The equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.
From the given options, the equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.
To determine the equation of a hyperbola, we examine the standard form:
For a hyperbola centered at (h, k), with vertical transverse axis, the standard form is:
(y - k)²/a² - (x - h)²/b² = 1
From the given graph, we can observe that the center of the hyperbola is (-2, 3). This corresponds to the values of (h, k) in the standard form.
Next, we need to determine the values of a and b, which are the lengths of the transverse and conjugate axes, respectively. Looking at the graph, we see that the transverse axis has a length of 2a = 4, so a = 2. The conjugate axis has a length of 2b = 10, so b = 5.
Plugging these values into the standard form, we obtain:
(y - 3)²/4 - (x + 2)²/25 = 1
The equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.
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The correct equation representing the hyperbola shown in the graph is:
(x + 2)²/25 - (y - 3)²/4 = 1.
The equation that represents the hyperbola shown in the graph is:
(x + 2)²/25 - (y - 3)²/4 = 1
Let's analyze the options provided:
(x - 2)²(v + 3)² = 1:
This equation is not a valid representation of a hyperbola because it contains a term (v + 3)², which is not consistent with the variable used in the graph.
(x + 2)²/25:
This equation represents a horizontal parabola, not a hyperbola.
(x - 2)²/25:
This equation represents a horizontal parabola, not a hyperbola.
(y - 3)²/1:
This equation represents a vertical line, not a hyperbola.
(y - 3)²/4:
This equation represents a hyperbola with a vertical transverse axis and a conjugate axis length of 2b = 4 (b = 2).
The equation is in the standard form for a hyperbola with a vertical transverse axis.
The equation is provided as a standard form assuming the given coordinates and graph match the standard form representation of a hyperbola.
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NO LINKS!! URGENT HELP PLEASE!!
Use the laws of sines and cosines for the missing variable
Answer:
x = 8
Step-by-step explanation:
The given diagram shows a triangle with the length of two sides and its included angle.
To find the value of the missing variable x, we can use the Law of Cosines.
[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]
From inspection of the given triangle:
a = 18b = 21c = xC = 22°Substitute the values into the formula and solve for x:
[tex]\begin{aligned}x^2&=18^2+21^2-2(18)(21)\cos 22^{\circ}\\x^2&=324+441-756\cos 22^{\circ}\\x^2&=765-756\cos 22^{\circ}\\x&=\sqrt{765-756\cos 22^{\circ}}\\x&=8.00306228...\\x&=8\end{aligned}[/tex]
Therefore, the value of the missing variable x is x = 8, rounded to the nearest hundredth.
X-2
5 = 8 using the change of base formula logby=
log y
log b
By using the change of base formula: The solution to the equation log(base y) (X-2) = 5 is [tex]X = y^5 + 2.[/tex]
To solve the equation log(base y) (X-2) = 5 using the change of base formula, we can rewrite the equation as log(base b) (X-2) / log(base b) y = 5.
Using the change of base formula, we can choose any base for b.
Let's choose base 10 for simplicity.
So the equation becomes log(base 10) (X-2) / log(base 10) y = 5.
We know that log(base 10) (X-2) represents the logarithm of (X-2) to the base 10, and log(base 10) y represents the logarithm of y to the base 10.
Now, to solve for X, we can isolate it by multiplying both sides of the equation by log(base 10) y:
log(base 10) (X-2) = 5 [tex]\times[/tex] log(base 10) y.
This simplifies to:
log(base 10) (X-2) [tex]= log(base 10) y^5.[/tex]
Since the logarithms on both sides have the same base, we can remove the logarithm and equate the arguments:
[tex]X - 2 = y^5.[/tex]
Now we can solve for X by adding 2 to both sides:
[tex]X = y^5 + 2.[/tex]
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b) Calculate the following using the order of operations and emphasizing factors of one.
2-(-7)+(-8)-(13) +-4-5
Answer:
-21
Step-by-step explanation:
2+7-8-13-4-5
9-30
-21
Hope it's helpful.
