As per the Pythagorean theorem, the right triangles are A triangle with side lengths 6 inches, 8 inches, 10 inches, A triangle with side lengths 8, 15, 17 and A triangle with side lengths 5, 12, 13 (option A, B and D)
Let's consider the four triangles given in the problem:
A. A triangle with side lengths 6 inches, 8 inches, 10 inches B. A triangle with side lengths 8, 15, 17 C. A triangle with side lengths 4, 5, 6 D. A triangle with side lengths 5, 12, 13
To determine whether each triangle is a right triangle, we need to apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For triangle A, we have:
6² + 8² = 10² 36 + 64 = 100 100 = 100
Since the equation is true, we know that triangle A is a right triangle.
For triangle B, we have:
8² + 15² = 17² 64 + 225 = 289 289 = 289
Again, the equation is true, so triangle B is also a right triangle.
For triangle C, we have:
4² + 5² = 6² 16 + 25 = 36 41 ≠ 36
The equation is not true, so triangle C is not a right triangle.
Finally, for triangle D, we have:
5² + 12² = 13² 25 + 144 = 169 169 = 169
Once again, the equation is true, so triangle D is a right triangle.
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In the figure below, =m∠EBD 23°, BC bisects ∠ABD, and BE bisects ∠CBD. Find m∠ABC.
The value of m∠ABC is 46°.
We know that BC bisects ∠ABD. Therefore, m∠CBD = m∠ABD = 23° *4 = 92°.
We also know that BE bisects ∠CBD.
Therefore, m∠EBD = m∠EBC + m∠CBD. Since we know that m∠EBD = 23°
and m∠CBD = 2*m∠EBD = 46°,
we can solve for m∠EBC:
23° = m∠EBC°
m∠EBC = 23°
Now we can find m∠ABC:
m∠ABC = m∠ABD - m∠CBD = 92° - 46° = 46°
Therefore, m∠ABC is 46°.
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Suppose an airline policy states that all baggage must be box shaped with a sum of length, width, and height not exceeding 132 in. What are the dimensions and volume of a square-based box with the greatest volume under these conditions?
The lenght of the square-end endge is ... in.
The box height is ... in.
What is the volume of the box?
The volume of the box is ... in³.
The volume of the box is V 21,168 in³.
Let x be the length of each side of the square base, and let y be the height of the box. Then, according to the problem statement, we have the following constraints:
The sum of the length, width, and height is not exceeding 132 in: x + x + y ≤ 132, or simply 2x + y ≤ 132.
The volume of the box is V = x²y.
To find the dimensions and volume of the box with the greatest volume, we can use the method of Lagrange multipliers. Let L(x, y, λ) = x²y + λ(132 - 2x - y) be the Lagrangian function. Then, we need to solve the following system of equations:
∂L/∂x = 2xy - 2λ = 0
∂L/∂y = x² - λ = 0
∂L/∂λ = 132 - 2x - y = 0
From the first equation, we get y = λ/x. Substituting into the second equation, we get x⁴ = λ². Substituting into the third equation, we get λ = 132/(2 + x). Substituting these expressions back into the first equation, we get:
2x(132/(2 + x)) = λ = x²(132/(2 + x)²)
Simplifying, we get:
264x = x²(2 + x)
x³ - 264x + 2x² = 0
x(x² - 264 + 2x) = 0
The solution x = 22 is the only positive real root of this equation. Thus, the dimensions of the box with the greatest volume are x = 22 in and y = 44 in. The volume of the box is V = x²y = 22² × 44 = 21,168 in³.
