Answer:
Step-by-step explanation:
To evaluate the expression (8y)², where y = 5, we substitute the value of y into the expression and perform the operations.
First, substitute y = 5:
(8y)² = (8 * 5)²
Next, perform the operation inside the parentheses:
(8 * 5)² = 40²
Now, calculate the square of 40:
40² = 1600
Therefore, when y = 5, (8y)² is equal to 1600.
=
Find the x and y intercepts of the graph of the equation: y= |3x-7|. Select all that
apply.
2. Sandra's house is located at the point (2,2). The school is located at the point (7, 10). Each
unit on the graph represents 1 mi. How far is the school from Sandra's house? Remember to
show your work.
Plot and label your points on the coordinate plane (1 point)
Use the Pythagorean Theorem to calculate the diagonal distance between the two
points, express your answer as a radical and as a decimal rounded to nearest
hundredths.
Answer:
Step-by-step explanation:
Which linear function has the greatest y-intercept?
y = 6 x + 1
On a coordinate plane, a line goes through points (0, 2) and (5, 0).
On a coordinate plane, a line goes through points (1, 2) and (0, negative 3).
y = 3 x + 4
The linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].
In a linear equation, the y-intercept is where the line crosses the y-axis.
It is represented by the constant term in the equation.
So, to determine which linear function has the greatest y-intercept, we need to look at the constant term of each equation.
Let's consider each equation: [tex]y = 6x + 1[/tex]
The constant term in this equation is 1.
So, the y-intercept is 1.
On a coordinate plane, a line goes through points (0, 2) and (5, 0).
To find the equation of this line, we can use the point-slope form:
[tex]y - y1 = m(x - x1)[/tex]
where m is the slope and (x1, y1) is a point on the line.
Using the points (0, 2) and (5, 0), we get:
[tex]m = \frac{(0 - 2)}{(5 - 0)} =-\frac{2}{5}[/tex]
So, the equation of the line is:
[tex]y - 2 = (\frac{-2}{5} )(x - 0)[/tex]
[tex]y = (\frac{-2}{5} )x + 2[/tex]
The constant term in this equation is 2.
So, the y-intercept is 2.
On a coordinate plane, a line goes through points (1, 2) and (0, -3).
To find the equation of this line, we can use the point-slope form:
[tex]y - y1 = m(x - x1)[/tex]
where m is the slope and (x1, y1) is a point on the line.
Using the points (1, 2) and (0, -3), we get:
[tex]m = \frac{ (-3 - 2) }{(0 - 1)} = -5[/tex]
So, the equation of the line is:
[tex]y - 2 = (-5)(x - 1)y = -5x + 7[/tex]
The constant term in this equation is 7.
So, the y-intercept is 7.
[tex]y = 3x + 4[/tex]
The constant term in this equation is 4.
So, the y-intercept is 4.
Therefore, we can see that the linear function that has the greatest y-intercept is [tex]y = 3x + 4[/tex].
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IfmZC = 142° and m LI = 48°, find mU B.
The minor arc UB in the circle measures 46 degrees,
What is the measure of arc UB?The external angles theorem states that "the measure of an angle formed by two secant lines, two tangent lines, or a secant line and a tangent line from a point outside the circle is half the difference of the measures of the intercepted arcs.
It is expressed as;
External angle = 1/2 × ( major arc - minor arc )
From the diagram:
External angle I = 48 degrees
Major arc ZC = 142 degrees
Minor arc UB = ?
Plug these values into the above formula and find the minor arc UB:
External angle = 1/2 × ( major arc - minor arc )
48 = 1/2 × ( 142 - minor arc )
Multiply both sides by 2:
2 × 48 = 2 × 1/2 × ( 142 - minor arc )
2 × 48 = ( 142 - minor arc )
96 = ( 142 - minor arc )
96 = 142 - Minor arc
Minor arc = 142 - 96
Minor arc = 46°
Therefore, the minor arc measures 46 degrees.
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Simplify the expression: 2a(4b + 3a) +2ab
"Twice the difference of M and 14 equals 64."