HELP I NEED ANSWER
Write an exponential decay function where the y-intercept is 4 and the y-values decrease by a factor of one-half as x increases by 1.
The exponential decay function that satisfies the given conditions is:
[tex]f(x) = 4 * (1/2)^x[/tex].
In this equation, the y-intercept is 4, which means that when x = 0, the function value is 4. As x increases by 1, the function decreases by a factor of one-half. This behavior is captured by raising 1/2 to the power of x in the equation.
The base of the exponent, 1/2, ensures that the function decreases exponentially. When x = 1, the exponent becomes 1, and[tex]1/2^1[/tex] equals 1/2. This means that the function value decreases to half of its previous value. Similarly, when x = 2, the exponent becomes 2, and[tex]1/2^2[/tex] equals 1/4. The function value decreases to one-fourth of its previous value, and so on.
By multiplying the exponential term by 4, we ensure that the y-intercept is 4. This scaling factor allows us to control the initial value of the function and match the given condition.
The exponential decay function[tex]f(x) = 4 * (1/2)^x[/tex] represents a decaying process where the y-values decrease exponentially as x increases, while starting at a y-intercept of 4.
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DC=x-2
Height=4
AB=2x+4
The area of the trapezoid ABCD shown above is 70 square units. Calculate x.
Answer:
Step-by-step explanation:To calculate the value of x, we can use the formula for the area of a trapezoid:
Area = (1/2) * (sum of the parallel sides) * height
Given that the area of the trapezoid ABCD is 70 square units, we can set up the equation as follows:
70 = (1/2) * (AB + DC) * Height
Substituting the given values:
70 = (1/2) * ((2x + 4) + (x - 2)) * 4
Simplifying the equation:
70 = (1/2) * (3x + 2) * 4
Multiplying both sides by 2 to remove the fraction:
140 = (3x + 2) * 4
Dividing both sides by 4:
35 = 3x + 2
Subtracting 2 from both sides:
33 = 3x
Dividing both sides by 3:
x = 11
Therefore, the value of x is 11.
please help i’m confused
The regression equation is y = 17.1643X - 2.47977
What is the equation of regression?To solve this problem, we have to calculate the equation of regression.
Sum of X = 2.97
Sum of Y = 28.66
Mean X = 0.33
Mean Y = 3.1844
Sum of squares (SSX) = 0.3552
Sum of products (SP) = 6.0959
Regression Equation = y = bX + a
b = SP/SSX = 6.1/0.36 = 17.1643
a = MY - bMX = 3.18 - (17.16*0.33) = -2.47977
y = 17.1643X - 2.47977
The line of best fit is y = 17.1643X - 2.47977
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NO LINKS!! URGENT HELP PLEASE!!
Find each indicated measure
Answer:
b. 160°
d. 55°
Step-by-step explanation:
The Inscribed Angle Theorem states that an inscribed angle is half of the central angle that subtends the same arc.
In other words, if an angle is inscribed in a circle and it intercepts an arc, then the measure of the inscribed angle is equal to half the measure of the central angle that also intersects that arc.
For question:
b.
By using above theorem:
m arc XW=2* m arc XYW
m arc XW= 2*80=160°
d.
m arc WV=125°
The Inscribed Angle Diameter Right Angle Theorem states that any angle inscribed in a circle that intercepts a diameter is a right angle.
By using this theorem:
m arc WV+m arc XV =180°
Now
m arc XV =180°-m arc WV
m arc XV=180°-125°
n arc XV=55°
Answer:
[tex]\text{b.} \quad m\overset{\frown}{XW}=160^{\circ}[/tex]
[tex]\text{d.} \quad m\overset{\frown}{XV}=55^{\circ}[/tex]
Step-by-step explanation:
An inscribed angle is the angle formed (vertex) when two chords meet at one point on a circle.
An intercepted arc is the arc that is between the endpoints of the chords that form the inscribed angle.
