An ellipse is defined as the set of all points such that the sum of the distances between the point and the two foci is constant. The equation of an ellipse can be represented in the standard form as:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) is the center of the ellipse, a is the distance from the center to a vertex, and b is the distance from the center to the focus.
Given that the center of the ellipse is at (2, 1), one vertex is at (2, -4), and one focus is at (2, -2), we can use this information to find the values of a and b, and represent the equation of the ellipse:
a = distance from center to vertex = |1 - (-4)|/2 = 2.5
b = distance from center to focus = |1 - (-2)|/2 = 1.5
The equation of the ellipse is:
(x-2)^2/2.5^2 + (y-1)^2/1.5^2 = 1
which can also be written as:
(x-2)^2/6.25 + (y-1)^2/2.25 = 1
This equation represents the ellipse with center at (2, 1), one vertex at (2, -4), and one focus at (2, -2)
What is the shape of the bases for the following polyhedron?
a. triangle
b. square
c.rectangle
d.circle
The shape of the bases for the following polyhedron depend on the type of polyhedron. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
The shape of the bases for a polyhedron depend on the type of polyhedron. Polyhedra can have a variety of shapes, such as triangular, square, rectangular, and circular. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
The shape of the bases for the following polyhedron depend on the type of polyhedron. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
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When an employer asks for "how many years of basic math experience do you have" on an application do you include school years? If so how many?
Answer:
Step-by-step explanation:
When an employer asks for "how many years of basic math experience do you have" on an application, it is generally understood that they are asking for the amount of time you have spent working with basic math concepts in a professional setting. This would typically include time spent working in roles that required the use of basic math skills, such as data entry, accounting, or similar positions.
However, if you have no professional experience in a math-related field but you have studied math in school, you can mention that you have a certain number of years of education in math and that you are familiar with basic math concepts.
It's up to you how many years of math classes you want to include, but it's not necessary to include all the years you've studied math since primary school, it's better to just mention that you have a certain level of education in math, or give an estimate of the number of years that you have studied it in a higher level education like college or university.
It's important to be honest and transparent in your application, but also to highlight the skills and experience that make you a good fit for the role.
Which expression can be used to find 6.3 x 4.2
Expression that can be used to find -6.3 x -4.2 are
(1): -4.7 - 2.3x + 0.5 - 4x
(2):-5.3x-4.2-x
(3):-3.4-6.3x-0.8
What is expression?Expression can be defined as a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction and multiplication
By Collecting like terms and adding the answer I get -6.3x - 4.2
(1):-4.7-2.3x+0.5-4x
Collect like terms
-2.3x-4x-4.7+0.5
-6.3x-4.2
(2):-5.3x-4.2-x
Collect like terms
-5.3x-x-4.2
-6.3x-4.2
(3): -3.4-6.3x-0.8
Collect like terms
-6.3x-3.4-0.8
-6.3x-4.2
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The option are:
-4.7 - 2.3x + 0.5 - 4x
-5.3x - 4.2 - x
-3.4 - 6.3x - 0.8
-6.2x - 4.2
-4.2x - 6.3
-3.7 - 5.3x + 0.5
multiple choice
On Rose's shelf are: 6 mystery books • 5 science books • 4 history books • 3 adventure books Rose will randomly choose 2 books to read one a time, without replacement. What is the probability that she will choose a science book first and an adventure book second?
Answer:
Step-by-step explanation:
The probability that ŕose will choose a science book first is 5/8 and the probability of the adventure book is 3/8
find the slope of the line passing through points -1,5 and -5,7
Answer:
-1/2
Step-by-step explanation:
use slope formula
it'd be:
\frac{5-7}{-1+5} =\frac{-2}{4} =\frac{x}{y} -1/2\\
cab fare in new york city is $7.50 for for the first mile, +$3 for each additional mile. what is the longest distance you can travel for $30
Answer:7
Step-by-step explanation:
from the smith chart, find the normalized input admittances corresponding to the following normalized input impedances:
The normalized input admittances are the inverse of the normalized input impedances and can be found on the Smith Chart.