Answer:
2(M - 14) = 64
M - 14 = 32
M = 46
An import company brought into the country some amplifiers that cost sh. 3,750-each. The government imposed an import duty of 125% and a sales tax of 20 %. If the company decided to make a 10% profit on the sales, calculate the selling price of each amplifier.
Answer:
To calculate the selling price of each amplifier, we need to consider the cost, import duty, sales tax, and the desired profit margin.
Cost of each amplifier: sh. 3,750
Import duty of 125% on the cost:
Import duty = 125% of sh. 3,750
= 125/100 * sh. 3,750
= sh. (125/100 * 3,750)
= sh. 4,687.50
Cost of each amplifier including import duty:
Total cost = Cost + Import duty
= sh. 3,750 + sh. 4,687.50
= sh. 8,437.50
Sales tax of 20% on the total cost:
Sales tax = 20% of Total cost= 20/100 * sh. 8,437.50
= sh. (20/100 * 8,437.50)
= sh. 1,687.50
Total cost including sales tax:
Total cost = Total cost + Sales tax
= sh. 8,437.50 + sh. 1,687.50
= sh. 10,125
Desired profit margin of 10% on the total cost:
Profit = 10% of Total cost
= 10/100 * sh. 10,125
= sh. (10/100 * 10,125)
= sh. 1,012.50
Selling price of each amplifier:
Selling price = Total cost + Profit
= sh. 10,125 + sh. 1,012.50
= sh. 11,137.50
Analyzing Wages-Plus-Commission Pay
Maybelle works in the shoe department for a high-end
store. She earns $15 per hour for at least 40 hours per
week. On top of that, she earns 10% commission for each
pair of shoes she sells. She is expected to sell $2,500
worth of shoes every week.
What will Maybelle earn in wages in a typical week?
How much commission will Maybelle earn if she sells
$3,000 of shoes in one week?
v
The next week, customers return $1,000 worth of
shoes. How much money will be deducted from
Maybelle's paycheck?
With an economic downturn, sales slump. As a result,
Maybelle will likely
a)In a typical week, Maybelle will earn a total of: $600 + $250 = $850
b) Her total earnings for the week would be: $600 + $300 = $900
c) Her paycheck will be reduced by $100.
d) Her wage earnings should remain the same assuming she still works at least 40 hours per week.
To calculate Maybelle's earnings in a typical week, we first need to determine the minimum number of hours she works. Since she earns $15 per hour for at least 40 hours per week, her minimum wage earnings are:
$15 x 40 = $600
In addition to her wage earnings, Maybelle earns a commission of 10% for each pair of shoes she sells. Since she is expected to sell $2,500 worth of shoes every week, her commission earnings are:
$2,500 x 0.10 = $250
b) If Maybelle sells $3,000 worth of shoes in one week, her commission earnings would be:
$3,000 x 0.10 = $300
On top of her wage earnings of $600 (based on working at least 40 hours at $15 per hour)
c) If customers return $1,000 worth of shoes the next week, Maybelle's commission earnings will not be affected. However, her wage earnings will be reduced by the amount of the returned shoes, which is $1,000 x 0.10 = $100.
d) With an economic downturn and sales slump, Maybelle's earnings from commission will likely decrease since she will sell fewer shoes. However, her total earnings will likely decrease due to the decrease in commission earnings.
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in a right triangle the hypotenuse is 17 and an adjacent side is 9. What is the measure of the angle opposite the adjacent sied?
★ Apply Phythagoras Theorem:-
[tex] \sf \longrightarrow \: {(adjacent \: side)}^{2} + {(opposite \: side)}^{2} = {(hypotenuse)}^{2} [/tex]
[tex] \sf \longrightarrow \: {(9)}^{2} + {(opposite \: side)}^{2} = {(17)}^{2} [/tex]
[tex] \sf \longrightarrow \: 81 + {(opposite \: side)}^{2} = {(17)}^{2} [/tex]
[tex] \sf \longrightarrow \: 81 + {(opposite \: side)}^{2} = 289[/tex]
[tex] \sf \longrightarrow \: {(opposite \: side)}^{2} = 289 - 81[/tex]
[tex] \sf \longrightarrow \: {(opposite \: side)}^{2} = 208[/tex]
[tex] \sf \longrightarrow \: opposite \: side= \sqrt{ 208}[/tex]
[tex] \sf \longrightarrow \: opposite \: side=14.422[/tex]
Devaughn's age is three times Sydney's age. The sum of their ages is 80 . What is Sydney's age?