[tex]\hrulefill[/tex]
Part bFrom inspection of the given circle:
The inscribed angle is m∠WRX = 80°The intercepted arc is arc XW.According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:
[tex]m \angle WRX = \dfrac{1}{2}\overset{\frown}{XW}[/tex]
[tex]80^{\circ}= \dfrac{1}{2}\overset{\frown}{XW}[/tex]
[tex]\boxed{m\overset{\frown}{XW}=160^{\circ}}[/tex]
[tex]\hrulefill[/tex]
Part dFrom inspection of the given circle:
The inscribed angle is m∠WVX = 90°The intercepted arc is arc WX.According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:
[tex]m \angle WVX= \dfrac{1}{2}\overset{\frown}{WX}[/tex]
[tex]90^{\circ}= \dfrac{1}{2}\overset{\frown}{WX}[/tex]
[tex]m\overset{\frown}{WX}=180^{\circ}[/tex]
The sum of the measures of the arcs in a circle is 360°.
[tex]m\overset{\frown}{VW}+m\overset{\frown}{WX}+m\overset{\frown}{XV}=360^{\circ}[/tex]
Therefore, so find the measure of arc XV, substitute the found measures of arcs VW and WX, and solve for arc XV:
[tex]125^{\circ}+180^{\circ}+m\overset{\frown}{XV}=360^{\circ}[/tex]
[tex]305^{\circ}+m\overset{\frown}{XV}=360^{\circ}[/tex]
[tex]\boxed{m\overset{\frown}{XV}=55^{\circ}}[/tex]
Use the formulas to answer this question.
One leg of a right triangle has length 11 and all sides are whole numbers. Find the lengths of the other two sides.
The other leg = and the hypotenuse =
The lengths of the other two sides of the right triangle are 36 and 85, respectively.
To find the lengths of the other two sides of a right triangle when one leg has a length of 11, we can use the Pythagorean theorem.
The Pythagorean theorem 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.
Let's denote the lengths of the other leg and the hypotenuse as x and y, respectively.
According to the Pythagorean theorem, we have:
x² + 11² = y²
To find the values of x and y, we need to find a pair of whole numbers that satisfy this equation.
We can start by checking for perfect squares that differ by 121 (11^2). One such pair is 36 and 85.
If we substitute x = 36 and y = 85 into the equation, we have:
36² + 11² = 85²
1296 + 121 = 7225
This equation is true, so the lengths of the other two sides are:
The other leg = 36
The hypotenuse = 85
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What is the percent of 1 - 3√(5/35) ?
Answer:
1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
0.0755 * 100 = 7.55%
Step-by-step explanation:
To find the percentage of 1 - 3√(5/35), we need to first evaluate the expression.
1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
To convert this decimal to a percentage, we simply multiply by 100:
0.0755 * 100 = 7.55%
what is the 20th term of the sequence that begins -4, 8, -16, 32...?
Answer:
-2097152 is the 20th term
Step-by-step explanation:
Write geometric sequence as an explicit formula
[tex]-4,8,-16,32\rightarrow-4(-2)^0,-4(-2)^1,-4(-2)^2,4(-2)^3\\a_n=a_1r^{n-1}\rightarrow a_n=-4(-2)^{n-1}[/tex]
Find the n=20th term
[tex]a_{20}=-4(-2)^{20-1}=-4(-2)^{19}=4(-524288)=-2097152[/tex]
Snow Fall (Inches)
2.75
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
4
O A. 1.25
OB. 0.75
O C. 2.5
O D. 1.5
●
1
2
3
4
Time (hours after Midnight)
5
12. The graph above depicts the amount of snow accumulation from midnight to 5:00 a.m. The x-axis represents time (hours after midnight), and the y-axis represents the number of
inches of snow on the ground. How many inches of snow accumulated between 2:00 a.m. and 5:00 a.m.?
The amount of snow accumulated between 2 am and 5 am is: 1.25 inches
How to Interpret Linear Equation Graphs?The general formula for the equation of a line in slope intercept form is:
y = mx + c
where:
m is slope
c is y-intercept
From the given graph attached, we see that the y-axis gives the amount of snow at different specific times.
