Answers
Solution:
Given:
[tex]\begin{gathered} focus=(1,2) \\ directrix,y=6 \end{gathered}[/tex]Step 1:
The equation of a parabola is given below as
[tex]\begin{gathered} y=\frac{1}{4(f-k)}(x-h)^2+k \\ (h,f)=focus \\ h=1,f=2 \end{gathered}[/tex]Step 2:
The distance from the focus to the vertex is equal to the distance from the vertex to the directrix:
[tex]\begin{gathered} f-k=k-6 \\ 2-k=k-6 \\ 2k=2+6 \\ 2k=8 \\ \frac{2k}{2}=\frac{8}{2} \\ k=4 \end{gathered}[/tex]Step 3:
Substitute the values in the general equation of a parabola, we will have
[tex]\begin{gathered} y=\frac{1}{4(f-k)}(x-h)^{2}+k \\ y=\frac{1}{4(2-4)}(x-1)^2+4 \\ y=-\frac{1}{8}(x-1)^2+4 \\ \end{gathered}[/tex]By expanding, we will have
[tex]\begin{gathered} y=-\frac{1}{8}(x-1)^{2}+4 \\ y=-\frac{1}{8}(x-1)(x-1)+4 \\ y=-\frac{1}{8}(x^2-x-x+1)+4 \\ y=-\frac{1}{8}(x^2-2x+1)+4 \\ y=-\frac{x^2}{8}+\frac{x}{4}-\frac{1}{8}+4 \\ y=-\frac{x^2}{8}+\frac{x}{4}-\frac{1+32}{8} \\ y=-\frac{x^2}{8}+\frac{x}{4}+\frac{31}{8} \end{gathered}[/tex]Hence,
The final answer is
[tex]\begin{gathered} \Rightarrow y=-\frac{x^{2}}{8}+\frac{x}{4}+\frac{31}{8}(standard\text{ }form) \\ \Rightarrow y=-\frac{1}{8}(x-1)^2+4(vertex\text{ }form) \end{gathered}[/tex]Related Questions
Use the formula for n^P_r to evaluate the following expression.
Answers
Use the following formula:
[tex]_nP_r=\frac{n!}{(n-r)!}[/tex]Then, for 11P6:
[tex]\begin{gathered} _{11}P_6=\frac{11!}{(11-6)!}=\frac{11!}{5!}=\frac{5!\cdot6\cdot7\cdot8\cdot9\cdot10\cdot11}{5!} \\ _{11}P_6=6\cdot7\cdot8\cdot9\cdot10\cdot11=332640 \end{gathered}[/tex]Hence, the result is 332640
A vase is in the shape of a cone. The height is 12 inches and the diameter is 4.4 inches.
What is the lateral surface area to the nearest tenth of a square inch?
O
O
24.3 square inches
149.1 square inches
168.6 square inches
99.5 square inches
Answers
84.27 square inch is the lateral surface area of cone.
Define lateral surface area.All of an object's sides, excluding its base and top, are considered its lateral surface. The size of the lateral surface is referred to as its area. This must be distinguished from the total surface area, which consists of the base and top areas as well as the lateral surface area. A figure's lateral area consists solely of the non-base faces. The lateral surface area of several forms, such as a cuboid, cube, cylinder, cone, and sphere, is discussed in this article.
Given,
Height = 12 inches
Diameter = 4.4 inches
Radius = 2.2 inches
Lateral surface area:
πr√h² + r²
3.14 × 2.2 √(12)² + (2.2)²
3.14 × 2.2 √144 + 4.84
3.14 × 2.2 √148.84
3.14 × 2.2(12.2)
3.14 × 26.84
84.27
84.27 square inch is the lateral surface area of cone.
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Write 0.000000000054 in scientific notation
Answers
Answer:
5.4 × 10^-11
Step-by-step explanation:
As you landscape a 4 leaf clover intersection, you will need to buy enough grass seed to cover all 4 circies. Each of the circles has the same diameter: 41 meters. Calculate the total area of all grass seed needed to cover all 4 circles.
