Answers
Step-by-step explanation:
(f○g)(x) = f(g(x))
so, we use the expression of g(x) as the "x" in f(x).
(f○g)(x) = f(g(x)) = 3(x² + 3) - 1 = 3x² + 9 - 1 = 3x² + 8
so,
(f○g)(3) = 3×3² + 8 = 27 + 8 = 35
check :
g(3) = 3² + 3 = 9 + 3 = 12
f(g(3)) = f(12) = 3×12 - 1 = 36 - 1 = 35
correct.
Related Questions
I = E/R solve for R
pls help
Answers
Step-by-step explanation:
I = E/R multiply both sides of the equation by R
I R = E / R * R
I R = E Now divide both sides by I
R = E/ I
What is the surface area of the triangular prism shown if a = 20 units, b = 29 units, c = 22 units, w = 15 units, and h = 21 units?
Answers
The surface area of the triangular prism is 1554 square units
Step-by-step explanation:
To find the surface area of a triangular prism, we need to add up the areas of all its faces. A triangular prism has two congruent triangular bases and three rectangular faces.
First, let's calculate the area of each triangular base. We can use the formula for the area of a triangle, which is A = 1/2 * base * height.
The base of each triangle is one of the sides of the triangle, and the height is the height of the prism, which is h = 21 units. We have:
Area of one triangular base = 1/2 * b * h = 1/2 * 29 * 21 = 304.5 square units
Since there are two congruent triangular bases, the total area of the bases is:
Total area of the bases = 2 * 304.5 = 609 square units
Next, let's calculate the area of each rectangular face. We can use the formula for the area of a rectangle, which is A = length * width.
The length of each rectangular face is one of the sides of the triangle, and the width is the height of the prism, which is h = 21 units. We have:
Area of one rectangular face = w * h = 15 * 21 = 315 square units
Since there are three rectangular faces, the total area of the rectangular faces is:
Total area of the rectangular faces = 3 * 315 = 945 square units
Finally, we can add up the areas of the bases and the rectangular faces to find the total surface area of the prism:
Total surface area = Total area of the bases + Total area of the rectangular faces
= 609 + 945
= 1554 square units
Therefore, the surface area of the triangular prism is 1554 square units.
Find the slope of the line shown in the graph. On a coordinate plane, a line goes through (0, negative 1) and (3, 0). a. 3 c. Negative one-third b. -1 d. One-third Please select the best answer from the choices provided
Answers
Answer:
The slope is [tex]\frac{-1}{3}[/tex]
Step-by-step explanation:
From the two points (0,-1) (3,0). The y values are 0 and -1. The x values are 3 and 0. The slope is the change in y over the change in x. You find the change by subtracting.
[tex]\frac{0-1}{3-0}[/tex] = [tex]\frac{-1}{3}[/tex]
Helping in the name of Jesus.
write an equation for the ellipse with vertices (-8, 5), (4, 5) and foci (-7,5),(3,5)
Answers
Answer:
Step-by-step explanation:
16
lbozo
What is the answer to this problem?
Answers
Thus, the event "Paulina gets injured playing soccer" is more likely to occur than the event "Paulina gets injured playing volleyball."
Answer:
C) neither. Both events are equally likely.
Step-by-step explanation:
help timed asap hurry
Answers
The value of y for the given quadrilateral is 5.
What is a quadrilateral?A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.
The Latin words quadra, which means four, and latus, which means sides, are combined to form the English term quadrilateral. A quadrilateral does not always have to have equal lengths on each of its four sides. As a result, depending on the sides and angles, we might have several sorts of quadrilaterals.
We know that the opposite sides of the quadrilateral have a sum of 180 degrees.
Thus,
23y + 13y = 180
36y = 180
y = 5
Hence, the value of y for the given quadrilateral is 5.
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Multiply and write the fraction or mixed number in simplest form 2/5x6/7=
Answers
Answer:
[tex]\frac{12}{35}[/tex]
Step-by-step explanation:
[tex]\frac{2}{5}[/tex] x [tex]\frac{6}{7}[/tex] = [tex]\frac{12}{35}[/tex]
[tex]\frac{12}{35}[/tex] is already in simplest form, so the answer is [tex]\frac{12}{35}[/tex]
Find the value of J.
Answers
The value of j is equal to 9√3 meters.
What is a 30-60-90 triangle?In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the sides are in the ratio 1:√3:2.
