AS BRAINLIEST
A graph has time driven (hours) on the x-axis, and Distance Driven (miles) on the y-axis. Points are grouped closely together an increase slightly. Points (2, 225) and (8, 75) are outside of the cluster.
The scatterplot shows the time driven on a trip compared to the distance driven. Inspect the scatterplot to determine if it has outliers.
How many outliers does the data set have?
The point
is an outlier in the data se
Answers
The data set has two outliers, namely the points (2, 225) and (8, 75).
Based on the given information about the scatterplot, we can observe that most of the points are grouped closely together and show a slight increase.
There are two points that lie outside of this cluster, specifically (2, 225) and (8, 75).
To determine if these points are outliers, we need to consider their deviation from the general pattern exhibited by the majority of the data points.
If these points deviate significantly from the overall trend, they can be considered outliers.
In this case, since (2, 225) and (8, 75) lie outside of the cluster of closely grouped points and do not follow the general pattern, they can be considered outliers.
These points are noticeably different from the majority of the data points and may have influenced the overall trend of the scatterplot.
The data set has two outliers, namely the points (2, 225) and (8, 75).
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Find the volume of the solid obtained by rotating the region
bounded by the graphs y=(x-4)^3,the x-axis, x=0, and x=5
about the y-axis? (Express numbers in exact form. Use symbolic
notation and fractions where needed.)
Answers
Answer:
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.
The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:
V = 2π ∫[a, b] x * f(x) dx
In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.
Substituting these values into the formula, we have:
V = 2π ∫[0, 5] x * (x - 4)^3 dx
To evaluate this integral, we can expand the cubic term and then integrate:
V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx
V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx
Integrating each term separately:
V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5
Now we can substitute the bounds of integration:
V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]
Simplifying:
V = 2π [(1/5 * 3125) - 0]
V = 2π * (625/5)
V = 2π * 125
V = 250π
Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.
Which expression is equivalent to 10f - 5f + 8 +6g +4?
Answers
The given expression, 10f - 5f + 8 + 6g + 4, simplifies to 5f + 12 + 6g when like terms are combined.
To simplify the expression 10f - 5f + 8 + 6g + 4, we can combine like terms by adding or subtracting coefficients that have the same variables:
10f - 5f + 8 + 6g + 4
Combining the terms with 'f', we have:
(10f - 5f) + 8 + 6g + 4
This simplifies to:
5f + 8 + 6g + 4
Next, we can combine the constant terms:
8 + 4 = 12
Thus, the simplified expression is:
5f + 12 + 6g
This expression is equivalent to 10f - 5f + 8 + 6g + 4.
In summary, the expression 10f - 5f + 8 + 6g + 4 simplifies to 5f + 12 + 6g after combining like terms.
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Ms. Florinda is a kindergarten teacher. She buys 100 pencils and wants to give 2 pencils to each of her students. She has 2 classes, a class with 22 students and a class with 19 students.
Part A
Write an expression for how many pencils she has left after giving them out to her students.
A.
100
−
2
×
(
22
−
19
)
B.
100
−
2
×
22
−
19
C.
100
−
2
×
22
−
2
×
19
D.
100
−
22
−
19
Part B
Does she have enough pencils to give each of her students 2?
Yes or no
, she has
15,18,37,59
More or fewer
than she needs.
Answers
Answer:
Part A:
The correct expression for how many pencils Ms. Florinda has left after giving them out to her students is:
A. 100 - 2 × (22 - 19)
Part B:
To determine whether Ms. Florinda has enough pencils to give each of her students 2, we can calculate the total number of pencils needed. The total number of students is the sum of the students in both classes, which is 22 + 19 = 41.
If each student needs 2 pencils, then the total number of pencils needed is 2 × 41 = 82.
Comparing this with the initial number of pencils Ms. Florinda bought (100), we can see that she has more than enough pencils to give each of her students 2.
Yes, she has enough pencils to give each of her students 2.
She has 18 more than she needs.
what is (0.3)0 in binominal distribution
Answers
Answer:
When p, the probability of success, is zero in a binomial distribution, the probability of getting exactly k successes in n trials is also zero for all values of k except when k is zero (i.e., when there are no successes).