The Smith Chart is a graphical tool used to quickly calculate the reflection coefficient, impedance, admittance, and other characteristics of a transmission line. To find the normalized input admittances corresponding to the normalized input impedances, we first need to find the normalized reflection coefficient for the given normalized input impedances. This can be done by locating the given normalized impedance on the Smith Chart and then determining the normalized reflection coefficient from the radial and angular coordinates. Once the normalized reflection coefficient has been determined, the normalized input admittance can be found by using the formula Y = 1/Z. The normalized input admittance is then located on the Smith Chart in the same manner as the normalized input impedance, using the radial and angular coordinates.
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(07.01 mc) a coolant is injected into engine fluid, causing the temperature of the engine fluid to decrease with respect to time at a rate that is proportional to the difference of the fluid's instantaneous temperature, t, and the fluid's original temperature, to. select the differential equation that represents the relationship.
The differential equation that represents the relationship is
dT/dt = - k * ( T - T0 ) from the condtion - dT/dt ∝ ( T - T0 ).
Given:
The engine fluid temperature rate of change, dT/dt
- dT/dt ∝ ( T - T0 ) [proportional to the difference of the fluid's instantaneous temperature, T, and the fluid's original temperature, T0.]
The Differential Equation that represents the relationship.
The framework is already set, there is only one thing that changes the proportionality to an equation. And that is a proportionality constant.
Let's call it 'k'
Since this is a directly proportional set up, the constant 'k' will be multiplied to the term of ( T - T0 )
- dT/dt = k * ( T - T0 )
And to make this look like a typical equation, multiply both sides by a -1 to get:
∴ dT/dt = - k * ( T - T0 )
As expected, this comes to represent the same thermodynamic effect you'd see in Newton's Law of Cooling; it is expected because most of all fluids exhibit this cooling effect when interacting with other fluids of differing temperatures.
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example 3
include critical numbers, sign chart , intervals of increasing / decreasing
show work and upload pics of work
include critical numbers, sign chart , intervals of increasing / decreasing
SHOW YOUR WORK
The critical points, sign chart and intervals of increase and decrease is plotted for the function f(x) = x³/4 - 3x.
Critical points - (−2,4) and (2,−4)
Interval of increase - (−∞,−2)∪(2,∞)
Interval of decrease - (-2,2)
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
A continuous function f with x in its domain has a critical point at that point xx if it satisfies either of the following conditions -
f'(x) = 0,
f'(x) is undefined
The given function is f(x) = x³/4 - 3x
So, f'(x) = 3x²/4 - 3
In the graph of the function, only two points are there on the graph where f'(x) = 0, which is -
(x,f(x))=(−2,4)
(x,f(x))=(2,−4)
A sign chart tells you when the value of a function f(x) is negative or positive, which is the same as when the graph of f(x) is below or above the x-axis, respectively.
Draw a solid line when the y-value of the function is greater than zero, which is when f(x)>0.
Draw a dashed line when the y-value of the function is less than zero, which is when f(x)<0.
In the graph it can be seen that the graph is below zero up to x = -3.464 and between x = 0 and x = 3.464. In these intervals the sign chart is dashed. The graph is above the x-axis between x = -3.464 and x = 0 and when x>3.464. In these areas the sign chart is solid.
In the graph it can be clearly seen that the graph is increasing from point -∞ up to -2 and from 2 to ∞.
Also in the graph it can be clearly seen that the graph is decreasing from point -2 up to 2 .
Therefore, the critical points, sign chart and intervals of increase and decrease is found.
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Karin wants to use the distributive property to mentally find the value of
19⋅42+19⋅58
.
Which expression can she use?
Answer:
19(42 + 58)
Step-by-step explanation:
19(100)
1900
Cual es la propiedad de 6÷1=6
Please help (Geometry)
PICTURE INCLUDED
Answer: [tex]\triangle TUS \cong \triangle QRP[/tex]
Step-by-step explanation:
When naming congruent triangles, corresponding vertices must have the same positions in the names of the triangles.
Given that alpha and beta are acute. Sin alpha = 1/√10 and Sin beta = 1/√5, prove without using calculator and table that Sin (alpha + beta) = 1/√2
If ∠A and ∠B are acute angles such that cos A = cos B we have proved that ∠A = ∠B since AC = BC and angles opposite to equal sides of a triangle are equal.
