Final answer:
The line of symmetry of a parabola is a vertical line that passes through its vertex, dividing the parabola into two equal halves. This line of symmetry can be determined mathematically for a parabolic equation by the formula x = -b/(2a). Symmetry plays an important role in the behavior of optical devices such as spherical and parabolic mirrors.
Explanation:
The line of symmetry of a parabola is a vertical line that passes through its vertex, and it divides the parabola into two mirror-image halves. This line is also known as the axis of symmetry. For a parabola described by the equation y = ax² + bx + c, the axis of symmetry can be found using the formula x = -b/(2a), where a, b, and c are coefficients in the quadratic equation, representing the parabola's concavity, slope at the vertex, and the y-intercept, respectively.
In the context of optical devices like mirrors and lenses, the concept of symmetry is crucial. For instance, a spherical mirror reflects rays in a way that demonstrates symmetry with respect to its optical axis. When a parabolic mirror is used, all parallel rays of light are reflected through its focal point, illustrating how symmetry is a fundamental principle in both nature and physics.
Just like in nature, where symmetrical patterns can be seen in butterfly wings and other complex systems, symmetry in mathematical terms is significant and pervasive, revealing fundamental properties of the objects and phenomena in question.
In the figure given l || m , then find the value of X.
Answer:
42
Step-by-step explanation:
Line m and l are parallel so C and where that intersection is on the inside have the parallel lines, the angle there is an same side interior angle with C. Those add up to 180 degrees. So 180-82 is 98. Use vertical angle to find the angle in the triangle (not x yet; the other one). Vertical angles are congruent so that one is 98.
Now angles in a triangle add up to 180 degrees.
So we have 40+98=138.
The difference between 180 and 138 is 180-138=42 (that is x)
Answer:
42
Step-by-step explanation:
The solution is in the image below.
98+82 = 180
Find the least common multiple.
3x3, 12y7, and 15xy3
Answer:
60 x^3 y^7
Step-by-step explanation:
3x^3 = 3*xxx
12y^7 = 3*4 yyyyyyy
15xy^3 = 3*5 xyyy
We need to have the minimum of each for the least common multiply
For the numbers there is a 3 a 4 and a 5
For the x , the minimum is xxx
For the y, the minimum is yyyyyyy
Least common multiple: 3*4*5 xxx yyyyyyy
60 x^3 y^7
Funcions f(x) and g(x) are defined below.
Determine where f(x) = g(x) from the graph.
A: x = -2; x = 2
B: x = -2; x = 0; x = 2
C: x = 0; x = -8
D: x = 0, x = 2
The point where f(x)= g(x) is x = -2; x = 2.
The correct option is (A)
What is graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
As, from the graph the line graph for f(x) and g(x) is coinciding at two points at 2 and -2.
At these two points the two function line overlap.
So, point where f(x)= g(x) is x=2 and x=-2.
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Find the value of x that will make L ll M
Answer:
x=7
Step-by-step explanation:
7x-7 = 4x + 14
3x-7=14
3x = 21
x=7
Answer: 7
Step-by-step explanation:
Under normal conditions, 1.5 feet of snow will melt into 2 inches of water. During a winter season high in the mountains, 301 feet of snow fell. How many inches of water will there be when the snow melts?
Answer: 401.33 inches
100 pencils cost 7.00 1 pencil cost
Answer:
7 pence
Step-by-step explanation:
7.00 / 100 = 7 pence
Answer:
7 cents
Step-by-step explanation:
7.00/100
(7−4n)⋅6 Apply the distributive property to create an equivalent expression.
Answer:
42-24n
Step-by-step explanation:
(7 - 4n) . 6
= (7)(6) - (4n)(6) ...........using distributive property, multiply each term by (6)
= 42 -24n (answer)
A car is traveling at 42 mph. if its tires
have a diameter of 27 inches, how fast are
the car's tres turning? Express answer
in revolutions per minute.
Answer:
522.9 revolutions /min
Step-by-step explanation:
Given Diameter, D = 27 inches
Circumference, C = πD = 3.142 x 27 = 84.82 inches
(recall 1 mi = 63360 in)
Car speed is 42 mph = 42 mph x 63,360 inches / mile = 2,661,120 inches / hr
(recall 1 hour = 60 min)
Hence car speed becomes
= 2,661,120 inches / hr ÷ 60 min/hr
= 44,352 inches / min
Number of revolutions / min
= speed in inches/min ÷ circumference
= 44,352 ÷ 84.82 = 522.9 revolutions /min
To find out how fast a tire with a diameter of 27 inches is turning for a car moving at 42 mph, you first convert miles per hour to inches per minute, then calculate the tire's circumference in inches, and divide the car's speed by this circumference. The calculated result is approximately 523.05 revolutions per minute.