[tex]\qquad\displaystyle \rm \dashrightarrow \: let \: \: Sydney's \: \: age \: \: be \: \: 'y'[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: Devaughn's \: \: age \: \: will \: \: be \: \: 3y[/tex]
Sum up ;
[tex]\qquad\displaystyle \tt \dashrightarrow \: 3y + y = 80[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: 4y = 80[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 80 \div 4[/tex]
[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 20[/tex]
So, Sydney's age is 20 years, n that of Devaughn is 20 × 3 = 60 years
Answer:
Sydney= 20, Devaughn= 60
Step-by-step explanation:
Let Sydney's age be 'x'
Devaughn's age = 3 times x = 3x
We Know That
The sum of their ages is 80.
So,
3x + x = 80
4x = 80
If we shift the 4 to the 80 side
x = 80/4
x = 20
So, Sydney's age is 20
Therefore, Devaughn's age =
3x = 3 times x
= 3 times 20
= 60
5 [11 marks] Question 4 - Exponents 1. A factory makes 1 000 000 000 yo-yos. Write the number of yo-yos in exponential form. (1) 2. Mrs. Penn has a vegetable garden. There are 8 rows and 8 columns in each bed. Each row and column is 1cm wide. Work out the number of small squares, each 1cm by 1cm in the grid and write it in exponential form. (1) [2 marks]
1.) The exponential form for the number of yo-yos, 1,000,000,000, is [tex]10^9[/tex]. This represents 10 raised to the power of the number of zeros in the original number.
2.) For Mrs. Penn's vegetable garden, with 8 rows and 8 columns of 1cm squares, the total number of small squares is 64. This can be expressed as [tex]2^6[/tex] , where 2 is the base number and 6 represents the exponent obtained from repeated multiplication.
1.) To write the number of yo-yos in exponential form, we can use the base 10 since we have a decimal system. The given number of yo-yos is 1,000,000,000. We can express it as 10 raised to the power of a certain exponent that represents the number of zeros in the original number.
In this case, the number has 9 zeros, so we can write it as [tex]10^9[/tex]. The exponential form for the number of yo-yos is [tex]10^9.[/tex]
2.) In Mrs. Penn's vegetable garden, there are 8 rows and 8 columns in each bed. Each row and column is 1cm wide. We need to calculate the number of small squares, each measuring 1cm by 1cm, in the grid.
Since there are 8 rows and 8 columns, we can multiply these two numbers together to find the total number of small squares. 8 multiplied by 8 equals 64.
The exponential form represents repeated multiplication of the base number. In this case, the base number is 64 since we have 64 small squares. To write it in exponential form, we need to determine the exponent that represents the number of times 64 is multiplied by itself.
Since 64 is 2 raised to the power of [tex]6 (2^6)[/tex], we can express it as [tex](2^6)^1[/tex]. Simplifying this, we get [tex]2^(6*1)[/tex] which equals [tex]2^6.[/tex]
Therefore, the exponential form for the number of small squares in the grid is [tex]2^6.[/tex]
In summary, the number of yo-yos in exponential form is [tex]10^9[/tex], and the number of small squares in the grid is [tex]2^6.[/tex]
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A dime is flipped, and a single die is rolled. Find the odds against obtaining a head.
Answer:
11 : 1
Step-by-step explanation:
The probability of obtaining a head when a dime is flipped is 1/2, since there are two possible outcomes (heads or tails) and each is equally likely.
The probability of rolling any particular number on a fair six-sided die is 1/6, since there are six equally likely outcomes (the numbers 1 through 6).
To find the odds against obtaining a head and rolling any number on the die, we need to multiply the probabilities of the two events. This gives us
(1/2) x (1/6) = 1/12
So the probability of obtaining a head and rolling any number on the die is 1/12.