So, in the case of (0.3)^0, the result would be 1, because any number raised to the power of 0 is equal to 1. Therefore, the probability of getting zero successes in a binomial distribution when the probability of success is 0.3 is 1.
Graph the function f(x)= 3+2 in x and its inverse from model 1.
Answers
The graph of the function and its inverse is added as an attachment
Sketching the graph of the function and its inverseFrom the question, we have the following parameters that can be used in our computation:
f(x) = 3 + 2ln(x)
Express as an equation
So, we have
y = 3 + 2ln(x)
Swap x and y in the above equation
x = 3 + 2ln(y)
Next, we have
2ln(y) = x - 3
Divide by 2
ln(y) = (x - 3)/2
Take the exponent of both sides
[tex]y = e^{\frac{x - 3}{2}}[/tex]
Next, we plot the graphs
The graph of the functions is added as an attachment
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The following is a list of shoe sizes for a group of 13 people.
4.5, 9.5, 8, 6.5, 10, 7, 8.5, 6, 7.5, 9, 6, 7, 11
Which of the following box plots best represents the numerical data?
A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 11.25 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.25. The lines outside the box end at 4.5 and 10. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 13 with tick marks every one-half unit. The box extends from 6.5 to 9 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 12. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
A box plot using a number line from 3 to 12.5 with tick marks every one-fourth unit. The box extends from 6.25 to 8.75 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 10.5. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
Answers
The box plot that best represents the numerical data is: A. A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.
How to complete the five number summary of a data set?In order to determine the five-number summary for the survey, we would arrange the data set in an ascending order:
4.5,6,6,6.5,7,7,7.5,8,8.5,9,9.5,10,11
Based on the information provided about the list of shoe sizes for a group of 13 people, we would use a graphical method (box plot) to determine the five-number summary for the given data set as follows:
Minimum (Min) = 4.5.
First quartile (Q₁) = 6.25.
Median (Med) = 7.5.
Third quartile (Q₃) = 9.25.
Maximum (Max) = 11.
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Given the function f(x) = 4 – 2x, find f(3r – 1).
Answers
Answer:
f(3r - 1) = -6r + 6
Step-by-step explanation:
To find f(3r - 1), we substitute 3r - 1 for x in the expression for f(x) and simplify:
f(x) = 4 - 2x
f(3r - 1) = 4 - 2(3r - 1)
= 4 - 6r + 2
= -6r + 6
So, f(3r - 1) = -6r + 6.
A company Charting its profits notices that the relationship between the number of units sold,x, and the profit,P, is a linear. If 170 units sold results in $20 profit and 220 units sold results in $2820 profit, write the profit function for this company.
P=
Find the marginal profit
$
Answers
Step-by-step explanation:
a linear relationship or function is described in general as
y = f(x) = ax + b
Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".
here f(x) is P(x) :
P(x) = ax + b
now we are using both given points (ordered pairs) to calculate a and b :
20 = a×170 + b
2820 = a×220 + b
to eliminate first one variable we subtract equation 1 from equation 2 :
2800 = a×50
a = 2800/50 = 280/5 = 56
now, we use that in any of the 2 original equations to get b :
20 = 56×170 + b
b = 20 - 56×170 = 20 - 9520 = -9500
so,
P(x) = 56x - 9500
need help please see attacged
Answers
The domain of f(x) is (0, +∞), and the range is (0, +∞). The graph of the function will have a vertical asymptote at x = 0 and will continuously increase as x approaches positive infinity.
To graph the given logarithmic function f(x) based on the table, we can use the information provided. The table presents pairs of values (x, y), where x represents the input and y represents the output of the function.
From the table, we can observe that the input values (x) are positive and non-zero. This indicates that the domain of the function is x > 0, meaning x is greater than zero. In interval notation, the domain would be written as (0, +∞).
Looking at the output values (y) in the table, we see that they are all positive. This suggests that the range of the function is y > 0, meaning y is greater than zero. In interval notation, the range would be expressed as (0, +∞).