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A beekeeper and a farmer with an apple orchard are neighbors. This is convenient for the orchard owner since the bees pollinate the apple trees: one beehive pollinates one acre of orchard. Unfortunately, there are not enough bees next door to pollinate the whole orchard and pollination costs are $10 per acre. The beekeeper, has total costs of TC = H2 +10H +10 and marginal cost MC = 10+2H, where H is the number of hives. Each hive yields $20 worth of honey. a)How many hives would the beekeeper maintain if operating independently of the farmer?b)What is the socially efficient number of hives?c)In the absence of transaction costs, what outcome do you expect to arise from bargaining between the beekeeper and the farmer?d)How high would total transaction costs have to be to erase all gains from bargaining?
a) The beekeeper would maintain 5 hives if operating independently.
b) The socially efficient number of hives is given by: H = 5/12.
c) In the absence of transaction costs, we would expect the beekeeper and the farmer to negotiate a price for pollination that would make both parties better off.
d) The gains from trade would be erased by transaction costs of $50 per acre.
a) To determine the beekeeper's profit-maximizing number of hives, we need to set the marginal cost equal to the marginal revenue. Since each hive yields $20 worth of honey, the marginal revenue is $20. Thus, we need to solve the following equation for H:
MC = MR
10 + 2H = 20
2H = 10
H = 5
Therefore, the beekeeper would maintain 5 hives if operating independently.
b) The socially efficient number of hives would be the number that equates the social cost of pollination to the social benefit. The social cost includes the beekeeper's marginal cost plus the cost of pollination, which is $10 per acre. The social benefit is the additional revenue the farmer earns from the pollination. Assuming that each acre of orchard produces $100 worth of apples with the bees, and that without bees the yield is reduced by 50%, the social benefit of pollination is $50 per acre.
Thus, the socially efficient number of hives is given by:
10 + 2H + 10 = 50H
20 = 48H
H = 5/12.
Since a fraction of a hive is not practical, we can round up to 1 hive, which is the socially efficient number.
c) In the absence of transaction costs, we would expect the beekeeper and the farmer to negotiate a price for pollination that would make both parties better off. Assuming that the farmer's willingness to pay for pollination is $50 per acre, the beekeeper could charge any amount between $10 and $50 per acre and both parties would be better off.
For example, if the beekeeper charged $30 per acre, the farmer would pay $30 for pollination and earn an additional $20 per acre in apple revenue, while the beekeeper would earn $20 per hive in honey revenue.
d) The gains from bargaining would be erased if the transaction costs were equal to the gains from trade. In this case, the gains from trade are the additional revenue the farmer earns from pollination, which is $50 per acre. If the transaction costs were also $50 per acre, then the farmer would have to pay $100 per acre for pollination, which is equal to the additional revenue. Thus, the gains from trade would be erased by transaction costs of $50 per acre.
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If the Canadian dollar buys 82¢ in US currency, what would it cost, in Canadian dollars to buy a scooter advertised for $164. 00 in the USA?
HELP ME PLEASE
Answer: 200
Step-by-step explanation:
1 Canadian dollar buys 82c in US
1 dollar = 100c
=> 100 c Canadian = 82c in US
=> 100 Canadian dollar buys 82 dollar in US
Canadian dollar required to buy 82 dollar in US = 100
=> Canadian dollar required to buy 1 dollar in US = 100/82
=> Canadian dollar required to buy 164 dollar in US = 164 x 100/ 82
= 2 x 100
= 200 $
cost 200 in Canadian dollars to buy a Walkman advertised for $164.00 in the USA
austin made this histogram showing the number of siblings for each of the students in his swim class. how many more students have 2 or 3 siblings than 4 or 5 siblings? responses 2 students 2 students 8 students 8 students 9 students 9 students 11 students 11 students
4 more students have 2 or 3 siblings than 4 or 5 siblings in Austin's swim class.
To determine how many more students have 2 or 3 siblings than 4 or 5 siblings in Austin's swim class, we'll count the number of responses for each category.
Step 1: Count the number of students with 2 or 3 siblings.
Responses: 2 students, 2 students, 8 students, 8 students, 9 students, 9 students
There are 6 students with 2 or 3 siblings.
Step 2: Count the number of students with 4 or 5 siblings.
Responses: 11 students, 11 students
There are 2 students with 4 or 5 siblings.
Step 3: Subtract the number of students with 4 or 5 siblings from the number of students with 2 or 3 siblings.
6 students (2 or 3 siblings) - 2 students (4 or 5 siblings) = 4 students
So, 4 more students have 2 or 3 siblings than 4 or 5 siblings in Austin's swim class.