What are the definitions of two acute angles?Image outcome
An acute angle has a degree that is less than 90 degrees, i.e. less than a right angle. Acute angle degrees include 12°, 35°, 48°, 65°, 80°, and 89°. An acute angle triangle (also known as an acute-angled triangle) is a triangle with all of its internal angles being acute angles. Remember that an acute angle is one that is less than 90°. Because all three angles are less than 90°, ABC is an acute angle triangle or acute-angled triangle.
The sine function for the angle yields the ratio of the length of the opposing side to the length of the hypotenuse.
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Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm^3. She painted the entire surface of the cylinder purple. How many square centimeters of the cylinder did Mariah paint in terms of π?
Enter the correct answer in the box in terms of π.
The square centimeters of the cylinder did Mariah paint in terms of π will be 24π cm².
What is the surface area of a right circular cylinder?Let r be the radius and h be the height of the cylinder. Then the surface area of the cylinder will be given as,
SA = 2πrh square units
Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm³. The height of the cylinder is calculated as,
V = πr²h
72π = π x 6² x h
72 / 36 = h
h = 2 cm
Then the surface area of the cylinder is given as,
SA = 2 x π x 6 x 2
SA = 24π square cm
The square centimeters of the cylinder did Mariah paint in terms of π will be 24π cm².
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SEE IMAGE!! PLS HELP ASAPP
Find the quotient. If possible, rename the quotient as a mixed number. Select the answer that is in simplest form.
2 3
4 ÷ 3 = ?
A.
7
12
B.
11
12
C.
3
4
Answer:11/12
Step-by-step explanation:
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The three true statements are:
The standard form of the equation is (x – 1)² + y² = 3.
The center of the circle lies on the x-axis.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
What is the center of the circle?
A circle is named by the point in the center. A radius is a line segment from the center of the circle to the edge. A diameter is a line segment that passes through the center of a circle.
The true statements are:
The standard form of the equation is (x – 1)² + y² = 3.
The center of the circle lies on the x-axis.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
Given an equation of x2 + y2 – 2x – 8 = 0, we can make the following observations:
We can complete the square to get the equation in standard form. (x – 1)² + y² = 3². This is true statement.
The center of the circle is (1,0) which is on the x-axis, this statement is true.
The radius is 3 units, which is the same as the radius of the circle x² + y² = 9 (3²). This statement is true.
The center of the circle does not lie on the y-axis as its x-coordinate is not 0.
Hence, The three true statements are:
The standard form of the equation is (x – 1)² + y² = 3.
The center of the circle lies on the x-axis.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
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5 1/8 equals 4 + 1 + 1/8 = 4 + blank 8 + 1/8 =
The blanks in the equation are filled as below
5 1/8 = 4 + 1 + 1/8 = 4 + 1/8 8 + 1/8 = 5 1/8
How to fill the blanks in the equationTo fill the blanks the equation is written completely in figures
5 1/8 = 4 + 1 + 1/8 = 4 + 8 + 1/8 =
From the knowledge of equality to maintains a true equation the last blank should be equal to 5 1/8.
With this information, we use addition operation to solve for the first blank as follows.
let the first blank be x
4 + 8 + 1/8 = 5 1/8
4 + x * 8 + 1/8 = 5 1/8
4 + 8x + 1/8 = 5 1/8
collection like terms
8x = 5 1/8 - 4 - 1/8
8x = 1
divide both sides by 8
x = 1/8
the blanks are 1/8 and 5 1/8
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ABC is straight angle.BD is drawn so that two adjacent angles are created. m ABC = 22x - 9 and m CBD = 4x + 7. does BD bisector ABC?show your work.
Yes, BD is the bisector of ∠ABC.
What is geometry?Geometry is a branch of mathematics that deals with shapes, sizes, angles, and dimensions of objects.
Given is that ABC is straight angle. BD is drawn so that two adjacent angles are created. It is given that m ∠ABD = 22x - 9 and m∠ CBD = 4x + 7.