To find the odds against this event, we need to compare the probability of the event happening to the probability of it not happening. The probability of the event not happening is 1 - 1/12 = 11/12.
Therefore, the odds against obtaining a head and rolling any number on the die are:
11 : 1
please answer i am stuck
(a) To find the assets in 2011 using the given information: A. To find the assets in 2011, substitute 11 for x and evaluate to find A(x).
In 2011 the assets are about $669.6 billion
(b) To find the assets in 2016 using the given information: B. To find the assets in 2016, substitute 16 for x and evaluate to find A(x).
In 2016 the assets are about $931.5 billion.
(c) To find the assets in 2019 using the given information: B. To find the assets in 2019, substitute 19 for x and evaluate to find A(x).
In 2019 the assets are about $1135.4 billion.
How to estimate the cost of the assets in 2011?Based on the information provided, we can logically deduce that the assets for a financial firm can be approximately represented by the following exponential function:
[tex]A(x)=324e^{0.066x}[/tex]
where:
A(x) is in billions of dollars.x = 7 corresponds to the year 2007.For the year 2011, the cost (in billions of dollars) is given by;
x = (2011 - 2007) + 7
x = 4 + 7
x = 11 years.
Next, we would substitute 11 for x in the exponential function:
[tex]A(11)=324e^{0.066 \times 11}[/tex]
A(11) = $669.6 billions.
Part b.
For the year 2016, the cost (in billions of dollars) is given by;
x = (2016 - 2007) + 7
x = 9 + 7
x = 16 years.
Next, we would substitute 16 for x in the exponential function:
[tex]A(16)=324e^{0.066 \times 16}[/tex]
A(16) = $931.5 billions.
Part c.
For the year 2019, the cost (in billions of dollars) is given by;
x = (2019 - 2007) + 7
x = 12 + 7
x = 19 years.
Next, we would substitute 19 for x in the exponential function:
[tex]A(19)=324e^{0.066 \times 19}[/tex]
A(19) = $1135.4 billions.
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Linear sequence of 35/100,5/10,65/100
The linear rule for the sequence is f(n) = 7/20 + 3/20(n - 1)
Finding the linear rule for the sequenceFrom the question, we have the following parameters that can be used in our computation:
35/100,5/10,65/100
In the above sequence, we can see that 15/100 is added to the previous term to get the new term
This means that
First term, a = 35/100
Common difference, d = 15/100
The nth term is then represented as
f(n) = a + (n - 1) * d
Substitute the known values in the above equation, so, we have the following representation
f(n) = 35/100 + 15/100(n - 1)
So, we have
f(n) = 7/20 + 3/20(n - 1)
Hence, the explicit rule is f(n) = 7/20 + 3/20(n - 1)
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Find the perimeter of the following square and the area of the shaded part if the radius of the
circle is 12 cm.
The perimeter of the square and the area of the shaded part are 48cm and 454.16cm² respectively²
How to determine the valueThe formula for calculating the area of a circle is expressed as;
Area = πr²
Now, substitute the value, we have;
Area = 3.14 × 12²
Find the square and multiply the values, we have;
Area = 3.14(144)
Multiply the values
Area = 452. 16 cm²
Perimeter of a square take the formula;
Perimeter = 4a
Substitute the value
Perimeter = 4(12) = 48 cm
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What is the coefficient in the expression
6-4x-8+2
Answer:
-4 is the coefficient or -4x whatever the answers are
Step-by-step explanation:
the coefficient in mathematics is basically whatever the number is infront of a variable in an expression, equation, etc.
100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answer:
Step-by-step explanation:
An electric utility company determines the monthly bill for a residential customer by adding an energy charge of 9.32 cents per kilowatt-hour to its base charge of $18.32 per month. Write an equation for the monthly charge y in terms of x, the number of kilowatt-hours used.
Answer: y = 18.32 + 0.0932x
Step-by-step explanation:
Given that the base charge per month is $18.32 and the charge per kilowatt-hour is 9.32 cents, which is assigned to the variable x, we can write an equation to find the total monthly charge based on kilowatt-hours.
total = monthly charge + kilowatt-hour x the amount of kilowatts
y = 18.32 + 0.0932x
here we see that y is the total cost per month, our base monthly charge is $18.32, our kilowatt hour charge is 9.32 cents, which we write in terms of dollar amounts for the sake of the equation (just divide by 100), and our variable x represents the number of kilowatts.