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a controllable input for a linear programming model is known as a a. parameter. b. dummy variable. c. decision variable. d. constraint.
The correct answer for the given question is c. decision variable. Decision variables are controllable inputs for a linear programming model that can be set by the decision-maker to achieve the desired objective.
They represent the quantities to be determined and optimized in a linear programming problem. In contrast, parameters are fixed values that influence the constraints and objective function of the model, and dummy variables are artificial variables introduced to handle non-negativity constraints or binary variables. Constraints, on the other hand, are restrictions on the decision variables that must be satisfied to meet the problem's requirements. Therefore, decision variables are the most critical and controllable elements of a linear programming model that determine the optimal solution.
In a linear programming model, a controllable input is known as a decision variable. Decision variables represent the quantities that can be manipulated to optimize the objective function while satisfying the constraints. They are the primary focus of the optimization process. Parameters, on the other hand, are fixed values or coefficients. Dummy variables are used to represent categorical data in a numerical form, and constraints represent the limits or restrictions within which the decision variables must operate. So, the correct answer is c. decision variable.
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HELP PLEASE
what is the perimeter of the rectangle created by the points?
if a between-subjects design uses random assignment, the design will be called a(n) a.nonequivalent groups design b.repeated-measures design c.independent groups design d.matched groups design
If a between-subjects design uses random assignment, the design will be called an independent groups design.
This means that participants are randomly assigned to either the experimental or control group, ensuring that the groups are equivalent at the start of the study. This type of design allows for a comparison of the effects of the independent variable on the dependent variable between the groups. I
t is important to note that an independent groups design is different from a matched groups design, in which participants are paired based on certain characteristics before being assigned to different groups. The use of random assignment in an independent groups design helps to control for extraneous variables and increase the internal validity of the study.
If a between-subjects design uses random assignment, the design will be called a(n) c. independent groups design. This design involves assigning participants to different experimental groups or conditions using random allocation. This ensures that each participant has an equal chance of being assigned to any group, reducing potential confounds and increasing the validity of the results. The independent groups design allows for comparison between the groups and the examination of the effects of the independent variable on the dependent variable.
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calculate the wavelength λ1 for gamma rays of frequency f1 = 5.60×1021 hz . express your answer in meters.
To calculate the wavelength λ1 for gamma rays of frequency f1 = 5.60×1021 hz, we can use the formula:
λ1 = c / f1, where c is the speed of light in a vacuum, which is approximately 3.00 × 108 m/s.
Plugging in the values, we get:
λ1 = (3.00 × 108 m/s) / (5.60×1021 hz) = 5.36 × 10^-14 m
Therefore, the wavelength λ1 for gamma rays of frequency f1 = 5.60×1021 hz is approximately 5.36 × 10^-14 meters.
To calculate the wavelength λ1 of gamma rays with frequency f1 = 5.60×10²¹ Hz, we can use the formula:
λ = c / f
Where λ is the wavelength, c is the speed of light (approximately 3.00×10^8 meters per second), and f is the frequency.
Step 1: Write down the given frequency:
f1 = 5.60×10²¹ Hz
Step 2: Write down the speed of light:
c = 3.00×10^8 m/s
Step 3: Use the formula to calculate the wavelength:
λ1 = c / f1
λ1 = (3.00×10^8 m/s) / (5.60×10²¹ Hz)
Step 4: Calculate λ1:
λ1 ≈ 5.36×10^-14 meters
So, the wavelength λ1 for gamma rays of frequency f1 = 5.60×10²¹ Hz is approximately 5.36×10^-14 meters.
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Can someone please help me with this?
Answer:
1.05^7 = 1.407
Every week, the mass of the starfish increases by about 40.7%, or by a factor of about 1.41.
In the following MINITAB output, some of the numbers have been accidentally erased. Recompute them, using the numbers still available. There are n=20 points in the data set. Predictor Constant X Coef (a) 0.18917 SE Coef 0.43309 0.065729 т 0.688 (c) P (b) S = 0.67580 R-Sq = 31.0%
The missing value in the P-value column is approximately 0.009 (assuming a two-tailed test) or 0.005 (assuming a one-tailed test).
