Answer:
[tex]\textsf{a)} \quad \dfrac{1}{2}[/tex]
[tex]\textsf{b)} \quad -2[/tex]
[tex]\textsf{c)} \quad a = -4,\;\; a = 6[/tex]
Step-by-step explanation:
Part (a)[tex]\boxed{\begin{minipage}{9cm}\underline{Gradient Formula}\\\\Gradient $=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}[/tex]
Given points:
P = (a, a-2)Q = (4-3a, -a)Substitute the given points into the gradient formula to find the gradient of line PQ:
[tex]\textsf{Gradient}=\dfrac{-a-(a-2)}{(4-3a)-a}=\dfrac{-a-a+2}{4-3a-a}=\dfrac{2-2a}{4-4a}=\dfrac{2(1-a)}{4(1-a)}=\dfrac{2}{4}=\dfrac{1}{2}[/tex]
Part (b)If two lines are perpendicular to each other, their gradients are negative reciprocals.
Therefore, the gradient of a line perpendicular to PQ is:
[tex]\implies \textsf{Gradient}=-2[/tex]
Part (c)[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]
Substitute the points and the given distance 10√5 into the formula and solve for a.
[tex]\implies 10\sqrt{5}=\sqrt{((4-3a)-a)^2+(-a-(a-2))^2}[/tex]
[tex]\implies 10\sqrt{5}=\sqrt{(4-3a-a)^2+(-a-a+2)^2}[/tex]
[tex]\implies 10\sqrt{5}=\sqrt{(4-4a)^2+(2-2a)^2}[/tex]
[tex]\implies 500=16-32a+16a^2+4-8a+4a^2[/tex]
[tex]\implies 500=20a^2-40a+20[/tex]
[tex]\implies 25=a^2-2a+1[/tex]
[tex]\implies a^2-2a-24=0[/tex]
[tex]\implies a^2-6a+4a-24=0[/tex]
[tex]\implies a(a-6)+4(a-6)=0[/tex]
[tex]\implies (a+4)(a-6)=0[/tex]
Apply the zero-product property:
[tex]\implies a+4=0 \implies a=-4[/tex]
[tex]\implies a-6=0 \implies a=6[/tex]
Therefore, the two possible values of a are:
a = -4, a = 6(03.02, 03.03, 03.04 MC)
The price of products may increase due to inflation and decrease due to depreciation. Marco is studying the change in the price of two products, A and B, over time.
The price f(x), in dollars, of product A after x years is represented by the function below:
f(x) = 0.69(1.03)*
Part A: Is the price of product A increasing or decreasing and by what percentage per year? Justify your answer. (5 points)
Part B: The table below
hows the price f(t), in dollars, of product B after t years:
t (number of years)
1
2
3
f(t) (price in dollars) 10,100 10,201 10,303.01
4
10,406.04
1) The price of part A is increasing by 3% per year.
2) Product A recorded a greater percentage change in price over the previous year.
How to calculate the percentage change in price?The percentage change in price is calculated by the formula,
% change = (New price - old price)/old price × 100
If the difference between the new price and the old price is positive, then it is said to be increasing and if it is negative, it is said to be decreasing.
Calculation:It is given that,
Maco is studying the change in the price of two products, A and B, over time.
Part A: For product A, the price in dollars is given by f(x) and the time is given by x.
I.e., f(x) = 0.69(1.03)ˣ
From this, product A has f(1) = 0.69(1.03)¹ = 0.71 and f(2) = 0.69(1.03)² = 0.73.
Then, the difference between f(2) and f(1) for product A is
f(2) - f(1) = 0.73 - 0.71 = 0.02
So, the % change of product A is increasing per year and it is
% increase = (f(2) - f(1))/f(1) × 100 = (0.02/0.71) × 100 = 2.89 ≅ 3%
Thus, the percentage of increase in product A is 3%.
Part B: For product B, the price in dollars is given by f(t) and the time is given by t.
I.e.,
t (in years) - 1 2 3 4
f(t) (price) - 10,100 10,201 10,303.01 10,406.04
From this, product B has f(1) = 10,100 and f(2) = 10,201
Then, the difference between f(2) and f(1) for product B is
f(2) - f(1) = 10,201 - 10,100 = 101
So, the % change of the product B is increasing per year and it is
% increase = (f(2) - f(1))/f(1) × 100 = (101/10100)× 100 = 1%
Therefore, product A has a greater percentage of increase than product B.
