(a) The vertex is (6, 144),(b) When the fireworks are launched at time x = 0 and return to the earth at time x = 12 seconds, respectively, these times are denoted by zeros in the function,.(c) this is the quadratic function that represents the situation f(x) = -4 + 144.
How to deal with this problem?a. To find the zeros and vertex of the quadratic function, we need to first write it in the standard form:
[tex]f(x) = ax^2 + bx + c[/tex]
where a, b, and c are constants.
The vertex of the function is given by:
x = -b/2a
Since the quadratic function is symmetrical around the vertex, we know that the time it takes to reach the maximum height is halfway between the launch time and the time it hits the ground again. So, the time to reach maximum height is (4 + 8)/2 = 6 seconds.
Therefore, we can set up a system of equations using the information given:
f(4) = 0 (the firework is launched from the ground)
f(6) = 256 (the firework reaches its maximum height)
f(12) = 0 (the firework hits the ground again)
Plugging in the values of x and f(x), we get:
16a + 4b + c = 0
36a + 6b + c = 256
144a + 12b + c = 0
Solving this system of equations, we get:
a = -4
b = 48
c = 0
Therefore, the quadratic function that represents the situation is:
[tex]f(x) = -x^2 + 48x[/tex]
The zeros of the function can be found by setting f(x) = 0:
[tex]f(x) = -4x^2 + 48x[/tex]
0 = x(-4x + 48)
x = 0 (the firework is launched from the ground)
x = 12 (the firework hits the ground again)
The vertex can be found using the formula:
x = -b/2a = -48/(-8) = 6
So the vertex is (6, 144).
b. Using the zeros and vertex, we can sketch a graph of f(x):
The quadratic function's graph
Considering the function's primary characteristics in light of the circumstances:
The peak height of the fireworks, which occurs at x = 6 seconds and f(6) = 256 feet, is represented by the function's vertex.
The firework is at ground level at x = 0 (launch time) and x = 12 seconds (when it reaches the ground again), which are represented by zeros in the function.
The graph's form suggests that the firework rises before falling again, which is consistent with the scenario given.
c. The quadratic function that represents the situation is:
[tex]f(x) = -4x^2 + 48x[/tex]
This function can be simplified by factoring out -4:
[tex]f(x) = -4( x^2- 12x)[/tex]
Completing the square:
[tex]f(x) = -4( x^2- 12x + 36 - 36)[/tex]
[tex]f(x) = -4( (x^2-6)- 36)[/tex]
[tex]f(x) = -4(x^2-6) + 144[/tex]
This form of the function shows that the vertex is (6, 144), and that the maximum height is 144 feet.
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Look at this expression.
Which of the following is the simplest form of the expression above?
A.
2a-2b-3
B.
2a-1b2
C.
2a-1/2b-3
D.
2a3b4
The simplest form of the expression "(14ab³)/(7a⁻²b⁻¹)" is represented by the expression (d) 2a³b⁴.
To simplify the expression (14ab³)/(7a⁻²b⁻¹), we first simplify the terms with the same base,
First, we simplify the coefficients "14" and "7" by dividing them to get:
⇒ (14ab³)/(7a⁻²b⁻¹) = 2ab³/a⁻²b⁻¹;
Next, we simplify the variables "a" and "b" by subtracting the exponents of "a" and "b" in the denominator respectively,
We get,
⇒ 2ab³/a⁻²b⁻¹ = 2a¹⁻⁽⁻²⁾b³⁻⁽⁻¹⁾ = 2a¹⁺²b³⁺¹ = 2a³b⁴,
Therefore, the simplest form of the expression (14ab³)/(7a⁻²b⁻¹) is 2a³b⁴, the correct option is (d).
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The given question is incomplete, the complete question is
Look at this expression. (14ab³)/(7a⁻²b⁻¹),
Which of the following is the simplest form of the expression above?