The perimeter of a basketball court is 96 meters and the length is 6 meters longer than twice the width, what are the length and width?
Answer:
the length of the basketball court is 34 meters and the width is 14 meters.
Step-by-step explanation:
According to the given information, the length is 6 meters longer than twice the width. Therefore, the length can be expressed as 2x + 6.
The perimeter of a rectangle is calculated by adding all four sides. In this case, the perimeter is given as 96 meters.
Perimeter = 2(length + width)
Plugging in the values, we have:
96 = 2((2x + 6) + x)
Simplifying the equation:
96 = 2(3x + 6)
96 = 6x + 12
6x = 96 - 12
6x = 84
x = 84/6
x = 14
So, the width of the basketball court is 14 meters.
To find the length, we can substitute the value of x back into the expression for the length:
Length = 2x + 6
Length = 2(14) + 6
Length = 28 + 6
Length = 34
What’s the temperature change in 120 to -20
Answer: 140
Step-by-step explanation: 120-(-20) to 120 + 20 goes to 140.
Given the expenses and income below, what is the back ratio? Monthly expenses: Mortgage (including all housing costs) = $1,982 Student loan = $258 Minimum credit card payments = $184 Home equity loan = $237 Monthly income: Salary = $4,115 Bonus = $700 Side business = $1,000 Dividends/interest = $95
Solve the missing element . use 3.14 for pi and Area = pi r2 ; C= pi D
We can solve for the missing elements as follows:
1. Radius - 10 inches
Diameter - 20
Circumference - 62.8
Area - 314
2. Radius - 6ft
Diameter - 12
Circumference - 37.68
Area - 113.04
3. Radius - 18
Diameter - 36 yards
Circumference - 113.04
Area - 1017.36
4. Radius 15
Diameter - 30 cm
Circumference 94.2
Area - 706.5
5. Radius - 5 mm
Diameter 10
Circumference 31.4
Area -78.5
6. Radius 20
Diameter - 40 inches
Circumference 125.6
Area -1256
How to solve for the valuesTo solve for the given values, we will use the formulas for area, circumference. Also, we can obtain the radius by dividing the diameter by 2 and the diameter is 2r. So we will solve for the values this way:
1. radius = 10 inches
diameter = 20
circumference = 2pie*r 2 *3.14*10 = 62.8
Area = 314
2. radius = 6ft
diameter = 12
circumference = 37.68
Area = 113.04
3. radius = 18
diameter = 36 yards
circumference = 113.04
Area = 1017.36
4. radius = 15
diameter = 30 cm
circumference = 94.2
Area = 706.5
5. radius = 5 mm
diameter = 10
circumference = 31.4
Area = 78.5
6. radius = 20 inches
diameter = 40 inches
circumference = 125.6
area = 1256
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The product of two irrational numbers is
rational. (Sometimes,Never,always)?
The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.
The product of two irrational numbers can be either rational or irrational, depending on the specific irrational numbers being multiplied. It is not always rational, nor is it never rational.
Consider the square root of 2 (√2) and the square root of 3 (√3), both of which are irrational numbers. When you multiply √2 and √3, you get √6, which is also an irrational number. In this case, the product of two irrational numbers is irrational.
However, there are cases where the product of two irrational numbers can be rational. For example, consider √2 and its reciprocal (1/√2), both of which are irrational. When you multiply these two numbers, you get 1, which is a rational number. So, in this case, the product of two irrational numbers is rational.
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f.) Use the regression line to calculate the best predicted value for x = 32. (Round to two decimal places)
y =
The best predicted value for x = 32 is -38333.33
Estimating the predicted value at x = 32 for the regression lineFrom the question, we have the following parameters that can be used in our computation:
The scatter plot (see attachment)
We have the following points from the line of best fit
(3, 10000) and (0, 15000)
The linear equation is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 15000
Using the other point, we have
10000 = 3m + 15000
So, we have
3m = -5000
Divide by 3
m = -5000/3
So, the equation is y = -5000/3x + 15000
When x = 32, we have
y = -5000/3 * 32 + 15000
Evaluate
y = -38333.33
Hence, the best predicted value for x = 32 is -38333.33
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Bobby bought 5 books for $215.15. If the books cost the same, what
amount did he pay for each book?