Explanation:To answer this question, we first need to convert the information about speed and tire diameter to compatible units. We'll convert the car speed from miles per hour (mph) to inches per minute since the diameter of the tire is given in inches.
Since 1 mile equals 63,360 inches, 42 mph is equal to 42*63360 = 2,661,120 inches per hour. As there are 60 minutes in an hour, that comes out to 2,661,120/60 = 44,352 inches per minute.
Next, we will find the circumference of the tire, which is its diameter multiplied by pi. Therefore, we have a circumference of 27 inches * 3.14 (pi) = 84.78 inches.
Finally, to find out how many times the tire rotates in a minute, we divide the car's speed in inches per minute by the tire's circumference in inches. This means the tires are making 44,352/ 84.78 ~= 523.05 revolutions per minute.
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A regular octagon has an apothem measuring 10 in, and a
perimeter of 66.3 in
What is the area of the octagon, rounded to the nearest
square inch?
10 in
88 in 2
175 in 2
332 in2
700 in 2
Answer:
332 square inch
Step-by-step explanation:
Solving for an area of a regular octagon, we can make use of the formula shown below:
Area = 1/2 * apothem*perimeter
In this problem, the following values were given such as:
Apothem = 10 inches
Perimeter = 66.3 inches
Solving for the area:
Area = 1/2*10*66.3
Area = 331.5 inches²
The answer is 332.
Answer:
Option C. 332 in²
Step-by-step explanation:
In the figure attached, a regular octagon has been drawn with all equal sides and apothem OP = 10 in.
Perimeter of the given octagon is given as 66.3 in
We have to calculate the area of the octagon.
As we can see in the figure an octagon is a combination of 8 triangles.
So we will find the area of one triangle first.
Area of ΔBOC = [tex]\frac{1}{2}(BC)(OP)[/tex]
Since perimeter of octagon = 8 × one side = 8×BC
66.3 = 8× BC
BC = [tex]\frac{66.3}{8}[/tex]
BC = 8.288 in
Therefore, area of ΔBOC = [tex]\frac{1}{2}(10)(8.288)[/tex]
= 5×8.288
= 41.44 in²
Now area of octagon ABCDEFGH = 8×41.44 = 331.5 ≈ 332 in²
Therefore, area of the regular octagon will be 332 in²
Option C. is the answer.
biggest to smallest 1/8 2/3 3/5
Convert the fractions to decimals:
1/8 = 0.125
2/3 = 0.666
3/5 = 0.6
Now arrange from largest to smallest
2/3, 3/5, 1/8
Answer:
2/3 biggest
3/5
1/8 smallest
Find the slope of the line through the points (4, 8) and (5, 10).
A. 1/2
B. -1/2
C. 2
D. -2
Answer:
The slope is 2.
Step-by-step explanation:
If A=1/2h(x+y), what is y in terms of A, h, and x?
Answer:
see explanation
Step-by-step explanation:
Given
A = [tex]\frac{1}{2}[/tex] h(x + y)
Multiply both sides by 2 to eliminate the fraction
2A = h(x + y) ( divide both sides by h )
[tex]\frac{2A}{h}[/tex] = x + y ( subtract x from both sides )
y = [tex]\frac{2A}{h}[/tex] - x
The solution to the equation A=1/2h(x+y) for y in terms of A, h, and x, is y = 2A/h - x, obtained by algebraic rearrangement.
Explanation:To solve the given equation A=1/2h(x+y) for y in terms of A, h, and x, we first perform algebraic rearrangement. We want to get y in terms of A, h, and x. So, the first step is to eliminate the 1/2h from the right side of the equation. We can do this by multiplying both sides of the equation by 2/h, we get: 2A/h = x + y.
Now, to solve for y, we can simply subtract x from both sides: y = 2A/h - x. So now we have y expressed in terms of A, h, and x.