(a) 2a⁻²b⁻³
(b) 2a⁻¹b²
(c) 2[tex]a^{-\frac{1}{2} }[/tex]b⁻³
(d) 2a³b⁴.
Please help with study island!!
The total volume of the wall units is 108 cubic feet
How to find the volume of the wallThe volume of the wall is solved by dividing the figure to give the various dimensions represented in the form length x width x depth
first division has the dimensions as 3 x 4 x 8 second division has the dimensions as 3 x 2 x 2The volume of each is solved then added to get the total volume
first division: 4 x 4 x 8 = 96
second division: 3 x 2 x 2 = 12
The total volume is
= 96 + 12
= 108 cubic feet
so the total volume is solved to be 108 cubic feet
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Consider the integral given below
Answer:
C ∫[-a,0] = -∫[0, a]
Step-by-step explanation:
You want an alternate representation of the given integral ...
∫(2x⁵ -5x³)dx
on the interval [-a, a].
Odd functionThe function f(x) = 2x⁵ -5x³ being integrated is an odd function. This means ...
f(-x) = -f(x) and f(0) = 0
Even functionAn even function is characterized by ...
f(-x) = f(x)
The integral F(x) of this odd function f(x) is an even function, so the parts either side of x=0 have the integrals ...
[tex]\displaystyle \int_{-a}^0{(2x^5-5x^3)}\,dx=F(0)-F(-a)=F(0) -F(a)[/tex]
[tex]\displaystyle \int_0^a{(2x^5-5x^3)}\,dx=F(a)-F(0)[/tex]
As we can see, these integrals are the opposites of each other, matching answer choice C.
__
Additional comment
The second attachment shows a numerical value for a=2.
Determine the rate of change of the function given by the table.
I WIL GIVE BRAINLYEST TO WHOEVER ANSWERS FIRST AND CORECTY
Answer:
Step-by-step explanation:
1
A scientist discovered a rock formation that grows at a rate of 0.01 meters per year. To predict the height, h, of the rock formation after t years, she used the formula h(t)=1.3+0.01t.
What’s the system of equations?
The solution set is {(x, y) | x = y + 2}, which represents a straight line with a slope of 1 passing through the point (2, 0).
Therefore, the correct answer is option 2, infinite solutions.
What is quadratic equation?Equations of the form ax2 + bx + c = 0, where a, b, and c are real numbers, and a 0, are known as polynomial equations in the variable x. Any equation that has the formula p(x) = 0, where p(x) is a polynomial of degree 2 with a single variable, is actually a quadratic equation.
The equation has the following form: ax2 + bx + c = 0, where x is the unknown variable and a, b, and c are constants, with "a" not equal to 0.
Numerous strategies, including factoring, completing the square, and the quadratic formula, can be used to solve quadratic equations.
To solve the system of equations:
x - y = 2 ---(1)
-3x + 3y = -6 ---(2)
We can use the first equation to isolate x in terms of y:
x = y + 2
-3(y + 2) + 3y = -6
Simplifying the equation, we get:
-3y - 6 + 3y = -6
Solving for y, we get:
0 = 0
For all values of y this equation is true-
So,the equations has infinite solutions.
To find the solution set, we can substitute the value of y into the expression for x:
x = y + 2
x = y + 2, for all values of y
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Does anybody know What is
11=4x
Answer:
2.75
Step-by-step explanation:
When you put that equation into a calculator, you get 2.75.
Answer:
Step-by-step explanation:
To solve for x in the equation 11=4x, we need to isolate x on one side of the equation.
We can start by dividing both sides of the equation by 4:
11/4 = (4x)/4
Simplifying the right side of the equation, we get:
11/4 = x
Therefore, x = 11/4 or 2.75.
Jason wants to find more information about the heights of players on his school's soccer team and his school's basketball team. Jason surveys 29 members of the soccer team and 16 members of the basketball team. The results of his study are shown in the dot plots below .
The average height of the soccer team is 77.45 inches, and the average height of the basketball team is 79 inches.