$32.82
$43.03
$52.84
$62.85
Answer:
Step-by-step explanation:
If bobby bought 5 books for 215.15 and they all cost the same, all you need to do is divide 215.15 by 5.
215.15/5=43.03
The answer to this problem is 43.03.
what steps are needed to find the equation of a line given the graph?
The equation of the line is y = x.
To find the equation of a line given its graph, you need to follow the steps below.
Step 1: Determine the slope of the line.The slope of the line can be determined using the formula: slope = rise/run or m = Δy/Δx. Rise is the change in the y-coordinates and run is the change in the x-coordinates.
Step 2: Determine the y-intercept of the line.The y-intercept is the point where the line intersects the y-axis. You can determine the y-intercept by looking at the point where the line crosses the y-axis on the graph. The y-intercept is denoted by the letter b.
Step 3: Write the equation of the lineThe equation of the line can be written in slope-intercept form, which is y = mx + b. The slope (m) and y-intercept (b) that were determined in steps 1 and 2 are used to substitute into this equation. Thus, the equation of the line becomes y = slope(x) + y-intercept.
Example:Let's say you are given the graph of a line below: .
Step 1: Determine the slope of the line.To determine the slope of the line, you need to choose two points on the line and calculate the rise and run. Let's choose the points (2, 1) and (4, 3). The rise is 2 (3 - 1) and the run is 2 (4 - 2). Therefore, the slope of the line is: m = 2/2 = 1.
Step 2: Determine the y-intercept of the lineThe line crosses the y-axis at the point (0, 0). Therefore, the y-intercept of the line is b = 0.
Step 3: Write the equation of the line.The equation of the line in slope-intercept form is y = mx + b. Substituting the slope and y-intercept into this equation gives: y = 1x + 0 or y = x.
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A zoo has categorized its visitors into three categories: member, school, and general. The member category refers to visitors who pay an annual fee to support the zoo. Members
receive certain benefits such as discounts on merchandise and trips planned by the zoo. The school category includes faculty and students from day care and elementary and
secondary schools; these visitors generally receive a discounted rate. The general category includes all other visitors. The zoo has been concerned about a recent drop in
attendance. To help better understand attendance and membership, a zoo staff member has collected the following data.
Attendance
School
Visitor
Category 2011
General 154,213 159,204 163,933 169,606
Member 114,023 103,295
96,937 79,717
82,385 79,376
81,470
80,790
Total 350,621 341,875 342,340
330,113
(0) Construct a bar chart of total attendance over time. Comment on any trend in the data.
(b) Construct a side-by-side bar chart showing attendance by visitor category with year as the variable on the horizontal axis.
(c) Comment on what is happening to zoo attendance based on the charts from parts (a) and (b).
2012
2013
2014
(a) Based on the total attendance data provided, a bar chart can be constructed to visualize the attendance trends over time. The horizontal axis represents the years (2011, 2012, 2013, 2014), and the vertical axis represents the total attendance. The bars will correspond to the attendance numbers for each year.
(b) A side-by-side bar chart can be created to compare the attendance by visitor category with the year as the variable on the horizontal axis. The visitor categories (school, general, member) will be represented by different colors or patterns, and the bars will show the attendance for each category in each year.
(c) By analyzing the charts from parts (a) and (b), we can observe the trends in zoo attendance.
(a) The bar chart of total attendance over time shows the following trend: In 2011, the attendance was 350,621. It decreased slightly in 2012 to 341,875, remained relatively stable in 2013 at 342,340, and then decreased again in 2014 to 330,113.
(b) The side-by-side bar chart showing attendance by visitor category with year as the variable on the horizontal axis provides a visual comparison of attendance for each category over the years.
The chart reveals that the general category consistently had the highest attendance throughout the years, followed by the school category. The member category consistently had the lowest attendance.