Answers
SOLUTION
Each of the circles has the same diameter: 41 meters.
If the diameter = 41 meters
Then the Radius =
[tex]\frac{41}{2}\text{ m}[/tex]Then we need to find the total area of the 4 circles =
[tex]\begin{gathered} 4\text{ X }\pi r^2 \\ =\text{ 4 X }\frac{22}{7\text{ }}\text{ X }\frac{41}{2}\text{ X}\frac{41}{2} \\ =\text{ }5283\text{ }\frac{1}{7}m^2 \end{gathered}[/tex]CONCLUSION: The total area of all grass seeds needed to cover all 4 circles =
[tex]5283\text{ }\frac{1}{7}m^2[/tex]
Use Polya's four-step problem-solving strategy and the problem-solving procedures presented in this section to solve the following exercise.A shirt and a tie together cost $68. The shirt costs $30 more than the tie. What is the cost of the shirt (in dollars).
Answers
Let x and y be the cost of a shirt and a tie, respectively; therefore, the two equations are
[tex]\begin{gathered} x+y=68 \\ \text{and} \\ x=30+y \end{gathered}[/tex]We have two variables and two equations; we need to solve the system of equations to find the values of x and y.
Solve using the substitution method.
Use the second equation into the first equation, as shown below
[tex]\begin{gathered} x=30+y \\ \Rightarrow(30+y)+y=68 \\ \Rightarrow30+2y=68 \\ \Rightarrow2y=68-30=38 \\ \Rightarrow y=\frac{38}{2} \\ \Rightarrow y=19 \end{gathered}[/tex]Now, use this value of y in the second equation
[tex]\begin{gathered} y=19 \\ \Rightarrow x=30+y=30+19 \\ \Rightarrow x=49 \end{gathered}[/tex]Remember that x is the cost of a shirt and y is the cost of a tie. Therefore, the answers are
Cost of a shirt: $49
Cost of a tie: $19
One can verify the answer by noticing that a shirt and a tie cost $49+$19=$68, and that a shirt costs $30+$19=$49
need help, what's the answer for the x and y?
Answers
Line equation in slope and y-intercept form:
y = mx + b
To calculate the slope, we use the first two points: (24,-15) and (28, -17)
m = (y2 - y1)/(x2 - x1)
m = (-17 - (-15))/(28 - 24)
m = (-17 + 15)/(4
m = -2/4 = -1/2
To find b we use the first point: (24, -15)
y = mx + b
b = y - mx = -15 - (-1/2)(24) = -15 + 12 = -3
b = -3
Answer:
y = (-1/2) x - 3
Rosa receives money from her relatives every year on her birthday. Last year, she received a total of $350. This year, she received $441. What is the percent of increase in Rosa’s annual birthday money?
Answers
Answer:
26%
Step-by-step explanation:
use a online percentage calculator
An ice cream truck began its daily route with 95 gallons of ice cream. The truck driver sold 58% of the ice cream. How many gallons of ice cream were sold? round to nearest gallon
Answers
Start making the percentage as a fraction
[tex]\begin{gathered} 58\text{\%}=\frac{58}{100} \\ \end{gathered}[/tex]multiply the fraction by the total of the daily routine
[tex]95\cdot\frac{58}{100}=55.1\approx55gallons[/tex]Shawn needs to reach a windowsill that is 10 feet above the ground. He placed his ladder 4 feet from the base of the wall. It reached the base of the window.
a. Draw a diagram of the right triangle formed by Shawn's ladder, the ground and the wall.
b. Find the length of Shawn's ladder to the nearest tenth of a foot.
Answers
If shown needs to reach a windowsill that is 10 feet above the ground and he placed his ladder 4 feet from the base of the wall.
Part a
The diagram of the right triangle formed by Shawn's ladder, the ground and wall has been plotted
Part b
The length of the Shawn's ladder is 10 foot
The distance between ladder base to the base of the wall = 4 feet
The distance between the wall base to the base of the window = 10 feet
Draw the right triangle using the given details
Part b
Using the Pythagorean theorem
[tex]AC^2= AB^2+BC^2[/tex]
Where AC is the length of the ladder
Substitute the values in the equation
AC = [tex]\sqrt{10^2+4^2}[/tex]
= [tex]\sqrt{100+16}[/tex]
= [tex]\sqrt{116}[/tex]
= 10.77
≈ 10 Foot
Hence, if shown needs to reach a windowsill that is 10 feet above the ground and he placed his ladder 4 feet from the base of the wall.