To find the average height of each team, we'll sum up the heights and divide by the number of players on each team.
Soccer team average height:
Sum of heights: 60+61+...+87+88 = 2,246 inches
Number of players: 29
Average height: 2246 / 29 = 77.45 inches
Basketball team average height:
Sum of heights: 72+73+...+86+87 = 1,264 inches
Number of players: 16
Average height: 1264 / 16 = 79 inches
So, the average height of the soccer team is 77.45 inches, and the average height of the basketball team is 79 inches.
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Soccer team heights (in inches):
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88
Basketball team heights (in inches):
72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87
Jason wants to find more information about the heights of players on his school's soccer team and his school's basketball team. Jason surveys 29 members of the soccer team and 16 members of the basketball team. The heights of the soccer team and basketball team players are given above. What is the average height of the soccer team and the basketball team?
Q6)Define the term 'Molar mass'. State its unit of measurement.
What is a ‘Mole’? Write the value for avogadro’s number
Answer:
The molar mass of a chemical compound is defined as the ratio between the mass and the amount of substance of any sample of said compound.
unit of measurement: g/mol.
A mole is the amount of substance that contains as many elementary entities as there are atoms in 12g of carbon-12.
Avogadro's number= 6.02*10^23 particles.
Dilate figure XYZ by a scale factor of 1/4 X(-2,4) Y(-4,-4) Z(4,2)
Therefore , the solution of the given problem of expressions comes out to be points X(-2,4), Y(-4,4), and Z(4,2) would be X'(-2,4), Y'(-3,-1), and Z'(0,3).
What is expression?Instead of using random estimates, shifting variable numbers should be employed instead, which can be growing, diminishing, or blocking. They could only help one another by transferring items like tools, knowledge, or solutions to issues. The explanations, components, or mathematical justifications for strategies like expanded argumentation, debunking, and blending may be included in the explanation of the reality equation.
Here,
Figure XYZ will be scaled down by 1/4, with the centres being X(-2,4), Y(-4,4), and Z(4,2)
=> Old_x - Center_x - Scale_factor * New_x + Center_x
=> Scale factor * (Old y - Centre y) + Centre y = New y
Regarding X:
=> New_x = 1/4 * (-2 - (-2)) + (-2) = -2
=> New_y = 1/4 * (4 - 4) + 4 = 4
Therefore, X'(-2,4) is the image of point X after dilation.
Regarding Y:
=> New_x = 1/4 * (-4 - (-2)) + (-2) = -3
=> New_y = 1/4 * (-4 - 4) + 4 = -1
Therefore, Y'(-3,-1) is the image of point Y following dilation.
Regarding Z:
=> New_x = 1/4 * (4 - (-2)) + (-2) = 0
=> New_y = 1/4 * (2 - 4) + 4 = 3
Therefore, Z'(0,3) is the image of point Z after dilation.
As a result, the image points for the dilated figure XYZ with the centre points X(-2,4), Y(-4,4), and Z(4,2) would be X'(-2,4), Y'(-3,-1), and Z'(0,3).
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The angle times three out of 10 of the circle what is the measure of the angle
The measure of the angle that turns through 3/10 of the circle is 108 degrees.
Define angleAn angle is a geometric figure that is formed by two rays that share a common endpoint, which is called the vertex of the angle. The rays are referred to as the sides or legs of the angle. Angles are typically measured in degrees, with a full circle measuring 360 degrees.
Angles can be classified based on their degree of measurement: acute angles are less than 90 degrees, right angles are exactly 90 degrees, obtuse angles are greater than 90 degrees but less than 180 degrees, and straight angles are exactly 180 degrees.
Since a full circle has 360 degrees, 3/10 of the circle can be represented as:
(3/10) x 360 degrees = 108 degrees
Therefore, the measure of the angle that turns through 3/10 of the circle is 108 degrees.
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The complete question is :
The angle turns through 3/10 of the circle. What is the measure of the angle?