This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side).
Note: j is the side opposite to the 60° angle.
The side opposite to the 60° angle, j = 9 × √3
The side opposite to the 60° angle, c = 9√3 meters.
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Describe the graph of the equation x=4. Is the equation a function?
Answers
Answer:
x = 4 is a straight, vertical line at 4 on the x axis.
Jane won 9 out of the 10 video games she played. What percent of the video games did Jane win?
Convince Me!
When using an equivalent fraction to find a percent, why do you write 100 as the denominator?
Answers
Jane won 90% of the video games. The solution has been obtained by using the arithmetic operations.
What are arithmetic operations?
The four fundamental operations, sometimes known as "arithmetic operations," are believed to adequately characterise all real numbers. Division, multiplication, addition, and subtraction are followed by the mathematical operations quotient, product, sum, and difference.
We are given that Jane won 9 out of the 10 video games she played.
Let the percentage be x.
Using the multiplication operation, we get,
⇒ 10 * x% = 9
⇒ 10 * [tex]\frac{x}{100}[/tex] = 9
⇒ [tex]\frac{x}{10}[/tex] = 9
⇒ x = 90
Hence, Jane won 90% of the video games.
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Since, there are multiple questions so, the question answered above is:
Question: Jane won 9 out of the 10 video games she played. What percent of the video games did Jane win?
As a Mathematics teacher, list and explain the intervention strategies you would
apply in your teaching in assisting a child who is like Peter and why do you think your
strategies will be effective? Substantiate your response by
Answers
Answer:
1. Differentiated Instruction: This intervention involves tailoring lessons and activities to meet the individual learning needs and preferences of students. For example, a teacher can design various instructional materials for students with different skill levels to ensure they engage in the lesson and learn effectively.
2. Explicit Instruction: This intervention involves breaking down a complex process or concept into smaller, more manageable parts, and explaining each part clearly and explicitly. The teacher provides ample opportunities for students to practice and apply what they have learned.
3. Formative Assessment: This intervention involves using ongoing, informal assessments to monitor students' progress and provide feedback to both students and teachers to make necessary adjustments. Teachers can use formative assessments to identify areas of difficulty and intervene early.
4. Collaborative Learning: This intervention involves encouraging students to work together in pairs or small groups to complete tasks or solve problems. Within these groups, students share ideas, clarify misunderstandings, and learn from each other.
Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback are the strategies to assist the child.
What is Teaching?Teaching can be defined as engagement with learners to enable their understanding and application of knowledge, concepts and processes
Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback:
These strategies are effective because they are research-based and have been shown to support students' learning in mathematics.
By catering to the specific needs of students, providing individualized support, and making connections to the real world, students are more likely to engage in mathematics, understand mathematical concepts, and develop positive attitudes towards mathematics.
Hence, Differentiated Instruction, Small Group Instruction, Formative Assessment, Peer Tutoring, Real-World Connections and Positive Feedback are the strategies to assist the child.
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Riley was a spectator at his town's air guitar competition. Contestants were allowed to play either the acoustic or electric air guitar, but not both. Riley recorded which type of guitar each contestant played. He also counted the number of contestants wearing different kinds of pants, as there were some interesting stylistic choices.
What is the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants?
show all steps
Answers
The conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, is 0.6 or 60%.
What is Probability ?
Probability is the measure of the likelihood or chance of an event occurring. It is a mathematical concept that is used to quantify uncertainty and provide a basis for making informed decisions in various fields such as science, engineering, economics, and finance.
To find the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, we need to use the formula for conditional probability:
P(A | B) = P(A and B) / P(B)
where A is the event that the contestant played an acoustic guitar, and B is the event that the contestant wore leather pants.
We are not given the total number of contestants in the competition, but we can use the information provided in the table to calculate the probabilities of A and B, as well as the joint probability of A and B.
Let's first calculate the probability of B:
P(B) = number of contestants wearing leather pants / total number of contestants
P(B) = 15 / 45
P(B) = 1/3
Now, let's calculate the joint probability of A and B, i.e., the probability that a randomly selected contestant played an acoustic guitar and wore leather pants:
P(A and B) = number of contestants who played acoustic guitar and wore leather pants / total number of contestants
P(A and B) = 6 / 45
Finally, we can use the formula for conditional probability to find the probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants:
P(A | B) = P(A and B) / P(B)
P(A | B) = (6 / 45) / (1/3)
P(A | B) = 0.6
Therefore, the conditional probability that a randomly selected contestant played an acoustic guitar, given they wore leather pants, is 0.6 or 60%.