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Katarina tosses a thumbtack on the ground 202020 times to see if it lands on its side or if it lands with the point facing up (there are no other possible outcomes). Let s=s=s, equals the number of times it lands on its side. Is sss a binomial variable? why or why not?.
Each prerequisite for a binomial variable is met in this circumstance, hence SSS has a binomial distribution.
What is a binomial variable?In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a series of n independent experiments, each asking a yes-or-no question and having its own Boolean-valued outcome: success (with probability p) or failure (with probability q=1-p). For a single trial, or n = 1, the binomial distribution is a Bernoulli distribution. A single success/failure experiment is also known as a Bernoulli trial or experiment, while a succession of results is known as a Bernoulli process. The widely used binomial statistical significance test is based on the binomial distribution.
True. Each trial has two possible results (it either lands on one side or it doesn't), each trial's findings are independent, there are a fixed number of trials (202020), and each trial has the same chance of success.
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if a person had an initial roland-morris score of 20 and a final score of 6, what would the percent improvement be?
70% percentage of the original roland-morris score of 20 was improved to the final score of 6.
Given that,
We have to find what percentage of the original roland-morris score of 20 was improved to the final score of 6.
We know that,
The original roland-morris score of 20 was improved to the final score of 6.
The initial score = 20
Final Score = 6
The improvement is 20 - 6 = 14
So,
The percent improvement is (14/20)* 100 = 70%
Therefore, 70% percentage of the original roland-morris score of 20 was improved to the final score of 6.
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If you took a 3 and 4, what are the two different fractions you could make?
Answer:
Step-by-step explanation:
If you took a 3 and 4 the two fractions you could make are 3/4 and 4/3
Find the volume of a cube with one edge of length 7.6 cm.
The required volume of the cube is 439cm³.
What is a cube?A cube is a three-dimensional solid object in geometry that is surrounded by six square faces, facets, or sides, three of which meet at each vertex.
It seems to be a hexagon when viewed from a corner, and its net is frequently portrayed as a cross.
One of the five Platonic solids, the cube is the only regular hexahedron.
So, the cube volume formula:
V = a³
We have the length of the side of the cube which is 7.6.
Now, insert it in the formula as follows:
V = a³
V = 7.6³
V = 7.6 * 7.6 * 7.6
V = 438.976
Rounding off: 439cm³
Therefore, the required volume of the cube is 439cm³.
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Please help!!! i need this done by tonight!!
Evaluate 11.5x + 10.9y when x = 6 and y =7
The value of the algebraic expression 11.5x + 10.9y at x = 6 and y = 7 is 145.3
What is an algebraic expression?
Algebraic expression consists of variables and numbers connected with addition, subtraction, multiplication and division
The given algebraic expression is 11.5x + 10.9y
We have to find the value of the algebraic expression at x = 6 and y = 7
Putting x = 6 and y = 7 in the algebraic expression,
[tex]11.5 \times 6 + 10.9 \times 7[/tex]
69 + 76.3
145.3
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What is the meaning of ASA SAS and SSS postulate?
The aforementioned postulates - ASA, SAS, SSS - are defined and explained below.
The given theorems are used in order to prove the congruency of triangles. If we are presented with a problem in which we are required to prove the congruency of two triangles, we can use these three postulates to prove it.
The three postulates are as thus -
SSS postulate: if 3 sides of the given triangles are equal, then the triangles are congruent.
SAS postulate: If 2 sides and an angle of the given triangles are equal, then the triangles are congruent.
ASA postulate: If 2 angles and a side (in the middle of the angles)of the given triangles are equal, then the triangles are congruent.
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5x-180x² factor completely
Answer:
-5(36x-1)x
Step-by-step explanation:
Margo was helping clean up after a class party. There were 3 boxes remaining with pizza in them. Each box had 3 of a pizza left. How much pizza was left in all?
Answer:
9 pizzas
Step-by-step explanation:
Since there were 3 boxes which each 3 pizzas, you're gonna need to solve using an easy multiplication problem; 3*3, which is 9. So it's 9 pizzas
A company orders 30 boxed lunches from a deli for $295.50. If each boxed lunch costs the same amount, how much do 9 boxed lunches cost?
Simplify fully
4x2 + 4x
—————
2x2 - 2
Answer:
2x
x-1
Step-by-step explanation:
Graph this line using interepts
2x-5y=10
Answer:
2-5 will give you -3 then 10 divide by -3 will give you 7
A group of friends wants to go to the amusement park. They have no more than $80 to spend on parking and admission. Parking is $14.75, and tickets cost $11.25 per person, including tax. Write and solve an inequality which can be used to determine xx, the number of people who can go to the amusement park.