Meanwhile the x-axis gives the time in hours after midnight
At 2am, the y-axis value is 1.25 inches, and as such at 2am snow accumulation was 1.25 inches.
At 5 am, the y-axis value reads 2.5 inches, and as such at 5am snow accumulation was 2.5 inches.
The difference in both snow accumulations is: 2.5 - 1.25 = 1.25
Hence, 1.25 inches snow accumulated between 2 am and 5 am.
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Pls help I’m stuck Tysm I can’t thank any more
Using the concept of perimeter of polygon, the perimeter of figure C is 27cm shorter than total perimeter of A and B
How much shorter is the perimeter of C than the total perimeter of A and B?To solve this problem, we have to know the perimeter of the polygon C.
The perimeter of a polygon is the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.
The perimeter of the figures are;
Using the concept of perimeter of a rectangle;
a. figure A = 2(4 + 11) = 30cm
b. figure B = 2(8 + 4) = 24cm
c figure C = 11 + 4 + 8 + 4 = 27cm
Now, we can add A and B and then subtract c from it.
30 + 24 - 27 = 27cm
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Determine the surface area and volume. Note: The base is a square.
Answer:
volume=60cm3, surface area=96cm2
Step-by-step explanation:
volume=1/3×(6×6)×5
=60cm3
surface area= 4(1/2×6×5)+(6×6)
=96cm2
If a pound of rolled oats costs $4
, how many ounces can be bought for $1.95
?
Answer:
7.80 ounces can be bought for $1.95
Step-by-step explanation:
Step 1: Determine how many ounces is in a pound:
Because we want our final answer to be in ounces, we first need to determine how many ounces is in a pound. 1 pound is equal to 16 ounces.Thus, 16 ounces cost $4.
Step 2: Create a proportion to determine how many ounces can be bought for $1.95.
Since you can get 16 ounces for $4, we can create a proportion to determine how many ounces can be bought for $1.95:
16 ounces / $4 = x ounces / $1.95
Step 3: Simplify on the left-hand side of the equation:
16/4 = x/1.95
4 = x/1.95
Step 4: multiply both sides by 1.95 to determine how many ounces can be bought for $1.95:
(4 = x/1.95) * 1.95
7.80 = x
Thus, 7.80 ounces can be bought for $1.95.
Find the area of the shaded portion if we know the outer circle has a diameter of 4 m and the inner circle has a diameter of 1.5 m.
A. 43.2 m2
B. 10.8 m2
C. 12.6 m2
D. 1.8 m2
True or false: f(x) is a function.
0
3
6
9
f(x)
0
1
3
Answer:
Step-by-step explanation:
If {0, 3, 6, 9} are are your x's or domain or input and there are no repeats, then yes TRUE it is a function.
Assume that random guesses are made for seven multiple choice questions on an SAT test, so that there are n=7 trials, each with probability of success (correct) given by p=0.45. Find the indicated probability for the number of correct answers.
Find the probability that the number x of correct answers is fewer than 4.
if there are 200 high school students in the district, how many would you expect to be in chemistry?
If there are 200 high school students in the district, the number of high school students expected to be in Chemistry is 60 because the percentage who offer Chemistry in the district is 30%.
How the number is determined:The number of high school students who offer Chemistry in the district can be determined by multiplying the total number of high school students and the percentage of students who offer Chemistry.
The result of a multiplication operation (multiplicand and multiplier), which is one of the basic mathematical operations, is known as the product.
The total number of high school students in the district = 200
The percentage of students who offer Chemistry in the district = 30%
The number of students likely to be offering Chemistry in the district = 60 (200 x 30%).
Thus, we can conclude that 60 high school students are in Chemistry based on the Chemistry percentage.
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Complete Question:The percentage of high school students in the district who offer Chemistry is 30%. If there are 200 high school students in the district, how many would you expect to be in Chemistry?