A rectangular patio has a length of (1 + 2x) yards and a width of (2 - 3x) yards. Which correctly describes the area and perimeter of the patio?
Area of the patio is -6x² + x + 2 square yards and perimeter of the patio is 6 - 2x yards.
What is a yard?A yard is a unit of measurement used to measure length or distance in both the US customary system and the British imperial system. One yard is equal to 3 feet or 36 inches. In the US customary system, a yard is equal to 0.9144 meters, while in the British imperial system, it is equal to 0.9144 meters or 3 feet. Yards are commonly used for measuring larger distances, such as in construction or landscaping, as well as in some sports like American football and cricket.
The area of the rectangular patio is given by multiplying its length by its width, so the area is:
A = (1 + 2x)(2 - 3x)
Now using distributive property we get,
A = 2 - 3x + 4x - 6x²
Simplifying further, we get:
A = -6x² + x + 2
Therefore, the area of the patio is described by the quadratic equation -6x² + x + 2.
To find the perimeter of the patio, we add up the lengths of all four sides. The length and width are given as (1 + 2x) and (2 - 3x), respectively, so the perimeter is:
P = 2(1 + 2x) + 2(2 - 3x)
Simplifying this expression, we get:
P = 2 + 4x + 4 - 6x
P = 6 - 2x
Therefore, the perimeter of the patio is described by the linear equation 6 - 2x.
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4) Jeanna sights the top of a building and the angle of elevation to be 35 degrees. She moves 100 feet closer and finds that the angle is now 40 degrees. What is the height of the building?
The vertical measurement of the structure corresponds to an estimated value of 119.53 feet.
How to Solve the Problem?The dimensions of the building can be represented by the variable h, whereas the proximity of Jeanna to the building in her initial stance can be identified as x. Utilizing the principles of trigonometry, it is feasible to express the following:
The trigonometric function involving the angle of 35 degrees and its corresponding acute triangle can be written as an equation in the form of tan(35) = h/x, which is denoted as equation 1.
Upon moving a distance of 100 feet from her initial position, Jeanna's distance from the building can be represented as x - 100. Additionally, the angle at which she must look up to view the top of the building is now 40 degrees. Through the application of comparable trigonometric reasoning, it is possible to articulate the following statement:
Equation 2 can be expressed as tan(40) = h/(x - 100), where h and x represent the height and horizontal distance, respectively.
Equation 1 can be manipulated in such a way as to express the variable x in terms of h. The value of x is obtained by dividing h by the tangent of 35 degrees.
The substitution of the given expression for variable x in equation 2 leads to the following result:
The equation tan(40) = h/(h/tan(35) - 100) is amenable for rephrasing in a more academic style of writing.
Upon performing reduction on this mathematical expression, it results in:
The given mathematical equation can be represented in an academic manner as follows: The equation tan(40) = tan(35)h/(h - 100tan(35)) holds true, where h denotes the height of an object located at an angle of 40 degrees to the horizontal plane. This equation specifies the relationship between the tangent values of two distinct angles, 40 degrees and 35 degrees, and the height of the object.
The present equation can be expressed in a more formal academic style as follows: "The function h is defined as the difference between the tangent of 40 degrees and the tangent of 35 degrees, i.e., h = tan(40) - tan(35). This formula can be simplified, resulting in h = -100tan(35)tan(40)."
By dividing each side of the equation by (tan(40) - tan(35)), we are able to obtain the following result:
The following expression denotes the value of h, computed using mathematical operations: h = -100tan(35)tan(40)/(tan(40) - tan(35)).
The value of h is approximately equal to 119.53 feet.
Hence, the vertical measurement of the structure corresponds to an estimated value of 119.53 feet.
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Find the mass of the triangular region with vertices (0, 0), (4, 0), and (0, 2), with density function ρ(x,y)=x^2+y^2.