Part a
The diagram of the right triangle formed by Shawn's ladder, the ground and wall has been plotted
Part b
The length of the Shawn's ladder is 10 foot
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write an equation in slope intercept form of the line that passes through the given point and is parallel to the graph of the equation(-3, -5); y = -5x+2
Answers
The equation is y = -5x-20.
GIven:
The equation is, y = -5x + 2.
A point on the line is (-3, 5).
The objective is to write an equation that passes throught the point and parallel to the given equation.
For parallel lines the product of slope values will be equal.
From the given equation, consider the slope of the equation as, m1 = -5.
Then, the slope of the parallel line will also be, m2 = -5.
Then, the equation of parallel line can be written as,
[tex]\begin{gathered} y=m_2x+b \\ y=-5x+b \end{gathered}[/tex]Here b represents the y intercept of the parellel line.
To find the value of b, substitute the given points in the above equation.
[tex]\begin{gathered} -5=-5(-3)+b \\ -5=15+b \\ b=-5-15 \\ b=-20 \end{gathered}[/tex]Now, substitute the value of b in the equation of parellel line.
[tex]y=-5x-20[/tex]Hence, the equation of parellel line is y = -5x-20.
Can u please help me solve ? I'm reviewing for a final, ty
Answers
Part A
we have that
Both students verify the identity properly
student A ----> expand the left side of the identity
student B ----> expand the right side of the identity
but the result is the same
both students proved that the given equation is an identity
Part B
Identities[tex]\begin{gathered} sin^2x+cos^2x=1\text{ ----> identity N 1 in step 3} \\ cos^2x=1-sin^2x \end{gathered}[/tex]and
[tex]cscx=\frac{1}{sinx}\text{ -----> identity N 2 step 5}[/tex]Which equation represents a line that passes through the two points in thetable?O A. y+3= (x+3)OB. y-3-(x-3)O G. y+3=(x+3)C.OD.y-3-(x-3)X36y35
Answers
The first step is to choose one option and rewrite it in the explicit form
I will choose the second option:
[tex]y-3=\frac{2}{3}(x-3)[/tex][tex]y=\frac{2}{3}(x-3)+3[/tex][tex]y=\frac{2}{3}x-2+3[/tex][tex]y=\frac{2}{3}x+1[/tex]Now replace the x points in the equation to verify if it satisfies their respective value in y
For x=3
[tex]y=\frac{2}{3}(3)+1=\frac{6}{3}+1=2+1=3[/tex]For x=3 satisfy y=3
Now x=6
[tex]y=\frac{2}{3}(6)+1=\frac{12}{3}+1=4+1=5[/tex]For x=6 satisfy y=5
So the answer is b.
I solved the attached equation as 7000 but it seems to ask for a “solution set” did I answer properly?
Answers
The solution set does have only one element: {7000}
What is the equation of the line that passes through the point (8,-6) and has a
slope of o?
Answers
The equation is y = -6
Given the following information about events A, B, and C, determine which pairs of events, if any, are independent andwhich pairs are mutually exclusive.
Answers
Let's begin by identifying key information given to us:
[tex]\begin{gathered} P(A)=0.39 \\ P(B)=0.42 \\ P(C)=0.19 \\ P(A|B)=0 \\ P(C|B)=0.19 \\ P(A|C)=0.39 \end{gathered}[/tex]When two events A and B are independent we have it thus:
[tex]undefined[/tex]If the inflation has been 2.7%, how much more do you have to pay this year foran item that cost $11.50 last year?
Answers
Given data:
The cost of the item is $11.50.
The inflation percentage is 2.7%.