How many boys are there in an introductory Chinese course if 352 students are enrolled and there are nine boys to every seven girls?
17x = 425
x = 25
8x = 200 boys
9x = 225 girls
Write a equation of the circle graphed below
Answer:
[tex](x+5)^2+(y+5)^2=25[/tex]
Step-by-step explanation:
Recall that the equation of a circle with center (h,k) and radius "r" is [tex](x-h)^2+(y-k)^2=r^2[/tex]
Since the center of the circle is (h,k)=(-5,-5) and the radius is r=5, then our equation will be [tex](x-(-5))^2+(y-(-5))^2=5^2[/tex] which can be simplified into [tex](x+5)^2+(y+5)^2=25[/tex]
6, 12, 24, 48, 96, … Each term is 6 more than the previous term. Each term is 12 more than the previous term. Each term is 1/2 the previous term. Each term is 2 times the previous term.
The given sequence can be generated by multiplying each term by 2, starting from the initial term of 6.
The pattern that fits the given sequence 6, 12, 24, 48, 96, ... is that each term is 2 times the previous term.
In the sequence 6, 12, 24, 48, 96, ... there are multiple possible patterns, each resulting from a different rule applied to generate the next term. Let's examine each of the proposed patterns:
Each term is 6 more than the previous term:
Starting with 6, if we add 6 to each term, we get:
6 + 6 = 12
12 + 6 = 18
18 + 6 = 24
24 + 6 = 30
30 + 6 = 36
...
This pattern does not match the given sequence since it does not produce the subsequent terms.
Each term is 12 more than the previous term:
Starting with 6, if we add 12 to each term, we get:
6 + 12 = 18
18 + 12 = 30
30 + 12 = 42
42 + 12 = 54
54 + 12 = 66
...
This pattern also does not match the given sequence.
Each term is 1/2 the previous term:
Starting with 6, if we multiply each term by 1/2, we get:
6 [tex]\times[/tex] 1/2 = 3
3 [tex]\times[/tex] 1/2 = 1.5
1.5 [tex]\times[/tex] 1/2 = 0.75
0.75 [tex]\times[/tex] 1/2 = 0.375
0.375 [tex]\times[/tex] 1/2 = 0.1875
...
This pattern does not match the given sequence.
Each term is 2 times the previous term:
Starting with 6, if we multiply each term by 2, we get:
6 [tex]\times[/tex] 2 = 12
12 [tex]\times[/tex] 2 = 24
24 [tex]\times[/tex]2 = 48
48 [tex]\times[/tex]2 = 96
96 [tex]\times[/tex]2 = 192
This pattern perfectly matches the given sequence. Each term is indeed 2 times the previous term, resulting in the next term.
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What is the name of the Platonic solid below
The name of the Platonic solid that resembles a cuboid is the hexahedron, or more commonly known as a cube.
The correct answer is option C.
The name of the Platonic solid that resembles a cuboid is the hexahedron, also known as a cube. The hexahedron is one of the five Platonic solids, which are regular, convex polyhedra with identical faces, angles, and edge lengths. The hexahedron is characterized by its six square faces, twelve edges, and eight vertices.
The term "cuboid" is often used in general geometry to describe a rectangular prism with six rectangular faces. However, in the context of Platonic solids, the specific name for the solid resembling a cuboid is the hexahedron.
The hexahedron is a highly symmetrical three-dimensional shape. All of its faces are congruent squares, and each vertex is formed by three edges meeting at right angles. The hexahedron exhibits symmetry under several transformations, including rotations and reflections.
Its regularity and symmetry make the hexahedron an important geometric shape in mathematics and design. It has numerous applications in architecture, engineering, and computer graphics. The cube, as a special case of the hexahedron, is particularly well-known and widely used in everyday life, from dice and building blocks to cubic containers and architectural structures.
Therefore, the option which is the correct is C.
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The question probable may be:
What is the name of the Platonic solid which resembles a cuboid?