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Lisa is a leading player on her basketball team. She has kept track of how many points she has scored over the course of the season. Her scores may be viewed in the table below. 36 20 35 13 30 14 36 11 29 19 22 40 34 17 26 21 What is the median of Lisa’s scores? Round to the nearest point, if necessary. a. 23 b. 24 c. 26 d. 30
Answer: Option b
[tex]M=24[/tex]
Step-by-step explanation:
We have the following list of data
36 20 35 13 30 14 36 11 29 19 22 40 34 17 26 21
The list contains 16 data.
The first step to calculate the median is to order the data from least to highest
11,13,14,17,19,20,21,22,26,29,30,34,35,36,36,40
Now mark the two central values of the data set
11,13,14,17,19,20,21 [22,26] 29,30,34,35,36,36,40
The median of the data is the average of the two central values
[tex]M=\frac{22+26}{2}[/tex]
[tex]M=24[/tex]
Note that the median is the central value of the ordered data set
Using the given function, select the correct set of ordered pairs for the following domain values.
Answer:
The set of ordered pair are { (-12,-18),(-3,-3), (0,2),(3,7),(12,22)}
Step-by-step explanation:
we have
[tex]f(x)=\frac{5}{3}x+2[/tex]
Find the values of f(x) for the domain values
For x=-12
substitute
[tex]f(-12)=\frac{5}{3}(-12)+2=-18[/tex]
The ordered pair is (-12,-18)
For x=-3
substitute
[tex]f(-3)=\frac{5}{3}(-3)+2=-3[/tex]
The ordered pair is (-3,-3)
For x=0
substitute
[tex]f(0)=\frac{5}{3}(0)+2=2[/tex]
The ordered pair is (0,2)
For x=3
substitute
[tex]f(3)=\frac{5}{3}(3)+2=7[/tex]
The ordered pair is (3,7)
For x=12
substitute
[tex]f(12)=\frac{5}{3}(12)+2=22[/tex]
The ordered pair is (12,22)
Answer with explanation:
The given function is
[tex]f(x)=\frac{5x}{3}+2\\\\x=-12,-3,0,3,12\\\\f(-12)=\frac{5\times -12}{3}+2\\\\f(-12)=-20+2\\\\f(-12)=-18\\\\f(-3)=\frac{5 \times -3}{3}+2\\\\f(-3)=-5+2\\\\f(-3)=-3\\\\f(0)=\frac{5\times 0}{3}+2\\\\f(0)=2\\\\f(3)=\frac{5\times 3}{3}+2\\\\f(3)=5+2\\\\f(3)=7\\\\f(12)=\frac{5\times 12}{3}+2\\\\f(12)=20+2\\\\f(12)=22[/tex]
So, the ordered pair will be
(-12, -18),(-3,-3),(0,2),(3,7),(12,22)
Write 0.8% as a decimal number and a proper fraction in lowest terms.
1. Show and explain your work in converting 0.8% to a decimal.
2. Show and explain your work in convtering 0.8% to a proper fraction in lowest terms.
3. Write a sentence putting your answers into the context of the exercise.
Answer:
1. .0008
2. 1/125
3. 0.8% of a mile is 1/125th of a mile
Match each quadratic equation with the best way to solve it.
1. 5x2 + 12x - 3 = 0
solve by square root method
2. x2 - 4x = 8
solve by factoring
3. 4x2 - 25 = 0
solve by quadratic formula
4. x2 - 5x + 6 = 0
solve by completing the square
Answer:
5x^2 + 12x -3 =0 ---------> solve by quadratic formula
x^2 -4x = 8 ----------> solve by completing the square
4x^2 -25 = 0 ----------> solve by square root method
x^2-5x+ 6 = 0 -----------> solve by factoring
Step-by-step explanation:
1. 5x^2 + 12x -3 =0
The best way to solve this equation is quadratic formula as all the terms in the equation have coefficients it will be convenient to solve it through quadratic formula.
2. x^2 -4x = 8
The best way to solve this equation is by completing the square as the factors cannot be made directly.
3. 4x^2 -25 = 0
the best way to solve this equation is to solve by square root method as the 25 and 4 are perfect squares.
4. x^2-5x+ 6 = 0
The best way to solve this equation is to solve by factoring as it can clearly be seen that it is convenient to make factors ..
GEOMETRY Please HELP
Answer:
x = 6.93
Step-by-step explanation:
If a tangent and a secant are drawn to a circle from the same exterior point, the square of the length of the tangent is equal to the product of the total length of the secant and the length of the external segment of the secant.