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The amount of money in a savings account increases by 0.2% every month.
Part A
Write a function for the amount of money in the account, B, after t months with an initial deposit of $100.
Part B
By what factor does the amount in the account increase every month? Every year? Every 5 years? Round answers to the nearest thousandth.
each month:
each year:
every five years:
Answers
Let B(t) be the amount of money in the account after t months with an initial deposit of $100.
Since the amount increases by 0.2% every month, the amount after t months can be found using the formula:
B(t) = 100 * (1 + 0.002)^t
Part B:
To find the factor by which the amount in the account increases every month, we need to calculate (1 + 0.002):
(1 + 0.002) ≈ 1.002
So the amount in the account increases by a factor of approximately 1.002 every month.
To find the factor by which the amount in the account increases every year, we need to calculate the factor for 12 months:
(1 + 0.002)^12 ≈ 1.026
So the amount in the account increases by a factor of approximately 1.026 every year.
To find the factor by which the amount in the account increases every 5 years, we need to calculate the factor for 60 months:
(1 + 0.002)^60 ≈ 1.136
So the amount in the account increases by a factor of approximately 1.136 every 5 years.
Part 1 of 3
O Points: 0 of 1
A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing
data are given in the table.
Labor-Hours per Ski
Trick Ski
9
1
Department
Fabricating
Finishing
Answer parts (A), (B), and (C) below.
Slalom Ski
6
1
Save
Maximum Labor-Hours
Available per Day
432
63
(A) If the profit on a trick ski is $60 and the profit on a slalom ski is $50, how many of each type of ski should be
manufactured each day to realize a maximum profit? What is the maximum profit?
The maximum profit is $. The maximum occurs when
trick skis and
slalom skis are produced.
Answers
The maximum profit of manufacturing company is = $3150
What do you mean by the term Linear Equation in one variable ?
If an equation is in ax + b = c and there is only one variable in the equation then it is called linear equation in one variable. a , b , c are real number and a is not equal to 0.
To maximize profit, we must determine the number of trick and slalom skis to produce each day, subject to the constraints of available labor hours.
Let x be the number of skis and y be the number of slalom skis produced per day.
The objective function to be maximized is:
Profit = 60x50 a
With the following limitations:
Production hours: 9x 6 a ≤ 432
Final hours: x y ≤ 63
Non-negativity constraints: x ≥ 0, y ≥ 0
We can solve this linear programming problem using graphical or algebraic methods. Here we use algebraic methods:
From the second limit, we get y ≤ 63 - x.
If this inequality is replaced by the first limit, we get:
9 x 6 (63 - x) ≤ 432
3x ≤ 54
x ≤ 18
This means that the maximum production rate is 18 pieces per day. Replacing x = 18 by the inequality y ≤ 63 - x, we get:
y ≤ 63 – 18 = 45
So the maximum production is 45 slalom skis per day.
To verify that this is the correct solution, we need to ensure that it satisfies the non-negativity constraints:
x ≥ 0 and y ≥ 0 are already given in the assignment statement.
Therefore, the maximum profit is:
Profit = 60(18) 50(45) = $3150
The biggest profit is when 18 trick skis and 45 slalom skis are made per day.
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IM BEGGING U SOMEONE
Answers
The dilation of the black figure to blue figure is a reduction with a scale factor of 0.5
How to find the scale of dilation?A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.
It can either be an enlargement or a reduction.
An enlargement means a scale factor greater than 1 while a reduction is a scale factor less than 1.
Thus, this dilation is a reduction.
Coordinates of initial figure are:
(-8, 10), (10, 10), (8, -8), (2, -8)
Coordinates of dilated figure are:
(1, -4), (4, -4), (-4, 5), (5, 5)
Thus, it is clear that the scale factor is 0.5
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convert to slope-intercept form (y=mx+b)
Answers
-3y = -5x + 15
Now divide completely isolate y by dividing -3 from both sides:
y = (-5/-3)x + (15/-3)
Simplify:
y = 5/3 x + (-5)
This becomes
y = 5/3 x - 5
or
y = 5x/3 - 5
A rectangular skating rink is 30 m by 20 m. It is to be doubled in area by adding a strip at one end and a strip of equal width along one side. Find the width of the strip.