For BD to be the bisector of ∠ABC, ∠ABD and m∠ CBD should be equal. So we can write -
22x - 9 = 4x + 7
22x - 4x = 16
18x = 16
x = 8/9
Now, we can write -
m ∠ABD = 22x - 9 = 22 x 8/9 - 9 = 10.55°
m ∠ABD = 10.55° ...... (1)
m ∠CBD = 4x + 7 = 4 x 8/9 + 7 = 10.55°
m ∠CBD = 10.55° ...... (2)
m ∠ABD = m ∠CBD
Yes, BD is the bisector of ∠ABC.
Therefore, yes, BD is the bisector of ∠ABC.
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8th grade geometry
First box options are is, is not. And the second box options are will, will not.
i. The tangent ratio for ΔRBK is 16/ 38.4
ii. The tangent ratio for ΔHMK is 8/ 19.2
What is a tangent of an angle?Trigonometric functions are set of functions that are applicable when the side or angle of a triangle is to be determined. Some basic trigonometric functions are: Sine, Cosine, Tangent etc.
The tangent of an angle relates its opposite side to the reference angle with the adjacent side to the same angle.
So that;
Tan θ = opposite/ adjacent
In the given question,
1. From ΔRBK, the tangent ratio is given as;
tan θ = opposite/ adjacent
= RB/ BK
but BK = BM + MK
= 19.2 + 19.2
= 38.4
So that;
tan θ = 16/ 38.4
2. From ΔRBK, the tangent ratio is given as;
tan θ = opposite/ adjacent
= HM/ MK
tan θ = 8/ 19.2
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is 3x-2y=-5 & 3x-2y=8 parallel perpendicular intersecting but not perpendicular
If you solve 3x - 2y = -5 for slope-intercept form, you'd have:
y = 3/2 x + 5/2
If you solve 3x - 2y = 8 for slope-intercept form, you'd have:
y = 3/2 x - 4
Since the slopes are the same, but the y-intercepts are different, these lines are parallel.
79=2x+3y solve for y
The results of the x-intercept and y-intercept results from above (39.5, 0) and (x1, y1) and (x2, y2) (0,26.333)
How to solve y?Here, we'll explain how to solve arithmetic issues with the formula 2x + 3y = 79 and offer answers.We'll start by figuring out the x-intercept and y-intercept of the equation 2x + 3y = 79 and then showing you the result.Then, we'll demonstrate how to obtain the coordinates for the graph plot for the equation 2x + 3y = 79 so that we may depict it on a graph.The slope of 2x + 3y = 79 will then be calculated and shown to you once we have solved 2x + 3y = 79 for both x and y.locate x-interceptWhere the graph veers off the x-axis is known as the x-intercept. We first set y1=0 and then solve for x to determine the x-intercept.2x + 3y = 79
2x + 3(0) = 79
x1 = 39.5 y1 = 0
the y-intercept
Where the graph crosses the y-axis is known as the y-intercept. We first solve for y before setting x2=0 to determine the y-intercept.
2x + 3y = 79
2(0) + 3y = 79
y2 = 26.333
x2 = 0
To draw a straight line on a graph, you must get two graph points. The format for the plot coordinates is (x1,y1) and (x2,y2).
Thus, we can obtain the graph plots for 2x + 3y = 79 as follows using the results of the x-intercept and y-intercept results from above:
(39.5, 0) and (x1, y1) and (x2, y2) (0,26.333)
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Express the area of the shaded region in the room shown algebraically
The weight gains of beef steers were measured over a 140 -day test period. The average daily gains (lb/day) of 9 steers on the same diet were as follows: 23
3.89 3.51 3.97 3.31 3.21 3.36 3.67 3.24 3.27
Determine the mean and median
The weight gains of beef steers were measured over a 140 -day test period .then mean is 3.545 and median is 3.435
Given
3.89 3.51 3.97 3.31 3.21 3.36 3.67 3.24 3.27
mean = 3.89+3.51+3.97+3.31+3.21+3.36+3.67+3.24+3.27 /10
= 35.45/10
= 3.545
average is the sum of all terms present in the question divided by the number of terms that is the way of calculating average
Solving (a): The mean = 3.545
This is calculated as:
median = 10+1 /2 term
= 11/2
=5.5 term
So, we have:
median = 3.435
The weight gains of beef steers were measured over a 140 -day test period .then mean is 3.545 and median is 3.435
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Let X be a number between 0 and 1 produced by a random number generator. Assuming that the random variable X has a uniform distribution, find the following probability: P(X > 0.49)
The probability of a random variable X with a uniform distribution being greater than 0.49 can be calculated using the formula P(X > 0.49) = 1 - P(X ≤ 0.49).