The mass of the triangular region with density function ρ(x,y) = [tex]x^2 + y^2 is 136/375.[/tex]
What is a triangle?A triangle is a three-sided polygon made up of three line segments that connect at three endpoints, called vertices. The study of triangles is an important part of geometry, and it has applications in various fields such as engineering, architecture, physics, and computer graphics.
According to the given information:The mass of a 2D region with variable density can be calculated using the double integral formula:
m = ∬R ρ(x,y) dA
where R is the region of integration, ρ(x,y) is the density function, and dA is the area element.
In this case, we have a triangular region with vertices (0, 0), (4, 0), and (0, 2), and the density function is ρ(x,y) = [tex]x^2 + y^2.[/tex]To set up the double integral, we need to determine the limits of integration for x and y.
Since the triangular region is bounded by the lines y = 0, y = 2, and x = (2/5)y, we can set up the integral as follows:
m = ∫0 ∫[tex]0^[/tex](2/5)y ([tex]x^2 + y^2)[/tex] dxdy
Integrating with respect to x first, we get:
m = ∫[tex]0^2 [(x^3/3) + xy^2]_0^(2/5[/tex])y dy
m = ∫[tex]0^2 [(8/375)y^5 + (4/15)y^3][/tex]dy
Evaluating the integral, we get:
m = [tex][(2/1875)y^6 + (2/5)y^4]_0^2[/tex]
m = (64/1875) + (16/5)
m = 136/375
Therefore, the mass of the triangular region with density function ρ(x,y) = [tex]x^2 + y^2 is 136/375.[/tex]
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i do not know the answer to this
Answer:
b
Step-by-step explanation:
b
Write an equation of the form y = mx for the line shown below.
Answer:
y = (5/3)x + 2/3
Step-by-step explanation:
find slope
rise/run
(4--1)/(2--1)
(5)/(3)
M = 5/3
then plug in a point to get the y-intercept
4 = (5/3) * 2 + b
4 = 10/3 + b
subtract 10/3 on both sides
2/3 = b
in y =mx +b the equation is
y = (5/3)x + 2/3
What inequality matches this graph?
A} x > -5
B} x ≥ -5
C} x < -5
D} x ≥-5
An inequality that matches this graph include the following: B} x ≥ -5.
What is a number line?In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.
This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).
Since the closed circle is at point -5 and increases to the right, an inequality that matches this graph is x ≥ -5.
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all the fifth grade students go outside for exercise. one-half of the students are on the soccer fields. One-fourth of the students are on the basketball courts. One-eight of the students are jogging around the track. Twelve of the students are tossing footballs to each other. How many fifth grade students are outside for exercise? How many students are playing each sport? Show all your mathematical thinking
Answer: 96 fifth grade students outside for exercise. 48 students playing soccer, 24 students playing basketball, 12 students jogging around the track, and 12 students tossing footballs.
Step-by-step explanation:
Let x be the total number of fifth grade students.
- 1/2x are on the soccer fields
- 1/4x are on the basketball courts
- 1/8x are jogging around the track
- 12 students are tossing footballs
x = 1/2x + 1/4x + 1/8x + 12
x = 4/8x + 2/8x + 1/8x + 12
x = 7/8x + 12
1/8x = 12
x = 12*8
x = 96
Therefore, there are 96 fifth grade students outside for exercise.
Soccer fields: 1/2x = 1/2 * 96 = 48 students
Basketball courts: 1/4x = 1/4 * 96 = 24 students
Jogging around the track: 1/8x = 1/8 * 96 = 12 students
Tossing footballs: 12 students (as given in the problem)
There are 48 students playing soccer, 24 students playing basketball, 12 students jogging around the track, and 12 students tossing footballs.