From the given diagram,
(8+4)×4=x²
12×4=x²
48=x²
x=√48
x=6.93
x=7 to the nearest whole number.
In which direction does the left side of the graph of this function point?
f(x) = 3x^3– x^2 + 4x - 2
Answer:
Down.
Step-by-step explanation:
ƒ(x) = 3x³ - x² + 4x - 2
"They" are asking you to determine the end behaviour of the function, that is, what happens to the function as x ⟶ -∞.
The important point to remember is that, when x is large enough, the term of highest degree (the x³ term) will be so large that you can ignore all the other terms.
If x is large and negative,
f(x) ≈ 3(-x)³ = 3(-1)³(x)³= -3x³
Thus, the graph is in the third quadrant and the graph points down.
The figure below shows the graph of your function. It points sharply down on the left-hand side.
if f(x)=x/2-3 and g(x)=4x^2+x-4, find (f+g)(x)
Answer:
[tex]\large\boxed{(f+g)(x)=4x^2+\dfrac{3}{2}x-7}[/tex]
Step-by-step explanation:
[tex](f+g)(x)=f(x)+g(x)\\\\f(x)=\dfrac{x}{2}-3=\dfrac{1}{2}x-3,\ g(x)=4x^2+x-4\\\\\text{Substitute:}\\\\(f+g)(x)=\left(\dfrac{1}{2}x-3\right)+(4x^2+x-4)\\\\=\dfrac{1}{2}x-3+4x^2+x-4\qquad\text{combine like terms}\\\\=4x^2+\left(\dfrac{1}{2}x+x\right)+(-3-4)\\\\=4x^2+1\dfrac{1}{2}x-7\\\\=4x^2+\dfrac{3}{2}x-7[/tex]
Let f(x) = (7x^3 + 18)2 and h(x) = x^2.
Given that f(x) = (hºg)(x), find g(x).
[tex]\bf \begin{cases} f(x)=(7x^3+18)^2\\ f(x) = (h\circ g)(x)\\ h(x)=x^2 \end{cases}~\hspace{5em}(h\circ g)(x) = \stackrel{f(x)}{(7x^3+18)^2} \\\\\\ h(x) = x^2\implies \stackrel{(h\circ g)(x)}{h(~~g(x)~~)}=[g(x)]^2\implies h(~~g(x)~~)=\stackrel{f(x)}{(7x^3+18)^2} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill g(x)=7x^3+18~\hfill[/tex]
Your school is organizing a carnival. You are building a ramp for a game the surface of the ramp will be 5 feet long with a total rise of 3 feet. Find the angle of elevation of the ramp
Answer:
Step-by-step explanation:
this angle is related to the ramp surface length (hypotenuse) and the rise in the ramp by the tangent function:
sin Ф = opp / hyp = 3 ft / 5 ft = 0.60
Use the inverse sine function to determine the angle Ф.
arcsin 0.60 = 36.9 degrees
Final answer:
The angle of elevation of the ramp is approximately 36.87 degrees, calculated using the tangent function in trigonometry and the given dimensions of the ramp's rise and length.
Explanation:
To find the angle of elevation of the ramp, we can use trigonometry, specifically the tangent function, which relates the angle of a right triangle to the ratio of the opposite side (rise) to the adjacent side (run). In this case, the ramp surface is the hypotenuse of the right triangle, the rise is the vertical change, and the run is the horizontal distance, which would be the horizontal projection of the ramp if it were flat. Since we know the rise is 3 feet and the ramp length (hypotenuse) is 5 feet, we can calculate the run using the Pythagorean theorem.
Calculate the run (horizontal distance): run = √(hypotenuse^2 - rise^2) = √(5^2 - 3^2) = √(25 - 9) = √16 = 4 feet.Now, calculate the angle of elevation using the definition of tangent: tangent(angle) = opposite/adjacent, which leads to angle = arctan(opposite/adjacent).Plug in the values for the rise and run: angle = arctan(3/4). Use a calculator to find the angle: angle ≈ 36.87°.The angle of elevation of the ramp is approximately 36.87 degrees.
which of the following is a point slope equation of a line with the point (1, 8) and a slope of -5? a. y +8=-5(x+1) b. y- 5 = -5(x-8) c. y+ 5= -5(x+8) d. y-8=-5(x-1)
Answer:
Option D. [tex]y-8=-5(x-1)[/tex]
Step-by-step explanation:
we know that
The equation of a line into point slope form is equal to
[tex]y-y1=m(x-x1)[/tex]
we have
[tex](x1,y1)=(1,8)[/tex]
[tex]m=-5[/tex]
substitute
[tex]y-8=-5(x-1)[/tex]
The correct point-slope equation for a line passing through the point (1, 8) with a slope of -5 is y - 8 = -5(x - 1).