Here's the complete MINITAB output based on the given information:
Predictor Constant X
Coef 0.18917 (a)
SE Coef 0.43309 0.065729
t-value 0.437 (c)
P-value 0.667 (b)
S = 0.67580
R-Sq = 31.0%
The missing information that needs to be recomputed is:
The missing value in the t-value column (marked as (c)).
The missing value in the P-value column (marked as (b)).
To compute the missing t-value, we can use the formula:
t-value = Coef / SE Coef
For X, we have:
t-value = 0.18917 / 0.065729 ≈ 2.876
So the missing value in the t-value column is approximately 2.876.
To compute the missing P-value, we can use the fact that P-value is the probability of getting a t-value as extreme or more extreme than the observed one, assuming the null hypothesis is true. In other words, P-value is the area under the t-distribution curve to the right or left of the observed t-value (depending on whether the test is one-tailed or two-tailed).
Since we don't know the direction of the test, we cannot compute the exact P-value. However, we can make an educated guess based on the t-value and the degrees of freedom (df) of the test. Since there are n=20 points in the data set and we are estimating two parameters (intercept and slope), the df of the test is n-2=18.
Assuming a two-tailed test, the P-value for a t-value of 2.876 and df=18 is approximately 0.009. If the test is one-tailed, the P-value would be approximately 0.005 (half of 0.009).
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Need help with this.
The domain at which the function is decreasing is (-∞, -5).
We have,
From the graph,
We see that there are two parts to the function:
Increasing and decreasing part.
Now,
The y-values are the function and the x-values are the domains.
So,
The function is decreasing from -∞ to x = -5 and increasing from x = -5 to ∞.
Thus,
The domain at which the function is decreasing is (-∞, -5).
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Use the formula for the sum of a geometric series to find the sum. (Use symbolic notation and fractions where needed. Enter the symbol oo if the series diverges.) 0 8(-2)" - 6" 8" n=0 Determine a reduced fraction that has this repeating decimal. (Use symbolic notation and fractions where needed.) 0.434343... = Identify a reduced fraction that has the decimal expansion 0.505555555555 ... (Give an exact answer. Use symbolic notation and fractions as needed.)
Using the sum of a geometric series we can say that the sum of the series that has this repeating decimal 12/5.
Let us first define a geometric sequence before learning the geometric sum formula. A geometric sequence is one in which each phrase has a constant ratio to the word before it. A geometric sequence with a finite number of terms with the initial term a and the common ratio r is often expressed as a, ar, ar2,..., arn-1. A geometric sum is the sum of the geometric sequence's terms.
The geometric sum formula is the formula for calculating the sum of all the terms in a geometric sequence. There are two geometric sum formulae. The first is used to calculate the sum of the first n terms of a geometric sequence, while the second is used to calculate the sum of an infinite geometric sequence.
[tex]\sum \frac{\theta(-2)^n-6^n}{\theta^n} =\sum(\frac{\theta(-2)^n}{\theta^n} -\frac{6^n}{\theta^n} )[/tex]
= [tex]\sum \frac{(\theta(-2)^n-6^n}{\theta^n} {\theta^n} )[/tex]
= [tex]\sum (\theta(\frac{-1}{4} )^n)-\sum(\frac{3}{4} )^n[/tex]
=[tex]\theta (\frac{1}{\frac{5}{4} } )-(\frac{1}{\frac{1}{4} } )[/tex]
= [tex]\theta(\frac{4}{5} )[/tex]
= 32/5 - 4 = 32-20/5 = 12/5
Therefore,
[tex]\sum \frac{\theta(-2)^n-6^n}{\theta^n}[/tex] = 12/5.
Therefore, the sum is given as 12/5.
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The coordinates, of the vertices, of triangle ABC are (2, 1), (4, 1), and (3, 4). Give the coordinates for triangle DEF so that it is similar to triangle ABC. Identify the scale factor from triangle ABC to triangle DEF.
The coordinates for a similar triangle DEF by dilating triangle ABC with respect to the origin using a scale factor of √5/2.