Answer:
To determine the number of people who can go to the amusement park, we need to write an inequality that represents the total cost of parking and admission, which is the maximum amount of money that the group has. Let x represent the number of people who can go to the amusement park. The total cost of parking and admission can be represented by the inequality 14.75 + 11.25x ≤ 80. To solve this inequality, we need to find the values of x that make the inequality true. We can do this by isolating the variable on one side of the inequality. To do this, we need to subtract 14.75 from both sides of the inequality: 14.75 + 11.25x ≤ 80
-14.75 -14.75
11.25x ≤ 65.25
Next, we need to divide both sides of the inequality by 11.25 to isolate the variable on one side of the inequality: 11.25x ≤ 65.25
x ≤ 5.77
Since the number of people must be a whole number, we can round down to find the maximum number of people who can go to the amusement park: x ≤ 5
This means that no more than 5 people can go to the amusement park if they want to stay within their budget.
Step-by-step explanation:
here is the graph of a linear equation. Select all true statements about the line and its equation one point is -1, 1 and another point is 3, 2
The linear function that goes through the points (-1, 1) and (3,2) is defined as follows:
y = 0.5x + 1.5.
How to define the linear function?The slope-intercept of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change of the function.b is the intercept, representing the value of y at which the graph of the function crosses the y-axis.The two points on the graph of the function are given as follows:
(-1,1) and (3,2).
Given two points, the slope is calculated as the change in y divided by the change in x, hence:
m = (2 - 1)/(3 - (-1)) = 2/4 = 0.5.
Hence:
y = 0.5x + b.
When x = -1, y = 1, hence the intercept b is obtained as follows:
1 = 0.5(-1) + b
b = 1.5.
Thus the equation is:
y = 0.5x + 1.5.
Missing InformationThe problem is incomplete, hence I will suppose that it asks for the definition of the equation, which will help you find whichever the correct statements are.
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Liam is solving the equation for `x`. Original equation: `12x+4=20x-12` The result of Liam's first step was `4=8x-12`. Describe the first step Liam made for the equation.
Liam has moved the variable 1x2 to the RHS and combined the like terms.
What is a linear function?A straight line on the coordinate plane is represented by a linear function. As an illustration, the equation y = 3x – 2 depicts a linear function because it is a straight line in the coordinate plane.
Given, Liam is solving the equation for 'x' for the equation:
12x + 4 = 20x - 12.
Now the result of his first step is 4 = 8x - 12.
So, what he did is moved 12x to the RHS and combined the like terms which is,
4 = 20x - 12x - 12.
4 = 12x - 12.
We can also solve for 'x'
4 = 12x - 12.
+ 12 + 4 = 12x.
16 = 12x.
x = 16/12.
x = 4/3.
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2(x- 4) + 3x- 17 for x = 3
Answer:
-10
Step-by-step explanation:
i just did it
Work out
Give your
(5 x 10^3) x (9 x 10^7)
answer in standard form.
Answer:
To find the product of 5 x 10^3 and 9 x 10^7, we need to multiply the two numbers. The first step is to multiply the two numbers without the exponents, which gives us 5 x 9 = 45. Next, we need to add the exponents to find the exponent of the final product. 10^3 and 10^7 have exponents of 3 and 7, respectively, so the final exponent is 3 + 7 = 10. This means that the final product is 45 x 10^10, or 4.5 x 10^11 in standard form.
Step-by-step explanation:
A line has a slope of -½ and a y-intercept of -6. What is an equation of the line in slope-intercept form? A y=-1/2x-6 B y-1/2x=-6 C y-6=-1/2x D y=1/2x-6
The equation of the line in slope-intercept form y = -1/2x + 6
What is the Slope Intercept Form ?The slope intercept form of a straight line is one of the most common ways to show the equation of a line. The slope intercept formula can be used to calculate the equation of a line when the y-intercept and slope of the straight line are known ( the y-coordinate of the point where the line intersects the y-axis). The equation of a line is the equation that each point on the line fulfils.
It is represented as y = mx + b
According to the given information
We are given that
Slope m = -1/2
y-intercept ( b ) = 6
Slope Intercept Form
y = mx + b
Putting the values in above equation
y = -1/2x + 6
The equation of the line in slope-intercept form y = -1/2x + 6
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a circle is inside a square. the radius of the circle is decreasing at a rate of 1 meter per day and the sides of the square are increasing at a rate of 2 meters per day. when the radius is 4 meters, and the sides are 18 meters, then how fast is the area outside the circle but inside the square changing? the rate of change of the area enclosed between the circle and the square is square meters per day.