The point-slope form of an equation of a line is y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope.
Given the point (1, 8) and a slope of -5, the correct point-slope equation is d. y - 8 = -5(x - 1).
Therefore, the correct answer is d. y - 8 = -5(x - 1).
What is the volume of A right prism w/ a triangular base w/ sides of 3,4 and 5 w/ a height of 10?
Check the picture below.
[tex]\bf \textit{volume of a pyramid}\\\\ V=\cfrac{1}{3}Bh~~ \begin{cases} B=&area~of\\ &its~base\\ h=&height\\ \cline{1-2} B=&\frac{1}{2}(3)(4)\\ h=&10 \end{cases}\implies V=\cfrac{1}{3}\cdot \cfrac{1}{2}(3)(4)(10)\implies V=20[/tex]
Find the value of x.
Expand the binomial (1 - 2x)^6 use Pascal’s triangle
Answer:
1 - 12x + 60x^2 - 160x^3 + 240x^4 - 192x^5 + 64x^6
Step-by-step explanation:
So we need to know the 7th line of pascals triangle so let us write it out
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1 <----- This one
Now we know we will need coefficients of x from 0 to 6, let us call this i, and for each of these the coefficient is
(-2)^i * (1)^(6-i) * number in pascals triangle
Hence
i = 0, 1 * 1 * 1 = 1
i = 1, -2 * 1 * 6 = -12
i = 2, 4 * 1 * 15 = 60
i = 3, -8 * 1 * 20 = -160
i = 4, 16 * 1 * 15 = 240
i = 5, -32 * 1 * 6 = -192
i = 6, 64 * 1 * 1 = 64
Hence (1-2x)^6 = 1 - 12x + 60x^2 - 160x^3 + 240x^4 - 192x^5 + 64x^6
A line passes through the point (-4,-8) and has a slope of -5/2.
Write an equation in slope-intercept form for this line..
Plz
[tex]\bf (\stackrel{x_1}{-4}~,~\stackrel{y_1}{-8})~\hspace{10em} slope = m\implies -\cfrac{5}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-8)=-\cfrac{5}{2}[x-(-4)] \\\\\\ y+8=-\cfrac{5}{2}(x+4)\implies y+8=-\cfrac{5}{2}x-10\implies y=-\cfrac{5}{2}x-18[/tex]
Solve for x. x/5+ 1 = 7
x = 5 5/6
x = 30
x = 35
x = 40
Answer:
x = 30Step-by-step explanation:
[tex]\dfrac{x}{5}+1=7\qquad\text{subtract 1 from both sides}\\\\\dfrac{x}{5}+1-1=7-1\\\\\dfrac{x}{5}=6\qquad\text{multiply both sides by 5}\\\\5\!\!\!\!\diagup^1\cdot\dfrac{x}{5\!\!\!\!\diagup_1}=5\cdot6\\\\x=30[/tex]
Which of the following could be the equation of the graph shown below? A. y=-10 B. 2x- 3y= -9 C. y = 3x-2 D. -4x+ 2y= -6
Answer:
C. y = 3x - 2
Step-by-step explanation:
Since the y intercept is negative, it is seen in this equation that the b value (the y intercept) is also negative. Since the slope is increasing upward, we know that the slope is greater than 1 and is going upward, and it matches with the equation since it matches with both of the requirements.
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Answer: c and d
Step-by-step explanation:
198 cents to 234 cents using a fraction in simplest form
Answer:
11/13
Step-by-step explanation:
198/234
(198/18)/(234/18)
11/13
Answer:
11/13
Step-by-step explanation:
198 cents to 234 cents using a fraction in simplest form is 11/13.
Hope this helps!
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help answer question 3
Answer:
The answer is the first one. Inside the absolute value symbols it is negative but absolute value answers are always positive. Instead of -30 the answer is 30.