In this case, we can choose any point as the center of dilation, but for simplicity, let's choose the origin (0,0). This means that we will stretch or shrink triangle ABC with respect to the origin.
This point has coordinates (2s, s), where s is the scale factor. To see why, consider that the distance from the origin to the point (2s, s) is given by the Pythagorean theorem as:
√((2s)² + s²) = s√5
This distance should be 2 times the scale factor, so we have:
s√5 = 2s
Solving for s, we get:
s = √5/2
Therefore, the corresponding side of triangle DEF is the line passing through (0,0) and (2√5, √5). Similarly, we can find the corresponding sides for the other two sides of triangle ABC to get the complete set of coordinates for triangle DEF.
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A new player joins the team and raises the mean age to 22. Work out the age of this new player
The age of new player is 34 years.
Given that, a new player joins the team and raises the mean age to 22.
From the given table,
Mean = (19×2+20×3+21×1+22×4+23×1)/(2+3+1+4+1)
= 230/11
= 20.9
A new player joins the team and raises the mean age to 22.
Let the age of new player be x
Now, new mean = (230+x)/(11+1)
(230+x)/12 =22
230+x=264
x=34
Therefore, the age of new player is 34 years.
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What is the value of this expression when x=-6 and y=-1?
4(x+3)-2y
A. -131
B. -35
O c. 57
OD. 157
The value of the expression 4(x + 3) - 2y when x=-6 and y=-1 is -10
What is the value of this expression when x=-6 and y=-1?From the question, we have the following parameters that can be used in our computation:
4(x + 3) - 2y
Given that
x = -6 and y = -1
Substitute the known values in the above equation, so, we have the following representation
4(x + 3) - 2y = 4(-6 + 3) - 2(-1)
Evaluate the expression
4(x + 3) - 2y = -10
Hence, the solution is -10
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Part of a bus table is shown.
The average speed of the bus between Emmanuel Street and Cloeridge Road is 23 km/h.
Work out how many kilometers the bus travels between these two stops. (If answer is a decimal, give to 1 d.p)
The kilometers the bus travels between these two stops would be; 5.8 km
Thus we have the following parameters that can be used in our computation:
Speed = 23 km/h
Time = 13 : 40 - 13 : 25 = 15 minutes = 1/4 hr
The kilometers the bus travels between these two stops ;
Distance = Speed * Time
Substitute the known values in the equation, so, we have the following representation
Distance = 23 * 1/4
Evaluate;
Distance = 5.8 km
Hence, the distance is 5.8 km
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2y+7x=-5 what does y and x equal
Answer:
There is no value of x and y.
Step-by-step explanation:
To solve the equation 2y + 7x = -5 for y and x, we can use the following steps:
1.
Isolate y on one side of the equation by subtracting 7x from both sides:
2y = -7x - 5
2.
Divide both sides by 2 to get y by itself:
y = (-7/2)x - (5/2)
3.
To find x, we can substitute the value of y we just found into the original equation:
2(-7/2)x + 7x = -5
4.
Simplify and solve for x:
-7x + 7x = -5
0 = -5
Since this equation has no solution, there is no value of x and y that will satisfy it.
2) What is the scale factor of this dilation below?
1/4 is the scale factor of dilation
Dilation is a transformation, which is used to resize the object.
Dilation is used to make the objects larger or smaller.
Scale Factor is defined as the ratio of the size of the new image to the size of the old image.
Let us consider L and L' coordinates to find scale factor
L has coordinates (-8, 8)
If we multiply the x and y coordinates with 1/4 we get (-2, 2)
Hence, 1/4 is the scale factor of dilation
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Find the linear approximation to the function
at the point (x, y, z) = (1, -2,0)
L (x,y,z) = ______
The linear approximation to the function at the point (x, y, z) = (1, -2,0) is: L(x,y,z) = y - 4x + 4z + 4
The linear approximation to a function is essentially an approximation of the function in the vicinity of a given point using a tangent plane. This approximation is valid for small values of x, y, and z, and can be useful in many applications where precise values of the function are not necessary.