Answers
The width of the added strip is 10 meters.
What is Area?
Area is a measure of the amount of surface enclosed by a two-dimensional shape or figure. It is usually expressed in square units, such as square meters or square inches, and is calculated by multiplying the length of a side or a diameter by itself or by using an appropriate formula for the shape.
If the area of the skating rink is doubled by adding a strip of equal width along one side and another strip at one end, then the new dimensions of the skating rink would be
=> (30 + x) × (20 + x),
where x is the width of the added strip.
According to the problem, the new area of the skating rink is double the original area, so we can set up the following equation:
=> (30 + x)(20 + x) = 2(30 ×20)
Expanding the left-hand side and simplifying, we get:
=> 600 + 50x + x² = 1200
Rearranging and solving for x, we get:
=> x² + 50x - 600 = 0
Factoring the left-hand side, we get:
=> (x + 60)(x - 10) = 0
Therefore, the possible values for x are x = -60 and x = 10. Since the width of the strip cannot be negative, we reject x = -60 and conclude that the width of the added strip is 10 meters.
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A surveyor is measuring the width of a pond. She chooses a landmark on the opposite side of the pond, and measures the angle to this landmark from a point 50 feet away from the original point. How wide is the pond to the nearest ft? A B 50 ft C
Answers
Answer: This issue can be resolved by employing trigonometry. Let's say that the surveyor is standing at point A, and the landmark on the other side of the pond is at point B. We can also make the assumption that the distance between points A and B is the width of the pond we need to find. Now, let's draw a line from point A to point B to represent the pond's width. Let's call this distance x. The surveyor's distance from the pond can also be represented by drawing a line 50 feet away from point A to point C.
The tangent of an angle is the opposite side (in this case, x) divided by the adjacent side (in this case, 50 feet), so we can now use the tangent function to determine x. We now have:
Rearranging this equation allows us to solve for x: tan(theta) = x/50, where theta is the angle between the lines from point A to point B and point A to point C.
We do not know the value of theta, but we are aware that the surveyor measured the angle to the landmark. x = 50*tan(theta) We should expect that the assessor estimated the point in degrees. Then, we can multiply the angle's tangent by 50 to get x using a calculator. For instance, if the surveyor measured an angle of 30 degrees, we would have:
The pond's width is therefore approximately 28.85 feet, rounded to the nearest foot, as x = 50tan(30) = 500.577. The response isn't given in the decisions, yet it is nearest to 29 feet
Step-by-step explanation:
HELPPP!!! ASAP SHOW WORK PLS
Answers
Therefore, only the points (4, 1) and (3, 8) can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem.
What is coordinate?In mathematics, a coordinate is a set of numbers that uniquely determines the position of a point or an object in space. The most common coordinate system is the Cartesian coordinate system, which uses two or three numerical values to locate points on a plane or in space.
Here,
To find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem, we need to use the coordinates of these two points to form a right-angled triangle, with the line segment connecting the two points as its hypotenuse. Therefore, the points that can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem are:
(4, 1) and (3, 8)
The point (0, 0) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.
The point (4, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.
The point (4, 3) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.
The point (1, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.
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A box contain 12 items of which 3 are defective.a sample of 3 items is selected at random from this box if x represents the number of defective items in the 3 selected items describe the random variable x
Answers
The random variable x can take on the values of 0, 1, 2, or 3, representing the number of defective items in the sample of 3 items.
Describing the random variable xThe random variable x represents the number of defective items in a sample of 3 items selected at random from the box.
Since there are 3 defective items out of 12 total items, the probability of selecting a defective item on the first draw is 3/12 or 1/4.
After the first draw, there are only 11 items remaining in the box, of which 2 are defective. Therefore, the probability of selecting a defective item on the second draw, given that the first draw was also defective, is 2/11.
Similarly, the probability of selecting a defective item on the third draw, given that the first two draws were also defective, is 1/10.
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We have been given that the ages of the winners of a cycling tournament are approximately bell-shaped. The mean age is 28.5 years, with a standard deviation of 3.5 years.
Answers
The z-score for an age of 30 years is 0.43.
What is confidence interval?A confidence interval (CI) for an unknown parameter in frequentist statistics is a range of estimations. The most popular confidence level is 95%, but other levels, such 90% or 99%, are occasionally used for computing confidence intervals. The fraction of related CIs over the long run that actually contain the parameter's true value is what is meant by the confidence level.