Since X has a uniform distribution, it means that the probability of any number between 0 and 1 occurring is the same. Therefore, P(X ≤ 0.49) = 0.49.
The probability of X being greater than 0.49 can be calculated using the formula for uniform distributions, which is P(X>a) = 1 - P(X≤a). In this case, P(X>0.49) = 1 - P(X≤0.49). The probability of X being less than or equal to 0.49 can be calculated by multiplying the probability of each individual outcome (since they are all equally likely) by the number of outcomes that satisfy the condition .Substituting this into the formula to calculate P(X > 0.49), we get P(X > 0.49) = 1 - 0.49 = 0.51. Therefore, the probability of a random number generated being greater than 0.49 is 0.51.
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Question 2 (1 point) The results of a survey of 70 Programming students on learning preferences were as follows: 50 liked to learn visually, 45 liked learning through listening and 35 liked learning through videos. 5 liked using all three channels, 32 liked to learn visually and through listening, 22 liked to learn both visually and videos, 14 liked to learn through listening and videos. How many preferred learning only through videos? options-(9, 17 ,13, 4)
9 students preferred learning only through videos.
What is inclusion-exclusion from the data provided?Inclusion-exclusion is a principle used in counting to find the total number of elements in a union of sets by subtracting the elements that are counted multiple times and adding back in the elements that are not counted at all.
The number of students who preferred learning only through videos is 17.
This can be found by using the principle of inclusion-exclusion:
35 students liked to learn through videos
14 students liked to learn through listening and videos
5 students liked all three channels
So, 35 - 14 + 5 = 26 students liked to learn through videos and listening
Therefore, 35 - 26 = 9 students preferred learning only through videos.
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f(x) is a direct variation function f(3)= 12 f(x)= ? f(2)= ?
The function f(x) will be : f(x) = 4x and f(2) = 8.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that f(x) is a direct variation function .
Direct variation of {y} with {x} means that as {x} increases, {y} increases uniformly with it. Mathematically -{K} = y/x = constant
Scale factor {K} is a dimensionless value that indicates the constant ratio value indicating direct variation.It is given that -
f(3) = 12
We can write the slope of f(x) as -
m = (12 - 0)/(3 - 0)
m = 12/3
m = 4
So, f(x) will be -
f(x) = 4x
and
f(2) will be -
f(2) = 8
Therefore, the function f(x) will be : f(x) = 4x and f(2) = 8.
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Please help me please
Hong rode the scooter for 61 minutes.
What is the algebra?
Algebra is a branch of mathematics that deals with mathematical equations and their properties. It is a way to represent and solve problems using mathematical symbols, letters and numbers.
We can use algebra to solve for the number of minutes Hong rode the scooter. Let x be the number of minutes Hong rode the scooter.
We know that the total bill is the sum of the start fee and the per-minute fee.
So, we can set up the equation as:
Total bill = start fee + per-minute fee
In this case:
10.88 = 6 + (x * 0.08)
We can then solve for x by isolating it on one side of the equation:
10.88 = 6 + (x * 0.08)
10.88 - 6 = x * 0.08
4.88 = x * 0.08
x = 4.88 / 0.08
x = 61
So, Hong rode the scooter for 61 minutes.
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From his eye, which stands 1.55 meters above the ground, Tyler measures the angle of elevation to the top of a prominent skyscraper to be 57^{\circ} ∘ . If he is standing at a horizontal distance of 112 meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.
Answer: ≈ 174.01
Thus, The Sky Scraper ≈ 174.01
Step-by-step explanation:
From his eye, which stands 1.55 meters above the ground, Tyler measures the angle of elevation to the top of a prominent skyscraper to be 57∘
∘. If he is standing at a horizontal distance of 112 meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.
1 - Draw Diagrams, Label Knowns:
Unknowns: X
2 - Identify: OPPOSITE, ADJACENT, HYPOTENUSE
3 - WHAT FUNCTION USES THE OPPOSITE AND THE ADJACENT?