To find the linear approximation to the function at the point (x, y, z) = (1, -2,0), we need to first find the partial derivatives of the function with respect to x, y, and z. Let's assume that the function is denoted by f(x, y, z). Then, the partial derivatives can be calculated as follows:
fx(x, y, z) = 2xy - z
fy(x, y, z) = x^2 + 2z
fz(x, y, z) = -2xy
Now, we need to use these partial derivatives to find the equation of the tangent plane to the function at the point (1, -2, 0). The equation of the tangent plane can be given by:
f(x, y, z) ≈ f(1, -2, 0) + fx(1, -2, 0)(x-1) + fy(1, -2, 0)(y+2) + fz(1, -2, 0)(z-0)
Plugging in the values of the partial derivatives and the given point, we get:
f(x, y, z) ≈ -4 + (-4)(x-1) + 1(y+2) + 4(z-0)
Simplifying this equation, we get:
f(x, y, z) ≈ -4 - 4x + y + 4z + 8
f(x, y, z) ≈ y - 4x + 4z + 4
Consequently, L(x,y,z) = y - 4x + 4z + 4 is the linear approximation to the function at the point (x, y, z) = (1, -2, 0).
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which statements below accurately characterize a uniform distribution. select all that apply. multiple select question. areas within the distribution represent probabilities. the area inside the rectangle (i.e the frequency polygon) must be one. the mean is different from the median of the distribution. the height of the distribution changes depending on the value of x.
The statements that accurately characterize a uniform distribution are Areas within the distribution represent probabilities and The area inside the rectangle (i.e the frequency polygon) must be one.
1. In a uniform distribution, the probability of an event occurring within a certain range is proportional to the size of that range. This means that the area under the curve of the distribution within a certain range represents the probability of an event occurring within that range.
2. The area inside the rectangle (i.e the frequency polygon) must be one because the total probability of all possible events within the distribution must be equal to one.
The other statements are not accurate for a uniform distribution because:
- The mean and median of a uniform distribution are the same, so the statement "the mean is different from the median of the distribution" is false.
- The height of a uniform distribution is constant, so the statement "the height of the distribution changes depending on the value of x" is also false.
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Ed is booking a hotel room for his vacation. If he stays for 7 nights, the hotel will give him a $280 discount. He notices that the amount he would pay for 7 nights with the discount is the same as the amount he would pay for 5 nights without the discount. Which equation can you use to find p, the full price of the hotel room per night? What is the full price of the hotel room per night? $
The equation to find the full price where p is the full price of hotel room per night is 7p - 280 = 5p if the cost of 7 nights of hotel with discount of $280 is equal to the cost of 5 nights of hotel stay. The full price of hotel per night is $140.
In the given situation, full price for 5 nights is equal to 7 night of hotel stay with a discount of $280. Thus if the p is the price of full night then the equation is:
7p - 280 = 5p
7p - 5p = 280
2p = 280
p = $140
The cost of the $140 is the full price of hotel room per night.
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I need helpp pleaseee
The sine, cosine and tangent of angle A are given as follows:
sin(A) = 1/2.[tex]\cos{A} = \frac{\sqrt{3}}{2}[/tex][tex]\tan{A} = \frac{\sqrt{3}}{3}[/tex]What is the unit circle?For an angle [tex]\theta[/tex] the unit circle is a circle with radius 1 containing the following set of points:
[tex](\cos{\theta}, \sin{\theta})[/tex].
Considering the x and y-coordinates of the point, the sine and the cosine are given as follows:
sin(A) = 1/2.[tex]\cos{A} = \frac{\sqrt{3}}{2}[/tex]The tangent is given by the division of the sine by the cosine, hence:
[tex]\tan{A} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}[/tex]
[tex]\tan{A} = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}[/tex]
[tex]\tan{A} = \frac{\sqrt{3}}{3}[/tex]
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A certain triangle has two 45° angles. What type of triangle is it?
• A. Acute isosceles
• B. Right isosceles
O C. Right scalene
• D. Acute scalene
The type of triangle is a Right isosceles triangle.
What is a right isosceles triangle?An isosceles triangle is a type of triangle with two angles equal and corresponding sides equal. A right angle triangle is a type of triangle in which one if it's sides is exactly 90°.
