Find the length of the arc shown in red, Leave your answer in terms of pi.
Answer:
The red arc has length (3/2) π
Step-by-step explanation:
* Lets revise some rules in the circle
- The measure of any arc = the measure of the central angle
subtended by this arc
- Equal central angles subtended by equal arcs
- The length of the arc is a part from the length of the circle,
it depends on the ratio between the measure of this arc
and the measure of the circle
- The measure of the circle = 360°
- The length of the circle = 2πr
* Lets solve our problem
∵ The measure of the arc = 30°
∴ The measure of its central angle 30°
∴ The central angle of the red arc = 30° by condition of
vertically opposite angles are equal in measures
∴ The measure of the red arc = 30°
- The ratio between measure the arc and the measure
of the circle is 30/360 = 1/12
∴ The arc is 1/12 from the circle
∴ The length of the arc is 1/12 from the length of the circle
∴ Length red arc = (1/12) × 2πr
∵ r = 9
∴ Length red arc = (1/12) × 2 × π × 9 = (3/2) π
* The red arc has length (3/2) π
QM Q9.) Write the standard form of the equation of the circle with the given center and radius.
You have the opportunity to lease space for your business with a fixed-rate lease. The property owner has proposed a five-year lease with a rent of $4,000 per month. How much is the rent per year? How much is the rent over the life of the lease?
Find the 15th term of the arithmetic sequence.
a+1, 2a+1, 3a+1
a. a+15
b. 15a+15
c. 15a + 1
d. 14a+14
What is the mean of 20, 37, 43, 54, 83, 102, 117, 160, 208?
Alex walked 1 mile in 15 minutes. Sally walked 3,520 yards in 24 minutes. In miles per hour, how much faster did Sally walk than Alex? (Note: 1 mile = 1,760 yards)
Dustin is stuck at the top of a ferris wheel. his mother is standing 38 feet from the base of the wheel watching him. if the angle of elevation from dustin's mom to Dustin is 73 degrees, how far off the ground is nick?
A. 118.2 ft
B. 120.9 ft
C. 124.3 ft
D. 126.5 ft
E. 128.1 ft
The correct option is A. [tex]118.2\ ft[/tex]. Dustin is approximately [tex]118.2\ ft[/tex] off the ground.
To find how far Dustin is off the ground, we can use trigonometry, specifically the tangent function.
Let [tex]\( h \)[/tex] denote the height of Dustin above the ground.
Given:
Angle of elevation [tex]\( \theta = 73^\circ \)[/tex]
Distance from Dustin's mother to the base of the ferris wheel [tex]\( d = 38 \)[/tex]feet
We can set up the tangent function:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \][/tex]
Substitute the given values:
[tex]\[ \tan(73^\circ) = \frac{h}{38} \][/tex]
Now, solve for [tex]\( h \)[/tex]
[tex]\[ h = 38 \times \tan(73^\circ) \][/tex]
Use a calculator to find [tex]\( \tan(73^\circ) \)[/tex]
[tex]\[ \tan(73^\circ) = 3.0985 \][/tex]
Therefore,
[tex]\[ h = 38 \times 3.0985 \][/tex]
[tex]\[ h = 117.839 \][/tex]
Rounding to the nearest tenth, Dustin is approximately [tex]\( 117.8 \) feet[/tex] off the ground.
I need help on this one please
F(x, y, z) = yz i + xz j + (xy + 18z) k c is the line segment from (1, 0, −1) to (5, 6, 1) (a) find a function f such that f = ∇f. f(x, y, z) = (b) use part (a) to evaluate c ∇f · dr along the given curve
c.
Through anti-derivatives, we find a function f(x, y, z) in part (a) of the problem. We then use the fundamental theorem of line integrals in part (b) to evaluate the line integral of ∇f ∙ dr along a curve c, using the function f we found.
Explanation:This problem lies within the field of vector calculus, specifically dealing with gradient fields and line integrals. Part (a) requires us to find a function f(x, y, z) such that the gradient of f (∇f) is equal to the field F. Since ∇f yields the vector field (Fx, Fy, Fz), we find the anti-derivatives of each component of F to solve for f. For this particular function f, the anti-derivatives are 1/2 * yz^2 + C1 for Fx = yz, 1/2 * xz^2 + C2 for Fy = xz, and 1/2 * xy^2 + 9z^2 + C3 for Fz = xy + 18z. Therefore, the function that satisfies the requirement is 1/2 * yz^2 + 1/2 * xz^2 + 1/2 * xy^2 + 9z^2 + C, where C is a constant.
Part (b) then requires the evaluation of the line integral of ∇f ∙ dr along a curve c, from point (1,0,-1) to (5,6,1). According to the fundamental theorem of line integrals, if a vector field is conservative (as it is in this case because we found a function f such that F = ∇f), the line integral of a vector field F along a curve c from a to b is simply f(b) - f(a). Therefore, using the function f found in part a, you can substitute the coordinates of points a and b to find the solution to the line integral equation.
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identify the type of conic section that has the equation 4x^2+25y^2=100 and identify its domain and range.