To convert an age to a z-score, we use the formula:
z = (x - μ) / σ
where:
x = the age we want to convert to a z-score
μ = the mean age
σ = the standard deviation of the ages
For example, if we want to convert an age of 30 years to a z-score using the given information with μ = 28.5 and σ = 3.5, we have:
z = (30 - 28.5) / 3.5
z = 0.43
So the z-score for an age of 30 years is 0.43.
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Complete question:
1. Explain the difference between an in-text citation and the reference list.
Answers
Answer:
In-text citations are the short citations you include in the written text that help a reader to understand which sources you are quoting or referring to in your writing example(Field, 2022). Reference is the full details of the source you have cited in your writing.
Need help, will give points.
Answers
The value of (f o g)(3) using composite function is 35
What is composite function?Composite function refers to the combining of functions in a manner where the output from one function becomes the input for the next function.
To calculate the composite function (f o g)(3), we use the techniques below.
From the question,
Given:
f(x) = 3x-1g(x) = x²+3Note that,
(f o g)(3) = f(g)(3)Step 1: Find g(3)
To find g(3) we suebtitute the value of x in g(x) by 3Therefore,
g(3) = 3²+3g(3) = 12Step 2: To find f(g)(3), we subetitute the value of x in f(x) by 12
f(g)(3) = 3(12)-1 f(g)(3) = 36-1f(g)(3) = 35Hence, (f o g)(3) is 35
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in an examination held it was discovered that 960 candidate took mathematics 780 took physics and 572 took chemistry .of these numbers 300 took both Mathematics and physics 340 took both Mathematics and chemistry 260 took both physics and chemistry while 130 candidates took all the three papers .how many candidates were there in that centre ?how many took Mathematics and physics? Only translate in a venn diagram
Answers
Answer 1452
Step-by-step explanation:
n(m n p n c) =n(m) + n(p) + n(c) - n(m n p) - n(m n c) - n(p n c) + n(m n p n c) = 960+ 780 + 572 - 300 - 340 - 260 + 130 = 1452
Translateeachexpression. Words
Expression
1. 2. 3. 4.
“five more than a number” “the quotient of a number and -2” “twice a number”
“a number decreased by 7”
“the product of a number and twelve” “three subtracted from a number” “earning $15 per hour”
“$5 less than the cost of the ticket”
“the sum of a number and 24”
“50 markers divided among several students” “two-fifths of a number”
“half a number minus nineteen”
“11 less than the product of a number and -4”
“the total of eight times a number and five”
“the difference of a number squared and 14”
“three-fourths of a number plus17
Answers
Answer:
Step-by-step explanation:
5 + x (where x represents the number)
x / (-2)
2x
x - 7
12x
x - 3
$15h (where h represents the number of hours worked)
C - $5 (where C represents the cost of the ticket)
x + 24
50 / n (where n represents the number of students)
(2/5)x
(1/2)x - 19
-4x - 11
8x + 5
x^2 - 14
(3/4)x + 17
----------------------------------
Five added to a number
The result of dividing a number by -2
Two times a number
A number decreased by 7
The result of multiplying a number by 12
A number decreased by 3
Earning $15 per hour
The cost of the ticket minus $5
The sum of a number and 24
Dividing 50 markers among several students
Two-fifths times a number
Half of a number minus 19
The result of subtracting 11 from the product of a number and -4
The sum of eight times a number and five
The difference between the square of a number and 14
Three-fourths times a number plus 17
after allowing 20% discount on the marked price of a smart boad, 15% vat was levied and sold it . If the different between the selling price with vat and selling price after discount is rs 4500, find the marked price of that smart board.
Answers
Answer:
mp = 37500
Step-by-step explanation:
let mp be x
D.A = 20×x/100
= 20x/100
=x/5
price after discount = mp - DA
= x-x/5 = 4x/5
VAT= 15%of 4x/5
= 15/100 × 4x/5
= 60x/500
sp with vat= 4x/5 +60x/500
= 460x/500
sp with vat - sp after discount = 4500
460x/500-4x/5=4500
60x= 4500×500
x= 37500
a rectangular park has a length of (x + 5) and a width of (x - 2). What is the area of the park?