Increase in the price is,
[tex]\begin{gathered} =11.50\times(\frac{2.7}{100}_{}) \\ =11.50\times0.027 \\ =0.3105 \end{gathered}[/tex]Total amount to be paid last year,
[tex]\begin{gathered} =11.50+0.3105 \\ =11.8105 \end{gathered}[/tex]Therefore you will have to pay $ 0.3105 more.
write the following comparison as a ratio reduced to lowest terms. 21 quarters to 13 dollars
Answers
In order to calculate the ratio of these values, let's divide them, using the fraction form:
[tex]\text{ratio}=\frac{21}{13}[/tex]Since the numbers 21 and 13 don't have any common factor, the fraction is already in the lowest terms.
So the ratio is 21:13
Slove for p 14 = -(p - 8)
Answers
Solve:
[tex]\begin{gathered} 14=-(p-8) \\ -14=p-8 \\ -14+8=p \\ p=-14+8 \\ p=-6 \end{gathered}[/tex]p=-6
The shorter leg of a right triangle is 9cm shorter than the longer leg. The hypotenuse is 9cm longer than the longer leg. Find the side lengths of the triangle.Length of the shorter leg: _ cmLength of the longer leg:__ cmLength of hypotenuse __ cm
Answers
Explanation:
let the longer leg = x
The shorter leg = 9cm shorter than the longer leg
The shorter leg = x - 9
hypotenuse = 9cm longer than the longer leg
hypotenuse = x + 9
Using pythagoras theorem:
hypotenuse² = shorter leg² + longer leg²
(x + 9)² = x² + (x - 9)²
Expanding:
x² + 9x + 9x + 81 = x² + x ² - 9x -9x + 81
x² + 18x + 81 = 2x² -18x + 81
collect like terms:
18x + 18x + 81 - 81 = 2x² - x²
36x + 0 = x²
x² - 36x = 0
x(x - 36) = 0
x = 0 or (x - 36) = 0
x = 0 or x = 36
if x = 0
shorter side = x - 9 = 0 - 9 = -9
Since the length cannot be negative, x = 36
The longer leg = x = 36 cm
The shorter leg = x - 9 = 36 - 9
The shorter leg = 27cm
The hypotenuse = x + 9 = 36 + 9
The hypotenuse = 45 cm
Find the perimeter of the rectangle. Write your answer in scientific notation.Area = 5.612 times 10^14 cm squared9.2 times 10^7cm is one side of the perimeter
Answers
Answer: Perimeter = 1.962 x 10^8 cm
Explanation:
The first step is to calculate the width of the rectangle. Recall,
Area = length x width
width = Area /length
From the information given,
Area = 5.612 times 10^14 cm squared
Length = 9.2 times 10^7cm
Thus,
width = 5.612 times 10^14 /9.2 times 10^7
width = 6.1 x 10^6
The formula for calculating the perimeter is
Perimeter = 2(length + width)
Thus,
Perimeter = 2(9.2 x 10^7 + 6.1 x 10^6)
Perimeter = 1.962 x 10^8 cm
writing equations in slope-intercept form common core algebra 1question 1
Answers
The equation of the line in the slope-intercept form is y = mx + b, where "m" is the slope and "b" is the y-intercept.
"b" is the point (0, yi).
"m" can be found using 2 points P₁ (x₁, y₁) and P₂ (x₂, y₂), according to the formula below:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]So, to solve this question, follow the steps below.
(a) First graph
Step 01: Find the y-intercept and another point in the graph.
To find the points in the graph, choose a x-value and find its corresponding y-value.
Choosing x = 0, y = 2.
P₁ = (0, 2).
Choosing x = -3, y = -2.
P₂ = (-3, -2).