The equation given represents an ellipse, a type of conic section. The domain of this particular conic section is -5 to 5, and its range is -2 to 2.
Explanation:The equation given represents an ellipse, which is one kind of conic section. The general form of the equation of an ellipse is[tex]x^2/a^2 + y^2/b^2 = 1,[/tex] where a and b are the lengths of the semi major and minor axes respectively. The equation you provided, [tex]4x^2+25y^2=100,[/tex]when arranged to fit into the general form of ellipse becomes [tex]x^2/25 + y^2/4 =1.[/tex] Thus, it is clear that the given equation represents an ellipse.
The domain of the function described by this elliptical equation includes all the x-values that the ellipse covers, and for this particular equation, is -5 to 5. The range includes all the y-values that the ellipse covers, which in this case is -2 to 2.
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Q #14 Solve the equation
What are some uses for the distance formula? Finding the perimeter of polygons. Finding the area of rectangles. Finding the equation of a circle. Finding the midpoint of segments. Finding how much gas you will need on a trip.
The Distance Formula is used widely in mathematics to calculate the distance between two points, which includes finding the perimeter of polygons, the equation of a circle and the midpoint of segments.
Explanation:The Distance Formula is a valuable tool in mathematics that has a wide range of practical uses and applications. It is primarily used to calculate the distance between two specific points on a coordinate plane. Some uses include the following:
Finding the perimeter of polygons: Distance Formula can be used to calculate the length of each side of the polygon, and then by summing these lengths, we get the perimeter. Finding the equation of a circle: By using Distance Formula, we can establish the radius of the circle - the distance from the center of the circle to any point on the circle. Finding the midpoint of segments: Distance Formula helps to identify the exact middle point between two defined points.
However, usage of the Distance Formula to determine the amount of gas needed for a trip would be incorrect as it requires additional factors like the fuel efficiency of your vehicle and the nature of your trip.
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PLZ HELP ASAP INSCRIBED SHAPES
For the particular problem raised in the introduction, assume that the total bill is $44. to answer the question "how should the bill be split?" we will create a linear equation. the unknown is how much money a single person (besides anika) must pay, so call that x. although four people (you plus three friends) went to dinner, only three are paying the unknown amount x for a total of 3x. since anika is paying $2, the total amount paid is 3x+2 dollars, which must equal the amount of the bill, $44. thus, the equation to find x is 3x+2=44. the steps for solving a linear equation are as follows: move all of the constants to the right side. move all of the variable terms (terms containing x) to the left side. divide both sides by the coefficient of the variable to isolate the variable. you will go through these steps one at a time to solve the equation and determine how much each person should pay.
if AB if the midsegment find x SHOW WORK
The surface inside the circles will be painted green. The surface outside the circles will be painted white. What is the ratio of green paint to white paint you will need to paint these tiles? π : (π – 4) (π – 4): π π2 : (π2 – 4) (π2 – 4): π2
Answer:
π : (π – 4)
Step-by-step explanation:
Consider the floor tile below.
The surface inside the circles will be painted green. The surface outside the circles will be painted white. What is the ratio of green paint to white paint you will need to paint these tiles?
π : (π – 4)
(π – 4): π
π2 : (π2 – 4)
(π2 – 4): π2
Odyssey
Is 7+8x=y proportional, if so what is the constant proportionality?
Is y=-2/5x proportional, if so what is the constant proportionality?
A student conducted a poll of 100 internet users and found the average time spent online per day was 2 hours. The student described 2 hours as a parameter. The student is incorrect because the value is a numerical measurement describing a characteristic of _____.
a ratio
a sample
a population
quantitative data
In statistics A sample is a part of a population .A sample statistic is any quantity we derive from a sample taken from a population . A sample refers to a set of observations drawn from a population.
We are given A student conducted a poll of 100 internet users and found the average time spent online per day was 2 hours .Here 2 hours is an observation derived from the set of observations made .So 2 hours is a characterstic of a sample.
The second option a Sample is the right answer.
Starting at home, ishaan traveled uphill to the grocery store for 121212 minutes at just 151515 mph. he then traveled back home along the same path downhill at a speed of 303030 mph. what is his average speed for the entire trip from home to the grocery store and back?
Answer: The average speed for the entire trip from home to the grocery store and back is 20 mph.
Step-by-step explanation:
Since we have given that
Ishan traveled uphill to the grocery store for 12 minutes at just 15 mph.
So, Time taken by him = 12 minutes
Speed = 15 mph
So, distance traveled = Speed × time
So, Distance becomes [tex]15\times \dfrac{12}{60}=\dfrac{180}{60}=3\ miles[/tex]
Again, he traveled back home along the same path downhill = 180 miles
Speed of covering downhill = 30 mph
so, Time taken [tex]\dfrac{Distance}{Speed}=\dfrac{3}{30}=\dfrac{1}{10}\ hours[/tex]
Since we know the formula for "Average speed ":
[tex]\dfrac{Total\ distance}{Total\ Time}\\\\\\=\dfrac{3+3}{\dfrac{12}{60}+\dfrac{1}{10}}\\\\\\=\dfrac{6}{\dfrac{1}{5}+\dfrac{1}{10}}\\\\\\=\dfrac{6\times 10}{2+1}\\\\\\=\dfrac{6\times 10}{3}\\\\\\=20\ mph[/tex]
Hence, the average speed for the entire trip from home to the grocery store and back is 20 mph.