A. Dodecaheron
B. Tetrahedron
C. Hexahedron
D. Octahedron
Cual es l diferencia entre -4 y 6
Hola!
-4 - 6
= -10
the answer is -10
Use the equation 20x+12y= 24 as an equation in three different linear systems. Write a second equation so that each system has a different number of solutions. Explain what you did for each system.
We have created three different linear systems using the equation 20x + 12y = 24.
System 1 has infinitely many solutions, System 2 has no solution, and System 3 has a unique solution.
Let's create three different linear systems using the equation 20x + 12y = 24 and ensure that each system has a different number of solutions.
System 1:
Equation 1: 20x + 12y = 24 (given)
Equation 2: 40x + 24y = 48
Explanation: In this system, we multiplied both sides of the given equation by 2 to create Equation 2.
By doing so, we have essentially created two equations that are multiples of each other.
Since the equations are equivalent, they represent the same line, and the system has infinitely many solutions.
Any values of x and y that satisfy the first equation will automatically satisfy the second equation as well.
System 2:
Equation 1: 20x + 12y = 24 (given)
Equation 2: 20x + 12y = 48
Explanation: In this system, we changed the constant term in Equation 2 to 48.
By doing so, we have created two parallel lines with the same slope. Since the lines are parallel, they will never intersect, and the system has no solution.
There are no values of x and y that satisfy both equations simultaneously.
System 3:
Equation 1: 20x + 12y = 24 (given)
Equation 2: 40x + 24y = 48
Explanation: In this system, we multiplied both sides of Equation 2 by 2 to create Equation 2.
By doing so, we have created two equations that have the same slope but different y-intercepts.
Since the lines are not parallel and have different y-intercepts, they will intersect at a single point, and the system has a unique solution.
There will be one specific pair of values for x and y that satisfy both equations simultaneously.
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Solve the system of equations.
y=x+5y=x2+5x−7
Enter your answers in the boxes.
Here's the answer for you guys if you need it (:
Answer:
(2, 7) and (-6, -1)
Step-by-step explanation:
y = x + 5
y = x² + 5x − 7
Equatig the above,
x² + 5x − 7 = x + 5
⇒ x² + 4x −12 = 0
⇒ x² + 6x - 2x - 12 = 0
⇒ x(x + 6) - 2(x + 6) = 0
⇒ (x - 2)(x + 6) = 0
⇒ x = 2 or x = -6
Eq(1) : y = x + 5 (given)
When x = 2
y = 2 + 5 = 7
Point : (2, 7)
When x = -6
y = -6 + 5 = -1
Point: (-6, -1)
Algebra Question
68% Oppose year round school
32% Favor year round school
Error +/- 5%
The error given in the graph represents the actual percent could be 5% more or 5% less than the percent reported by the survey.
A. Write and solve an absolute value equation to determine the least and greatest percent of students who could be in favor of year-round school.
B. A classmate claims that ⅓ of the student body is actually in favor of year-round school. Does this conflict with the survey data? Explain.
*can't add graph for some reason
A. To determine the least and greatest percentage of students who could be in favor of year-round school, we can use the error given in the survey, which is +/5%. Let's denote the actual percentage of students in favor of year-round school as x.
The least percentage can be found by subtracting 5% from the reported percentage of 32%:
32% - 5% = 27%
So, the least percentage of students in favor of year-round school is 27%.
The greatest percentage can be found by adding 5% to the reported percentage of 32%:
32% + 5% = 37%
Therefore, the greatest percentage of students in favor of year-round school is 37%.
Hence, the least percentage is 27% and the greatest percentage is 37%.
B. A classmate claiming that ⅓ of the student body is actually in favor of year-round school conflicts with the survey data. According to the survey, the reported percentage in favor of year-round school is 32%, which is not equal to 33.3% (⅓). Therefore, the classmate's claim contradicts the survey results.
It's important to note that the survey provides specific data regarding the percentages of students in favor and opposed to year-round school. The claim of ⅓ being in favor does not align with the survey's findings and should be evaluated separately from the survey data.