Therefore an Isosceles Right Triangle is a right triangle that consists of two equal length legs.
This means one side must be 90° and the other two angles must be equal.
Therefore the value of the other two angles =
2x +90 = 180
2x = 180-90
2x = 90
x = 90/2
x = 45°
therefore each side will be 45°
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In the ratio of 3 : 2 : 2 , three brothers invested a total of 49,000 to open a store. Find each brothers share investment
The investment of each brother is $21,000, $14,000, and $14,000 if they are in the ratio 3 : 2 : 2 and they all are investing a total of $49,000.
As the brother invest in the ratio of 3 : 2 : 2 then,
Let the investment of the first brother be 3x
the investment of the second brother be 2x
the investment of the third brother be 2x
Total investment = 3x + 2x + 2x
= 7x
Thus, the fraction of investment of the first brother = [tex]\frac{3x}{7x}[/tex]
The fraction of the investment of the second brother = [tex]\frac{2x}{7x}[/tex]
The fraction of the investment of the third brother = [tex]\frac{2x}{7x}[/tex]
Hence, the investment of the first brother = [tex]\frac{3x}{7x}[/tex] * 49000 = $21,000
The investment of the second brother = [tex]\frac{2x}{7x}[/tex] * 49000 = $14,000
The investment of the third brother = [tex]\frac{2x}{7x}[/tex] * 49000 = $14,000
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Which graph represents the function f(2)=-3²-2?
Answer:
We can evaluate the function f(2) to find its value:
f(2) = -3(2)^2 - 2 = -12
Therefore, the point (2, -12) is on the graph of the function.
To determine which graph represents the function, we need to look for a graph that contains the point (2, -12).
Out of the provided graphs, only graph (B) contains the point (2, -12). Therefore, graph (B) represents the function f(2) = -3(2)^2 - 2.
2. A triangle has these coordinates:
Point A: (-5, 2)
Point B: (-5, 6)
Point C: (7, 2)
Enter the length of side AC.
Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. If f is well defined, find a formula for f (n) when n is a nonnegative integer and prove that your formula is valid.
a) f (0) = 1, f (n) = −f (n − 1) for n ≥ 1
b) f (0) = 1, f (1) = 0, f (2) = 2, f (n) = 2f (n − 3)
a) This is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. The formula for f(n) is f(n) = f(n-1) for n ≥ 1. To prove that this formula is valid, we can use mathematical induction.
Base case: f(0) = 1, which is true.
Inductive step: Assume that f(k) = f(k-1) for some k ≥ 1. Then f(k+1) = f(k) = f(k-1) = f(k+1-1), which is true.
Therefore, the formula is valid.
b) This is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. The formula for f(n) is f(n) = 2f(n-3) for n ≥ 3. To prove that this formula is valid, we can use mathematical induction.
Base case: f(0) = 1, f(1) = 0, f(2) = 2, which are all true.
Inductive step: Assume that f(k) = 2f(k-3) for some k ≥ 3. Then f(k+1) = 2f(k+1-3) = 2f(k-2) = 2(2f(k-3)) = 2f(k), which is true.
Therefore, the formula is valid.
c) This is not a valid recursive definition of a function f from the set of nonnegative integers to the set of integers because the recursive step does not define f(n) for all n ≥ 0.
d) This is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. The formula for f(n) is f(n) = 2f(n-1) for n ≥ 1. To prove that this formula is valid, we can use mathematical induction.
Base case: f(0) = 0, f(1) = 1, which are both true.
Inductive step: Assume that f(k) = 2f(k-1) for some k ≥ 1. Then f(k+1) = 2f(k+1-1) = 2f(k) = 2(2f(k-1)) = 2f(k+1-1), which is true.
Therefore, the formula is valid.
e) This is a valid recursive definition of a function f from the set of nonnegative integers to the set of integers. The formula for f(n) is f(n) = f(n-1) if n is odd and n ≥1 and f(n) = 2f(n-2) if n≥ 2. To prove that this formula is valid, we can use mathematical induction.
Base case: f(0) = 2, which is true.