An airliner travels 45 miles in 5 minutes. What is it’s speed in miles?
A rectangle has a width of 8 centimeters and an area of 160 square centimeters a similar rectangle has an area of 250 square centimeters. What are the dimensions of the larger rectangle?
The dimensions of the lager rectangle are;
length = 25cm
width = 10cm
Similar shapes are shapes with equal corresponding angles and equal side ratio. This side ratio is called scale factor.
scale factor = √ area factor
area factor = area of new shape/ area of old shape
area factor = 250/160
= 25/16
scale factor = √25/√16 = 5/4
Therefore the width of the larger rectangle = 8 × 5/4 = 10cm
the length = 250/10 = 25 cm
Therefore the dimensions of the larger rectangle are;
length = 25cm
width = 10cm
Given the parent function of y=|x|, state the type of transformation that occured to get the function below. y=1/4|x|
the transformation that occurred to the parent function[tex]\( y = |x| \)[/tex] to get the function[tex]\( y = \frac{1}{4}|x| \)[/tex] is a vertical compression by a factor of 4.
The parent function ( y = |x| ) represents the absolute value function, which takes the absolute value of \( x \) and outputs its positive value.
The function [tex]\( y = \frac{1}{4}|x| \)[/tex] is a transformation of the absolute value function. Specifically, it involves the following transformations:
1. **Vertical Compression:**
The coefficient[tex]\( \frac{1}{4} \)[/tex] before[tex]\( |x| \)[/tex] compresses the graph vertically. It shrinks the height of the graph by a factor of 4 compared to the parent function [tex]\( y = |x| \).[/tex]
So, the transformation that occurred to the parent function[tex]\( y = |x| \)[/tex] to get the function[tex]\( y = \frac{1}{4}|x| \)[/tex] is a vertical compression by a factor of 4.
Twelve people are entered in a race. If there are no ties, in how many ways can the first three places come out?
[tex] 12\cdot11\cdot10=1320 [/tex]
The number of ways the first three places in a race with twelve competitors can be filled is 1320, calculated by the permutations formula which multiplies the choice for each of the three places -- 12 for first, 11 for second, and 10 for third.
The student's question asks for the number of different ways the first three places in a race can be filled when there are twelve competitors. This is a problem of permutations where we do not consider the remaining positions after the third place. To solve this, we calculate the number of permutations for the first three places, which is a sequence of choices. We have 12 choices for the first place, 11 choices for the second place after the first place has been filled, and 10 choices for the third place after the first two places have been filled.
The total number of permutations can be calculated as:
First place: 12 possibilitiesSecond place: 11 possibilities (since one person is already in the first place)Third place: 10 possibilities (as two contestants are in the first and second place)By the counting principle, we multiply these choices together to find the total number of permutations for the first three places, which is 12 x 11 x 10.
Therefore, the total number of ways the first three places can be awarded is 12 x 11 x 10 = 1320 different permutations.
0.88 cm equals how many mm
John paid 5% sales tax on his television, which cost $769 before taxes. What did the television cost in total?
Two students, Ann and Max, factored the trinomial 4x2 − 6x − 4. Ann factored it as 2(x − 2)(2x + 1) and Max factored it as (x − 2)(4x + 2). Indicate which student factored the trinomial completely and which student did not, and explain why.
Is 28 cups greater or less than 14 Pints
Help with this one please
The Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality (RMS-AM-GM-HM), is an inequality of the root-mean square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers that says:
with equality if and only if . This inequality can be expanded to the power mean inequality.
As a consequence we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving.
Grace has one $10 bill and three $5 bills she spends 9.30 on a belt and $4.20 on snack how much money does grace have left
Trevor just received a jar of mixed nuts containing 325 almonds, 250 pistachios, 75 cashews, and 350 pecans for his birthday. He's going to start randomly choosing some nuts from the jar, and he's not going to do anything that would favor him choosing any particular type of nut over another. Let's use this information to calculate some probabilities. (4 points: Part I - 1 point; Part II - 1 point; Part III - 1 point; Part IV - 1 point)
What is the probability that the first nut Trevor chooses is a pecan?
Part II: What is the probability that the first nut Trevor chooses is an almond or a cashew?
Final answer:
The probability of choosing a pecan first is 35%, and choosing either an almond or a cashew first is 40%.
Explanation:
Probability Calculation in Combinatorics
The student's question involves calculating the probability of certain outcomes when choosing nuts from a jar. To answer Part I: the probability that the first nut Trevor chooses is a pecan, we'll need to divide the number of pecans (350) by the total number of nuts in the jar (1000). Therefore, the probability of choosing a pecan first is 350/1000 or 0.35 (35%).
For Part II, we calculate the probability of choosing either an almond or a cashew. The total of almonds and cashews is 325 almonds + 75 cashews = 400. So, the probability is 400/1000, which simplifies to 0.4 (40%).