Answers
The area of a rectangle is given by the formula:
Area = length x width
In this case, the length of the park is x + 5 and the width is x - 2. So, we can substitute these expressions into the formula to get:
Area = (x + 5) x (x - 2)
Expanding the product, we get:
Area = x^2 + 5x - 2x - 10
Simplifying, we have:
Area = x^2 + 3x - 10
Therefore, the area of the rectangular park is given by the expression x^2 + 3x - 10.
A rectangular room is 2 times as long as it is wide, and its perimeter is 48 meters. Find the dimensions of the room.
Answers
Answer: Let's assume that the width of the rectangular room is "x" meters. Then, according to the problem, the length of the room is 2 times the width, which means the length is "2x" meters.
The perimeter of the room is the sum of all four sides, so we can write:
Perimeter = 2 × (length + width)
Substituting the values, we get:
48 = 2 × (2x + x)
Simplifying the expression, we get:
48 = 2 × 3x
Dividing both sides by 2, we get:
24 = 3x
Solving for "x", we get:
x = 8
Therefore, the width of the room is 8 meters and the length is 2 times the width, which is 2 × 8 = 16 meters.
So the dimensions of the room are 8 meters by 16 meters.
Step-by-step explanation:
sherita is painting the hallway of her apartment she draws a graph illustrating the proportional relationship of the minutes she works x to the passes through the point(16,18)
Part A: which ordered pair on the line would display Sherita’s unit rate of square feet painted per minute.
Part B:select all of the points that lie on the line
(2,2.5)
(3.5,3.9375)
(4,5)
(32,36)
(10,11.5)
(0,0)
Answers
The answer of the given question based on graph is Part A: the unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125) and Part B: the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).
What is Slope?Slope refers to the steepness of a line or a surface. More specifically, slope is defined as the ratio of the vertical change (rise) between two points to the horizontal change (run) between the same two points on a line or a surface. In other words, slope measures how much a line or a surface rises or falls as it moves horizontally.
Since the relationship between the minutes worked and the square feet painted is proportional, we can use the slope-intercept form of a linear equation to represent it:
To find the equation of the line, we can use the given point (16,18) and the fact that the relationship is proportional. If we let the unit rate be r (in square feet per minute), then we have:
18 = r(16)
r = 18/16
r = 1.125
So the equation of the line is:
y = 1.125x
Part A: The unit rate of square feet painted per minute is 1.125, so the ordered pair that displays this unit rate is (1, 1.125), since the slope represents the unit rate in a proportional relationship.
Part B: To determine which of the points lie on the line, we can plug in the x-values into the equation y = 1.125x and see if the resulting y-values match.
(y - 18) / (x - 16) = slope
Using the slope calculated above, we can simplify this equation to:
y = 1.125(x - 16) + 18
Now, we can plug in the x and y values for each point and check if they satisfy this equation:
(2,2.5): 2.5 = 1.125(2 - 16) + 18, not on the line
(3.5,3.9375): 3.9375 = 1.125(3.5 - 16) + 18, on the line
(4,5): 5 = 1.125(4 - 16) + 18, on the line
(32,36): 36 = 1.125(32 - 16) + 18, on the line
(10,11.5): 11.5 = 1.125(10 - 16) + 18, not on the line
(0,0): 0 = 1.125(0 - 16) + 18, not on the line
Therefore, the points that lie on the line are (3.5,3.9375), (4,5), and (32,36).
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Answers
Hello!
Lets consider what the question asks for:
--> graphs of either line intersecting/parallel/identical
--> # of solutions the system has
The information given:
6x + y = 25
x + 3y = 3
Let's solve the FIRST problem
--> graphs of either line intersecting/parallel/identical
--> let's put both equations into slope-intercept form, where we
isolate the y-variable all by itself to one side
[tex]6x+3y=25\\3y=-6x+25\\\\y=-2x+\dfrac{25}{3}[/tex]
[tex]x+3y=3\\3y=-x+3\\y=-\dfrac{1}{3}x+1[/tex]
--> both equations AREN'T parallel as the slope (coefficient of x-
variable) aren't equal
--> both equations AREN'T identical as they don't look the same
--> both equations ARE INTERSECTING as they have different
slopes and aren't identical
Let's solve our second question:
--> how many solutions does the system have:
--> both lines intersect
-->both lines are linear as the highest power that the x-variable
has are 1
--> thus there is only ONE SOLUTION
Answer:
Intersecting
Has ONE SOLUTION