Graphically, the function f(x) is logarithmic and will have a vertical asymptote at x = 0. As x approaches positive infinity, the function increases without bound. The graph starts at y = 125 when x = 1, and it intersects the y-axis at y = 5 when x = 1.5. The graph of the function will resemble a curve that approaches but never touches the x-axis.
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Similar Triangles
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
I need help on number 1 and 2
Answers
The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;
1. ΔAJK ~ ΔSWY according to the SAS similarity postulate
2. ΔLMN ~ ΔLPQ according to the AA similarity postulate
3. ΔPQN ~ ΔLMN
LM = 12, QP = 8
4. ΔLMK~ΔLNJ
NL = 21, ML = 14
What are similar triangles?
Similar triangles are triangles that have the same shape but may have different sizes.
1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;
24/16 = 3/2
18/12 = 3/2
The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.
2. The ratio of the corresponding sides in each of the triangles are;
MN/LN = 8/10 = 4/5
PQ/LQ = 12/(10 + 5) = 12/15 = 4/5
The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule
3. The alternate interior angles theorem indicates;
Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;
ΔPQN ~ ΔLMN by the AA similarity postulate
LM/QP = (x + 3)/(x - 1) = 18/12
12·x + 36 = 18·x - 18
18·x - 12·x = 36 + 18 = 54
6·x = 54
x = 54/6 = 9
LM = 9 + 3 = 12
QP = x - 1
QP = 9 - 1 = 8
4. The similar triangles are; ΔLMK and ΔLNJ
ΔLMK ~ ΔLNJ by AA similarity postulate
ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)
ML/NL = LK/LJ = (24 - 8)/24
(24 - 8)/24 = (6·x + 2)/((7·x + 7)
16/24 = (6·x + 2)/(7·x + 7)
16 × (7·x + 7) = 24 × (6·x + 2)
112·x + 112 = 144·x + 48
144·x - 112·x = 32·x = 112 - 48 = 64
x = 64/32 = 2
ML = 6 × 2 + 2 = 14
NL = 7 × 2 + 7 = 21
MN = 2 + 5 = 7
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solve this system of equations by using the elimination method x-5y=16 4x-2y=-8
Answers
Answer:
(- 4, - 4 )
Step-by-step explanation:
x - 5y = 16 → (1)
4x - 2y = - 8 → (2)
multiplying (1) by - 4 and adding to (2) will eliminate x
- 4x + 20y = - 64 → (3)
add (2) and (3) term by term to eliminate x
(4x - 4x) + (- 2y + 20y) = - 8 - 64
0 + 18y = - 72
18y = - 72 ( divide both sides by 18 )
y = - 4
substitute y = - 4 into either of the 2 equations and solve for x
substituting into (1)
x - 5(- 4) = 16
x + 20 = 16 ( subtract 20 from both sides )
x = - 4
solution is (- 4, - 4 )
¿Cuál es el costo de un plátano si el racimo de 22 plátanos cuesta $23.10?
Answers
The cost of a single unit is given as follows:
$1.05.
El costo de un plátano es el seguiente:
$1.05.
How to obtain the cost of a single unit?The cost of a single unit is obtained applying the proportions in the context of the problem.
The cost of 22 units is of $23.10, hence the cost of a single unit is obtained dividing the total cost by the number of units, as follows:
23.1/22 = $1.05.
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how can you write the expression with a rationalized denominator?
3 sqrt 2 / 3 sqrt 6
see photo attached for answers
Answers
The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.
To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):
(3√2 / 3√6) * (-√6 / -√6)
This simplifies to:
-3√12 / (-3√36)
Further simplifying, we have:
-3√12 / (-3 * 6)
-3√12 / -18
Finally, we can cancel out the common factor of 3:
- 3√9 / - 6.
Simplifying further, we get:
3√9 / 6.
Option C is the correct answer.
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On a line graph, time is usually represented on the vertical axis.
O True
O False
--
Answers
HELPPPPPP ME PLEASEEEEE!!
Answers
Answer:
Step-by-step explanation:
The quadratic formula is y=ax^2+bx+c
If we move everything to the left side of the equation,
-6x^2=-9x+7 becomes
-6x^2+9x-7=0
a=-6, b=9, c=-7, so the third answer choice