SOH - CAH - TOA
NOW SHOWING WORK:
tan 57 = opposite/adjacent = x/112
tan 57 = x/112
tan 57/1 = x/112
112 tan 57 = x ( Cross Multiply)
x = 172.464875 (Now, type into the calculator)
So, Now we ADD:
1.55, the distance from the ground to his eye.
172.464875 + 1.55 = 174.014875
≈ 174.01 (Round to The Nearest HUNDREDTH)
Thus, the Sky Scraper is ≈ 174.01 meters tall.
So, your answer would be :
Sky Scraper ≈ 174.01
Hope I have explained it and helped you in your studies!
The height of the skyscraper is given by H = 174.01 m
What are trigonometric relations?Trigonometry is the study of the relationships between the angles and the lengths of the sides of triangles
The six trigonometric functions are sin , cos , tan , cosec , sec and cot
Let the angle be θ , such that
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
tan θ = sin θ / cos θ
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Given data ,
We can start by drawing a right triangle, where the horizontal distance from Tyler to the base of the skyscraper is one leg, the height of the skyscraper is the other leg, and the line of sight from Tyler's eye to the top of the skyscraper is the hypotenuse.
Then we can use trigonometry to solve for the height of the skyscraper.
Let the height above the ground from his eye = 1.55 m
Now , the angle of elevation θ = 57°
The horizontal distance from the base of the skyscraper = 112 m
Now , from the trigonometric relations ,
tan θ = opposite / adjacent
So , tan 57° = x / 112
Multiply by 112 on both sides , we get
x = 172.46 m
Now , the total height of the skyscraper = 172.46m + 1.55 m
So , height H = 174.01 m
Hence , the skyscraper has a height of 174.01 m
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A dinner was held to raise money for a children's museum. A ticket for one person cost $200 and a ticket for a couple (two people) cost $350. A total of 130 people attended the dinner, and the ticket sales total was $24,000. What is the total number of tickets that were sold?
There were 50 single tickets sold and 40 couples tickets were sold.
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that is equal.
Let's call the number of single tickets sold "x". Then the number of couple tickets sold would be "y".
We know the total number of attendees is 130, so we can write the equation:
x + 2y = 130
We also know that the total revenue from the ticket sales is $24,000, so we can write another equation using the cost of the tickets:
200x + 350y = 24,000
We now have a system of two equations that we can solve for x and y then we get the values are:
x = 50 and y = 40
So, 50 single tickets were sold and 40 couples tickets were sold.
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Let f and g be the functions given by f (x) = e^x and g (x) = x^4. On what intervals is the rate of change of f (x) greater than the rate of change of g (x)? (A) (0.831, 7.384) only(B) (0.831) and (&.384)(C) (0.816) and (1.430, 8.613)(D) (0.816, 1.430) and (8.613)
The rate of change of a function at a point is given by its derivative. To compare the rate of change of f(x) and g(x) at different points, we need to find the derivative of both functions and compare them.as explained below :
The derivative of f(x) = e^x is f'(x) = e^x
The derivative of g(x) = x^4 is g'(x) = 4x^3
Now we need to compare the two derivatives to find on which intervals the rate of change of f(x) is greater than the rate of change of g(x). To do this, we will find the point of intersection between the two functions and compare them.
e^x = 4x^3
Taking the natural logarithm of both sides we get:
x = ln(4x^3) / ln(e)
Approximating we get x ≈ 0.816 and x ≈ 1.430 and x ≈ 8.613
For x < 0.816, e^x < 4x^3, so the rate of change of f(x) is smaller than the rate of change of g(x)
For x > 8.613, e^x > 4x^3, so the rate of change of f(x) is greater than the rate of change of g(x)
Between 0.816 < x < 1.430 and 1.430 < x < 8.613, e^x = 4x^3, so the rate of change of f(x) is equal to the rate of change of g(x).
So the answer is (D) (0.816, 1.430) and (8.613)
Note that the answer (B) (0.831) and (&.384) is not possible as the point of intersection is a number and not an interval and the answer (A) (0.831, 7.384) is not correct as the point of intersection is 1.430 and 8.613 not 0.831 and 7.384
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