Step 02:
Use the functions f(x) = 8x + 11 and g(x) = 4x² + 7x - 2 to evaluate the following:a. f(8) =b. f(-8)=c. g(6) =d. g(-7)=e. g(a) =
Answers
Given:
f(x) = 8x + 11
g(x) = 4x² + 7x - 2
We are asked to evaluate using the following:
(a) f(8)
f(8) = 8(8) + 11
f(8) = 64 + 11
f(8) = 75
(b) f(-8)
f(-8) = 8(-8) + 11
f(-8) = -64 + 11
f(-8) = -53
(c) g(6)
g(6) = 4(6)² + 7(6) - 2
g(6) = 4(36) + 42 - 2
g(6) = 144 + 42 - 2
g(6) = 184
(d) g(-7)
g(-7) = 4(-7)² + 7(-7) - 2
g(-7) = 4(49) - 49 - 2
g(-7) = 196 - 49 - 2
g(-7) = 145
(e) g(a)
g(a) = 4(a)² + 7(a) - 2
g(a) = 4a² + 7a - 2
Write the equation of a line that is parallel to y = 1/2x -4 and that passes through the point (9, -6)
Answers
The most appropriate choice for equation of line in slope intercept form will be given by-
[tex]y = \frac{1}{2}x - \frac{21}{2}[/tex] is the required equation of line
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by [tex]y = mx + c[/tex]
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If [tex]\theta[/tex] is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = [tex]tan\theta[/tex]
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line
Here,
The given equation of line is [tex]y = \frac{1}{2} x - 4[/tex]
Slope of this line = [tex]\frac{1}{2}[/tex]
Slope of the line parallel to this line = [tex]\frac{1}{2}[/tex]
The line passes through (9 , -6)
Equation of the required line =
[tex]y - (-6) = \frac{1}{2}(x - 9)\\2y + 12 = x - 9\\2y = x - 9 -12\\2y = x -21\\y = \frac{1}{2}x - \frac{21}{2}[/tex]
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A survey stopped men and women at random to ask them where they purchased groceries, at a local grocery store or online.Grocery OptionsStoreOnlineTotalWomen231235Men221537Total452772What percent of the people surveyed shop at a local grocery store? Round your answer to the nearest whole number percent
Answers
Given:
A survey stopped men and women at random to ask them where they purchased groceries, at a local grocery store or online. Grocery Options
Store Online Total
Women 23 12 35
Men 22 15 37
Total 45 27 72
Required:
To find the percentage of the people surveyed shop at a local grocery store.
Explanation:
The total number of people is 75.
And the total number of people surveyed shop at a local grocery store is 45.
Now the percentage of the people surveyed shop at a local grocery store is,
[tex]=\frac{45}{72}\times100[/tex][tex]\begin{gathered} =62.5\% \\ \\ \approx63\% \end{gathered}[/tex]Final Answer:
63% of the people surveyed shop at a local grocery store.
9. The Elite Vacuum Company has determined its cost for making vacuums to beC = 24V + 1000, where C is the cost in dollars and V is the number of vacuums.If the cost must be between $49,000 and $121,000, how many vacuums can they makeper week? (You must set up and solve an inequality.)
Answers
We are given the relationship between the cost in dollars (C) and the number of vacuums (V) to be:
[tex]C\text{ = 24V + 1000}[/tex]From the constraint, we have that the cost(C) must be greater than $49000 and less than $121000
Writing this as inequality:
[tex]\begin{gathered} 24V\text{ + 1000 }\ge\text{ 49000 } \\ 24V\text{ + 1000 }\leq\text{ 121000} \end{gathered}[/tex]Solving the linear inequalities for V:
[tex]\begin{gathered} 24V\text{ + 1000 }\ge\text{ 49000} \\ 24V\text{ }\ge\text{ 49000 - 1000} \\ 24V\text{ }\ge\text{ 48000} \\ \text{Divide both sides by 24} \\ \frac{24V}{24}\text{ }\ge\text{ }\frac{48000}{24} \\ V\text{ }\ge\text{ 2000} \end{gathered}[/tex]Similarly for the second inequality:
[tex]\begin{gathered} 24V\text{ + 1000 }\leq\text{ 121000} \\ 24V\text{ }\leq121000\text{ - 1000} \\ 24V\text{ }\leq\text{ 120000} \\ \text{Divide both sides by 24} \\ \frac{24V}{24}\text{ }\leq\text{ }\frac{120000}{24} \\ V\text{ }\leq5000 \end{gathered}[/tex]Hence, the number of vacuums they can make per week can be between 2000 and 5000 or in inequality:
[tex]2000\text{ }\leq\text{ V }\leq\text{ 5000}[/tex]Answer:
Between 2000 and 5000 vacuums
Write an expression for the sequence of operations described below.1)) multiply 7 by 8, then divide f by the resultDo not simplify any part of the expression.Submit
Answers
We need to write an expression for the operations:
[tex]\begin{gathered} \text{ multiply 7 by 8} \\ \\ \text{dived f by the result} \end{gathered}[/tex]The first operation (multiplication) can be represented as:
[tex]7\cdot8[/tex]The second operation (the division of f by the previous result) can be represented as:
[tex]f\div(7\cdot8)[/tex]Notice that we need the parenthesis to indicate that the product is the first operation to be done.