(c) Based on the charts from parts (a) and (b), it can be observed that there has been a general decline in zoo attendance over the years, with a notable decrease from 2011 to 2014.
The drop in attendance could indicate a potential issue that the zoo needs to address in order to attract more visitors.
Additionally, the consistently lower attendance in the member category suggests that the zoo may need to reconsider its membership benefits and strategies to retain and attract more members.
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Which type of conic section is defined by the equation:... 100pts
Answer:
This is an equation of a parabola.
[tex](y+6)^2=4(x+1)[/tex]
Step-by-step explanation:
A conic section is a curve obtained by the intersection of a plane and a cone. The three major conic sections are parabola, hyperbola and ellipse (the circle is a special type of ellipse).
The standard equations for hyperbolas and ellipses all include x² and y² terms. The standard equation for a parabola includes the square of only one of the two variables.
Therefore, the equation y² - 4x + 12y + 32 = 0 represents a parabola, as there is no x² term.
As the y-variable is squared, the parabola is horizontal (sideways), and has an axis of symmetry parallel to the x-axis.
The conic form of a sideways parabola is:
[tex]\boxed{(y-k)^2=4p(x-h)}[/tex]
where:
(h, k) is the vertex.(h+p, k) is the focus.x = h-p is the directrix.To write the given equation in conic form, we need to complete the square for the y-variable.
Rearrange the equation so that the y-terms are on the left side:
[tex]y^2 + 12y = 4x - 32[/tex]
Add the square of half the coefficient of the y-term to both sides of the equation:
[tex]y^2 + 12y+\left(\dfrac{-12}{2}\right)^2 = 4x - 32+\left(\dfrac{-12}{2}\right)^2[/tex]
[tex]y^2 + 12y+\left(-6\right)^2 = 4x - 32+\left(-6\right)^2[/tex]
[tex]y^2 + 12y+36 = 4x - 32+36[/tex]
[tex]y^2 + 12y+36 = 4x +4[/tex]
Factor the perfect square trinomial on the left side of the equation:
[tex](y+6)^2=4x+4[/tex]
Factor out the coefficient of the x-term from the right side of the equation:
[tex](y+6)^2=4(x+1)[/tex]
Therefore, the equation of the given conic section in conic form is:
[tex]\boxed{(y+6)^2=4(x+1)}[/tex]
where:
(-1, -6) is the vertex.(0, -6) is the focus.x = -2 is the directrix.The conic section of the equation y² - 9x + 12y + 32 = 0 is a parabola
Selecting the conic section of the equationThe given equation is
y² - 9x + 12y + 32 = 0
The above equation is an illustration of a parabola equation
The standard form of a parabola is
(x - h)² = 4a(y - k)²
Where
(h, k) is the center
While the general form of the equation is
Ax² + Dx + Ey + F = 0
In this case, the equation y² - 9x + 12y + 32 = 0 takes the general form
Read more about conic section at
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An isosceles triangle has two angles both equal to x. The third angle is 45 degrees bigger than either of these. Find the value of x.
Let's use the fact that the sum of the angles of a triangle is always 180 degrees to solve this problem. Let the two equal angles be x, then the third angle is x + 45.Let's add all the angles together:x + x + x + 45 = 180Simplifying this equation, we get:3x + 45 = 180Now, we need to isolate the variable on one side of the equation. We can do this by subtracting 45 from both sides of the equation:3x = 135Finally, we can solve for x by dividing both sides of the equation by 3:x = 45Therefore, the value of x is 45 degrees.
Answer:
45°
Step-by-step explanation:
An isosceles triangle has two angles both equal to x. The third angle is 45 degrees bigger than either of these. Find the value of x.Let's turn the question into an equation
180 = x + x + x + 45
180 - 45 = 3x
135 = 3x
x = 135 : 3
x = 45°
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check
180 = 45 + 45 + 45 + 45
180 = 180
same value the answer is good
What is the perimeter of a rectangle with workings
Answer:
See below
Step-by-step explanation:
The perimeter of a rectangle is [tex]P=2L+2W[/tex] where L is the length and W is the width.