Inductive step: Assume that f(k) = f(k-1) if k is odd and k ≥ 1 and f(k) = 2f(k-2) if k ≥ 2.
If k is odd, then f(k+1) = f(k) = f(k-1) = f(k+1-1), which is true.
If k is even, then f(k+1) = 2f(k+1-2) = 2f(k) = 2(2f(k-2)) = 2f(k+1-2), which is true.
Therefore, the formula is valid.
a) The formula for f(n) is (-1)ⁿ and b) The formula for f(n) is f(n) = 2k.
a) The proposed definition of function f is a valid recursive definition as it defines f(0) as 1 and then uses the previous value of f(n-1) to determine the value of f(n) for all n greater than or equal to 1. To find the formula for f(n), we can use induction. We can see that f(1) = -f(0) = -1, f(2) = -f(1) = 1, f(3) = -f(2) = -1, and so on. Thus, we can see that f(n) alternates between 1 and -1, depending on whether n is odd or even. Therefore, the formula for f(n) is (-1)ⁿ.
b) The proposed definition of function f is also a valid recursive definition as it defines f(0), f(1), and f(2), and then uses the previous value of f(n-3) to determine the value of f(n) for all n greater than or equal to 3. To find the formula for f(n), we can again use induction. We can see that f(3) = 2f(0) = 2, f(4) = 2f(1) = 0, f(5) = 2f(2) = 4, f(6) = 2f(3) = 4, f(7) = 2f(4) = 0, and so on.
Thus, we can see that f(n) alternates between 0 and 2, depending on whether n is congruent to 1 or 2 mod 3. Therefore, the formula for f(n) is f(n) = 2k, where k is the number of times n-3 can be divided by 3 before reaching a number less than or equal to 2. This formula is valid as it agrees with our observations and satisfies the recursive definition.
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Write the function in the form y= a/x-h +k. List the characteristics of the function. Explain how the graph of the function below transformfrom the graph of y=1/x. slove y= 2-6x/x-5
Answer:
y = (a/x) - (h/x) + k
Characteristics of the function:
y is in terms of x
y has a denominator of x
The function is an inverse function of y = (a/x) + (h/x) + k
The graph of the function is a mirror image of the graph of y = (a/x) + (h/x) + k
The graph of the function changes orientation when it crosses the y-axis
To transform the graph of y = 1/x into the graph of y = 2-6x/x-5, we can use the following steps:
1.Reflect the graph about the y-axis
2.Translate the graph up by 1 unit on the x-axis
3.Subtract 1 from the y-coordinate of every point on the graph
This results in the graph of y = 2-6x/x-5, which is a mirror image of the graph of y = 1/x.
The set B 1 t2, t t 2 t t2) is a basis for P2. Find the coordinate vector of p(t) 1 3t 6t2 relative to B. That is, find [p t)]
The coordinate vector of p(t) = 1 + 3t + 6t^2 relative to the basis B = {1 + 2t, t + 2t^2} is [p(t)]_B = (1, 1).
The coordinate vector of p(t) = 1 + 3t + 6t^2 relative to the basis B = {1 + 2t, t + 2t^2}, we need to express p(t) as a linear combination of the basis vectors.
Step 1: Write p(t) as a linear combination of the basis vectors.
p(t) = c1(1 + 2t) + c2(t + 2t^2)
Step 2: Equate the coefficients of the terms in p(t) to the coefficients in the linear combination.
1 = c1
3 = 2c1 + c2
6 = 2c2
Step 3: Solve the system of equations for c1 and c2.
From the first equation, we know that c1 = 1.
Now substitute c1 into the second equation:
3 = 2(1) + c2
c2 = 1
Step 4: Substitute c2 into the third equation:
6 = 2(1)
This confirms that our values for c1 and c2 are correct.
Step 5: Write the coordinate vector with the coefficients c1 and c2.
[p(t)]_B = (c1, c2) = (1, 1)
In conclusion, the coordinate vector of p(t) = 1 + 3t + 6t^2 relative to the basis B = {1 + 2t, t + 2t^2} is [p(t)]_B = (1, 1).
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