Answer:
96 students in general
Step-by-step explanation:
I drew an Euler circle to make it more clear. so we can see that 1/8 class = 12
so 1/4 class is 12×2=24
1/2 class is 12×4=48
12+12+24+48=96
I hope this helped
which expresions are equivalent to 1/2³
1/8
(1/2)^3 1^3 2^3 1/8 or:(1/2)^3 (1/2) (1/2)(1/2)1×1×1/2×2×1/8
Connor has 3 3/4 feet of brown fabric and 3/4 yard of green to make a costume for the school play. How many more brown than green does Connor have?
what is the next 3 numbers in this sequence 3, 2, 6, 5, 15, 14
Answer:
42, 41, 123
Step-by-step explanation:
3, 2, 6, 5, 15, 14
- 1 x3. - 1. x3. - 1
you just gotta look for the pattern thats the same in the sequence so the next three numbers would be x3, then - 1 then x3 again onto the numbers.
Answer:
The next three numbers are 42, 41, and 123
Step-by-step explanation:
3 - 1 = 2
2 * 3 = 6
6 - 1 = 5
5 * 3 = 15
15 - 1 = 14
The pattern is subtracting one and then multiplying by three.
14 * 3 = 42
42 - 1 = 41
41 * 3 = 123
Find the size of angle p. Give your answer
in degrees (°).
137⁰
60°
60°
112°
р
134
Not drawn accurately
PLEASE HELPPP
Answer:
76°
Step-by-step explanation:
The sum of the interior angles of any quadrilateral is 360°.
p = 360 - 60 - 134 - 90 = 76
A company sells snow globes made out of glass that are in the shape of hollow
spheres. Khloe measures the outer diameter of a snow globe to be 30 cm. If the glass
has a thickness of 0.35 cm, find the total volume of glass that makes up the globe.
Round your answer to the nearest hundredth if necessary. (Note: diagram is not
drawn to scale.)
The total volume of glass that makes up the snow globe is approximately 1,757.84 cubic centimeters.
What is the inner sphere?
The Inner Sphere is a region of interstellar space surrounding Earth to a radius of roughly 450 - 550 light-years, generally demarcated by the outer borders of the "Great Houses."
The outer sphere has a diameter of 30 cm, so its radius is 15 cm. The volume of the outer sphere is given by the formula:
V outer = (4/3)πr_outer³
V outer = (4/3)π(15 cm)³
V outer ≈ 14,137.17 cm³
The inner sphere has a diameter of 30 cm minus the thickness of the glass on both sides, so its diameter is:
30 cm - 2(0.35 cm) = 29.3 cm
Therefore, the radius of the inner sphere is:
r inner = 29.3 cm / 2 = 14.65 cm
The volume of the inner sphere is given by the formula:
V inner = (4/3)πr_inner³
V inner = (4/3)π(14.65 cm)³
V inner ≈ 12,379.33 cm³
Finally, the volume of glass that makes up the snow globe is the difference between the volumes of the outer and inner spheres:
V glass = V outer - V inner
V glass ≈ 1,757.84 cm³
Therefore, the total volume of glass that makes up the snow globe is approximately 1,757.84 cubic centimeters.
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Monica has a dog food dispenser that can hold 25 cups of food. Sometimes, she likes to mix different flavors of food for her dog. This morning, Monica noticed there was some chicken-flavored food in the dispenser already. After Monica added 11 cups of beef-flavored food to the dispenser, it was completely full.
C - 11 = 25 or c + 11 = 25.
Answer:
I got you
Step-by-step explanation:
c-11=25
c=25+11
c=36
Consider the information below for Postman Builders Inc. Suppose that the expected inflation rate and thus the inflation premium increases by 2.0 percentage points, and Postman acquires risky assets that increase its beta by the indicated percentage. What is the firm's new required rate of return? Beta 1.5, Required Return 10.2, RPm 6%, Percentage increase in beta 65
Therefore, Postman Builders Inc’s new required rate of return is 23.7%1.
SET A RATE?Rate might signify two different things. As a noun, it denotes an amount, frequency, or measure that is typically compared to another quantity or measure. Or, it can be used as a verb to indicate "assign a standard or value to" using a specific scale.
Postman Builders Inc. has a necessary return of 10.2% and a beta of 1.5, according to the information given.