Thus, the area is decreasing at a rate 12+20*π
What is area?The mathematical concept of "area" describes how big a region is on a planar or curved surface. A surface area is the area of an open surface or the border of a three-dimensional object, whereas a plane region's or plane's area's refers to a shape's or planar lamina's area.
Here,
Assume the circle's radius is r and the square's sides are s each.
The issue informs us that the radius is expanding at a pace of 2 m/hour. Dr/dt thus equals 2.
Four sides are shrinking simultaneously as the sides get smaller. ds/dt = -1 is the rate of change.
(why? because 4m/4 sides equals 1.
Find the area's formula right now. The region between the circle and the square is
A(t) = s^2 - π* r^2.
S(t) and R(t) are both time functions, thus you may compute a derivative with respect to t for both sides of the
equation. It follows that A'(t) = 2*s*s'(t) - 2**r*r' (t)
When r = 5 and s = 24/4 sides = 6, we are interested in the pace at which the area changes. Consequently, you
A'(t) = 2*6*(-1) - 2*π*5*2 = -12 - 20*π = -(12+20*π)
Thus, the area is decreasing at a rate 12+20*π
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Write System
Horatio has 57 coins consisting of nickels, dimes, and quarters. He has twice as many dimes as nickels. If
the coins are worth a total of $7.25, which system of equations is used to determine how many of each coin
Horatio has?
A n+d+q=57
n = 2d
0.05n+ 0.10d+0.25g = 7.25
B. n+d+q=57
n = 2d
5n+ 10d 25g = 7.25
C. n+d+q=57
d=2n
0.05n+0.10d+ 0.25g = 7.25
D. n+d+g 57
d=2n
5n+ 10d 25g = 7.25
The system of equations of Horatio's coins is (c) d = 2n, n + d + q = 57 and 0.05n + 0.10d + 0.25q = 7.25
How to determine the system of equations?From the question, we have the following parameters that can be used in our computation:
Number of coins = 57
Worth of coins = $7.25
Type of coins = Nickel, dimes and quarters
Twice as many dimes as nickels
Represent the nickels with n, the dimes with d and the quarters with q
So, we have the following representation
n + d + q = 57
When the nickels, dimes and quarters are converted to dollars, we have the following representations
Nickels = $0.05
Dimes = $0.10
Quarters = $0.25
So, we have the following representation
0.05n + 0.10d + 0.25q = 7.25
Recall that
He has twice as many dimes as nickels
This means that
d = 2n
Hence, the system of equations is d = 2n, n + d + q = 57 and 0.05n + 0.10d + 0.25q = 7.25
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If 2x2 – 28x = –98, then x could equal which of the following?
The value of x in the quadratic equation is as follows:
x = 7 and x = 7
How to solve quadratic equation?A quadratic equation is a polynomial equation in one unknown that contains the second degree, but no higher degree, of the variable.
The standard form of a quadratic equation is ax² + bx + c = 0, when a ≠ 0.
Let's solve for x in the quadratic equation 2x² – 28x = –98.
2x² – 28x = –98
2x² – 28x + 98 = 0
divide through by 2
x² - 14x + 49 = 0
The two numbers you can add get -14 and multiply to get 49 are -7 and -7.
Hence,
x² - 14x + 49 = 0
x² - 7x - 7x + 49 = 0
x(x - 7) -7(x - 7) = 0
(x - 7)(x - 7) = 0
Therefore,
x = 7 and x = 7
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there are water lilies on a lake. each day, the amount of water lilies doubles. after 20 days, there are so many water lilies that the entire lake is covered. after how many days was half of the lake covered?
We are aware that by day 20, the blossom will have covered half of the lake, or 1/2. It doubles the following day to become 1, which equals the entire lake. Knowing that the following day is 21.
Given that:
Each day, the amount of water lilies doubles. after 20 days, there are so many water lilies that the entire lake is covered.
we have something that doubles so 2 times something
after 20days = 20 × 2 = 40
However, we are aware that by day 20, the blossom will have covered half of the lake, or 1/2. It doubles the following day to become 1, which equals the entire lake. Knowing that the following day is 21.
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what is the quotient of 10/16 ÷ 5/16
Answer: 2
Step-by-step explanation:
Answer:
Step-by-step explanation:
The answer is 2.