Solve the problem. Use what you learned from the example.
Use the information
in the tree diagram.
Write a statement that
is always true about
obtuse triangles. Write
a statement that is
sometimes true about
obtuse triangles.
Show your work. Use pictures and words to explain.
Acute
Equilateral
Triangles
Right
Isosceles
Obtuse
Scalene
C
Statement that is always true about obtuse triangles:
An obtuse triangle always has one angle that measures more than 90 degrees.
In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.
An obtuse angle is an angle that measures more than 90 degrees. Since an obtuse triangle is defined as having one obtuse angle, it will always have an angle that measures more than 90 degrees.
Therefore, the statement that an obtuse triangle always has one angle that measures more than 90 degrees is always true.
Statement that is sometimes true about obtuse triangles:
An obtuse triangle can have different side lengths.
In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.
The "Scalene" category represents triangles with different side lengths. Therefore, it is possible for an obtuse triangle to have different side lengths, making the statement "An obtuse triangle can have different side lengths" sometimes true.
However, it is also possible for an obtuse triangle to have two or more sides with the same length, which would make it an isosceles or equilateral triangle.
Hence, the statement is only sometimes true and not always true.
In summary, an always true statement about obtuse triangles is that they always have one angle that measures more than 90 degrees.
A sometimes true statement about obtuse triangles is that they can have different side lengths.
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Charimaya is running a race around a square track of length 75 m. Find the distance covered by her at the end of her fifth round.
At the end of her fifth round, Charimaya would have covered a distance of 1500 meters.
To find the distance covered by Charimaya at the end of her fifth round, we need to calculate the total distance covered in one round and then multiply it by five.
Given that the track is square-shaped with a length of 75 m, we know that all four sides of the track are equal in length.
To calculate the distance covered in one round, we need to find the perimeter of the square track. Since all sides are equal, we can simply multiply the length of one side by 4.
The length of one side of the square track is 75 m. Therefore, the perimeter of the track is:
Perimeter = 4 × 75 m = 300 m
So, Charimaya covers a distance of 300 m in one round.
To find the distance covered at the end of her fifth round, we multiply the distance covered in one round by 5:
Distance covered in 5 rounds = 300 m × 5 = 1500 m
Therefore, at the end of her fifth round, Charimaya would have covered a distance of 1500 meters.
It's worth noting that since the track is square-shaped, each round consists of running along all four sides of the track.
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A water slide is a straight ramp 20 m long that starts from the top of a tower 18 m high. Find the angle the slide forms with the tower. Approximate to the nearest degree.
The angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).
To find the angle the slide forms with the tower, we can use trigonometric ratios. Let's consider the right triangle formed by the height of the tower (18 m), the length of the slide (20 m), and the angle we want to find.
Using the tangent function, we have:
tan(angle) = opposite/adjacent
In this case, the opposite side is the height of the tower (18 m) and the adjacent side is the length of the slide (20 m). Therefore:
tan(angle) = 18/20
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(18/20)
Using a calculator, we find that arctan(18/20) is approximately 40.56 degrees.
Therefore, the angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).
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What is the sum of the series?
∑k=14(2k2−4)
Enter your answer in the box.
Answer:
44
Step-by-step explanation:
The sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.
The series is: [tex]\sum_{k=1}^4[/tex] (2k²−4)
Let's find the value of each term for k=1, k=2, k=3, and k=4, and then add them up:
For k=1:
2(1)² - 4 = 2(1) - 4 = 2 - 4 = -2
For k=2:
2(2)² - 4 = 2(4) - 4 = 8 - 4 = 4
For k=3:
2(3)² - 4 = 2(9) - 4 = 18 - 4 = 14
For k=4:
2(4)² - 4 = 2(16) - 4 = 32 - 4 = 28
Now, let's add all the terms:
-2 + 4 + 14 + 28 = 44
So, the sum of the series [tex]\sum_{k=1}^4[/tex] (2k²−4) is 44.
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