Answer:
[tex]f\div(7\cdot8)[/tex]2. The area of the arena is 2160 in.2 a) Will the arena fit on the rug? Show your work and explain your answer below. b) If the length of the arena is 60 inches, what is the width? c) If the arena fits, and is placed exactly in the middle of the rug, how much standing room on the rug could a drive have? Use your measurements from above to help you. ? 3. If 15 robots can fit on the arena floor at one time, how much space does each robot take up?
Answers
Answers:
2a. The arena will fit on the rug
b. Width = 36 in
c. Standing room: 4 in
3. 144 in²
Explanation:
2. Part a.
First, we need to convert the measures of the rug to inches, so taking into account that 1 ft = 12 in, we get
Length = 6 ft x 12 in/ 1ft = 72 in
Width = 4 ft x 12 in/ 1 ft = 48 in
Then, the area of the rug will be
Area = Length x Width
Area = 72 in x 48 in
Area = 3456 in²
Therefore, the area of the arena, which is 2160 in² is lower than the area of the rug. It means that the area will fit on the rug.
Part b.
The area of the arena is equal to
Area = Length x Width
To find the width of the area, we need to solve the equation for the width, so
Width = Area/Length
So, replacing Area = 2160 in² and Length = 60 in, we get
Width = 2160 in² / 60 in
Width = 36 in
Therefore, the width of the area is 36 in.
Part c.
The measures that we get from parts a and b can be represented as
Therefore, the missing length can be calculated as:
(48 in - 36 in)/2 = 12 in/ 2 = 6 in
Therefore, a drive will have 6 in of standing room.
3.
Finally, to know how much space each robot take up, we need to divide the area of the arena by 15, so
2160 in²/ 15 = 144 in²
Therefore, each robot take 144 in²
8% of the students at Jemerson Middle School are absent because of illness. If there are 150 students in the school, how many are absent? 12015128
Answers
12 students
Explanation
when you have 8% , it means 8 of every 100 students are absent
find the decimal form
[tex]8\text{ \% = }\frac{8}{100}=0.08[/tex]then, to find the 8% of any number, just multiply the number by 0.08
Step 1
If there are 150 students in the school, how many are absent?
[tex]\begin{gathered} \text{absent}=\text{total}\cdot0.08 \\ \text{absent}=150\cdot0.08 \\ \text{absent}=12 \end{gathered}[/tex]so, 12 students are absent
Solve this system of equations by substitution. Enter your answer as an ordered pair (x,y). Do not use spaces in your answer.
Answers
We have the following:
[tex]\begin{gathered} y=-\frac{1}{2}x+4 \\ y=2x-1 \end{gathered}[/tex]Solving by substitution
[tex]\begin{gathered} -\frac{1}{2}x+4=2x-1 \\ 2x+\frac{1}{2}x=4+1 \\ \frac{5}{2}x=5 \\ x=\frac{2\cdot5}{5} \\ x=2 \end{gathered}[/tex]Now for y
[tex]\begin{gathered} y=2\cdot2-1=4-1=3 \\ \end{gathered}[/tex]Therefore, the answer is:
[tex](2,3)[/tex]During a game, 65% of the pitches Tina threw were strikes. She threw 120 2 poi total pitches during the game. How many throws were strikes? * a) 92 O b) 65 c) 78 d) 44