(10/16)/(5/16)=2
10/5=2
16/16=1
2/1=2
y=3x-4 what's the answer pleas and thank you
Answer:
-12 (?)
Step-by-step explanation:
You have to multiply to find out what Y =
3x-4
A positive x a negative = a negative number (the negative sign doesn't effect how you multiply only changes the answer to a negative)
3 x (-4) =
-12
so Y = -12 ( 3 x -4 )
Muhammad bought 4 movies for a total of $25. At this rate, what is the cost of 12 movies?
Answer:
$75
Step-by-step explanation:
25÷4= 6.25
6.25×12= 75
The total of 12 movies would be $75.
[tex]4 * 3 = 12[/tex]
[tex]25*3=75[/tex]
Thus, you're answer is $75.
Write an equation of the line passing through point $p\left(-2,\ \ \ 6\right)$ that is parallel to the line $x=-5$.
The equation of a line passing through point P(-2,6) and parallel to x = -5 is x = - 2.
Slope: The pace at which the y-axis changes in relation to the x-axis is known as the slope.
Because tan90° is undefinable, the slope of a vertical line is also undefinable.
First, a line with the equation x = -5 is a vertical line that crosses the y-axis at x = -5 for all values of y from zero to infinity, with no defined slope.
For every real value of y, a line that is perpendicular to the line at x = -5 and contains coordinate points P(-2,6) is also a vertical line at x = -2.
Thus, it has the equation x = -2.
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there are 51 houses on a street. each house has an address between 1000 and 1099, inclusive. show that at least two houses have addresses that are consecutive integers.
according to the question,
given data in the question'
there are 51 houses on a street'
each hoses has an address between 1000 and 1099
There are 51 residences and 1000 possible addresses, according to the pigeon hole principle. There must be at least one address between each house in order for there to be no houses with consecutive addresses. To do this, only give houses even numbers (leaving odd addresses as the buffer address)
Giving houses even numbers (leaving odd addresses as the buffer address) There are now 1000=50 addresses that can be used.
According to the pigeon hole principle, there must be at least one box that contains at least N
k things if N objects are put into k boxes.
Because of this, it is impossible to assign 50 distinct addresses to 51 separate homes without utilising a single consecutive integer.
So, at least one instance of a home must have consecutive integers, which.
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There are 2 houses that have at least consecutive integers .
What is meant by the pigeonhole principle?According to the pigeonhole principle, at least one container must hold more than one item if n things are placed into m containers, where n > m.
Why is pigeonhole principle important?According to the pigeonhole principle, if there are more pigeons than there are pigeonholes, then at least one pigeonhole must hold more than one pigeon. Although the underlying idea is clear, the ramifications are staggering. The explanation is that the principle establishes the reality (or absence) of a certain phenomena.
There are potential residences and addresses. There must be at least one address between each house in order for there to be no houses with consecutive addresses. To do this, only give houses even numbers (leaving odd addresses as the buffer address)
Houses are given even numbers, leaving odd addresses as the buffer address. As a result, we now have
50
addressable addresses
According to the "pigeon hole concept," if items are put into boxes, then there must be at least one box that contains at
[N/k]object.
As a result, it is impossible to assign distinct addresses to several dwellings without using at least one consecutive number.
Since there must be at least two homes with consecutive integer addresses, there must be at least one instance of consecutive integer addresses for houses.
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Jude Scored 10 out of 35 points playing basketball while Tom scored 18 out of 46 points in a different game. Who got the highest score? Use percentage to find out. Show your working (please also answer in %)
Answer:
To find the percentage score for Jude, we need to divide the number of points scored by the total number of points and multiply the result by 100. This can be written as:
Percentage score for Jude = (10/35) * 100 = 28.57%
To find the percentage score for Tom, we need to divide the number of points scored by the total number of points and multiply the result by 100. This can be written as:
Percentage score for Tom = (18/46) * 100 = 39.13%
Since Tom scored a higher percentage of points compared to Jude, Tom got the highest score.
big bobs bakery charges $9 for 1/2 dozen donuts. what is the price per dozen?
The charge per dozen donuts will be $18 according to the given ratio.
What are the ratio and proportion?The ratio is the division of the two numbers.
For example, a/b, where a will be the numerator and b will be the denominator.
Proportion is the relation of a variable with another. It could be direct or inverse.
As per the given,
Big bobs bakery charges $9 for 1/2 dozen donuts.
$9 for 1/2 dozen donuts
For one dozen,
$9 x 2 = $18
Hence "The above ratio will result in a fee of $18 for a dozen donuts.".
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