We may get the new necessary rate of return as follows if Postman purchases risky assets that increase its beta by 65% and the predicted inflation rate and inflation premium increase by 2 percentage points.
New Required Rate of Return
= Risk-free rate + Beta * (Expected Market Return - Risk-free rate)
= 2% + (1.5 * (6% + 2% + (65% * 6%))) = 23.7%
New Required Rate of Return equals 8% plus 2.47 x (-2%) to 3.85%.
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Find the circumference of the circle. Round to the nearest tenth.
A. 148.4 m
B. 169.6 m
C. 84.8 m
D. 54.0 m
Answer:
251.2 in
Step-by-step explanation:
Circumference of circle = 2 · π · r
π = 3.14
r = 40 in
Let's solve
2 · 3.14 · 40 = 251.2 in
So, the circumference of the circle is 251.2 in
Un deportista se ejercita y recorre 350 metros, retrocede 125 metros avanza 80 metros. Si el punto de partida es su casa a que distancia de la misma se encuentra
The sportsman has a final distance with respect to his home equal to 305 meters.
How to determine the distance of a sportsman with respect to a home
In this problem we must the final distance of the sportsman with respect to his home. This can be determined by the following expression:
X = ∑ xₙ, for n = {1, 2, 3, ..., N}
Where the sportsman goes far away for his home for x > 0, in meters.
Now we proceed to determine the final distance:
X = + 350 m - 125 m + 80 m
X = + 305 m
The final distance of the sportsman with respect to his home is 305 meters.
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2x-y=3 and 7x-y=13 solve the simultaneous equation
Answer:
x=4
y=5
Step-by-step explanation:
Suppose Thom’s route from his house to school consists of two straight legs: First, he drives straight for 6 miles, then he makes a right turn of 60 degrees and drives another 8 miles. How far (as the crow flies) is Thom’s school from his house?
The direct distance between Thom's house and his school, as the crow flies, is 7.21 miles.
How do you calculate direct distance?To find the direct distance between Thom's house and his school, we can use the Law of Cosines.
The Law of Cosines is a formula used to find the length of one side of a triangle when the lengths of the other two sides and the angle between them are known. In this case, we have a triangle with sides of length 6 miles, 8 miles, and the angle between them is 60 degrees.
The Law of Cosines formula is:
c² = a² + b² - 2ab * cos(C)
where:
a and b are the lengths of the known sides of the triangle
C is the angle between those two sides
c is the length of the unknown side (the distance we want to find)
In this case:
a = 6 miles
b = 8 miles
C = 60 degrees
First, convert the angle from degrees to radians:
C (radians) = (60 degrees * π) / 180 = 1.047 radians
Now, apply the Law of Cosines:
c² = 6² + 8² - 2 x 6 x 8 x cos(1.047)
c² = 36 + 64 - 96 x cos(1.047)
c² ≈ 36 + 64 - 96 x 0.5
c² ≈ 36 + 64 - 48
c² = 52
Finally, find the square root to get the distance:
c = √52 = 7.21 miles
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4<7 Multiply both sides by 7 , then by 6, then by 3, then by 10??
When you multiply an inequality by a positive number, the direction of the inequality does not change. So if 4 < 7, then:
what is inequality?
Inequality is a mathematical statement that describes a relationship between two values or expressions that are not equal. In an inequality, we use the symbols "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "≠" (not equal to) to indicate the relationship between the values or expressions.
For example, "x < 5" is an inequality that means "x is less than 5", and "y ≥ 10" is an inequality that means "y is greater than or equal to 10". Inequalities are often used in algebra and other branches of mathematics to express relationships between variables or to solve problems.
Multiplying both sides by
7 gives: 28 < 49
Multiplying both sides by
6 gives: 24 < 42
Multiplying both sides by
3 gives: 12 < 21
Multiplying both sides by
10 gives: 40 < 70
All of these inequalities are still true, because we are multiplying both sides by a positive number.
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