Jessica sold the drinks for $1 each to raise money as she is running for class president.
What is elimination method?The elimination method is a technique used to solve a system of linear equations in two variables. In this method, we multiply one or both equations by appropriate constants to obtain equivalent equations in which one of the variables has the same coefficient but with opposite signs. Then we add the equations to eliminate one of the variables, and solve for the remaining variable.
According to question:Let the price of one candy bar be represented by x, and the price of one drink be represented by y.
From the first equation, we know that:
50x + 40y = 65
From the second equation, we know that:
100x + 50y = 100
We want to find the value of y, so we can solve for y using either substitution or elimination method. Here, we will use elimination method.
Multiplying the first equation by 2, we get:
100x + 80y = 130
Subtracting the above equation from the second equation, we get:
100x + 50y - (100x + 80y) = 100 - 130
Simplifying the left side and right side of the equation, we get:
-30y = -30
Dividing both sides by -30, we get:
y = 1
Therefore, Jessica sold the drinks for $1 each.
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Roberto has a tool box that is shaped like a rectangular prism with a volume of 4,500 cubic inches what is the width of the toolbox
The width of the rectangular prism shaped toolbox is 10 inches.
What exactly is a prism?A prism is a three-dimensional geometric shape that has two parallel and congruent bases that are connected by a set of parallelograms or rectangles. A prism is named according to the shape of its base, such as triangular prism, rectangular prism, or hexagonal prism.
A prism is characterized by its length, width, and height, and it has a volume equal to the product of its base area and its height. The lateral faces of a prism are parallelograms or rectangles, and the edges of a prism where the lateral faces meet the bases are perpendicular to the bases.
Now,
To find the width of the toolbox, we can use the formula for the volume of a rectangular prism, which is:
Volume = length x width x height
We are given that the volume of the toolbox is 4,500 cubic inches, the length is 30 inches, and the height is 15 inches.
Then
4,500 = 30 x width x 15
Simplifying the right side, we get:
4,500 = 450 * width
Dividing both sides by 450, we get:
width = 10 inches
Therefore, the width of the toolbox is 10 inches.
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it is 4/10 of a mile to john's house from frank's house. How much less than a mile is the distance from frank's house to john's house and then back to frank's house
The distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.
How do you figure out the overall distance?While travelling from one location to another and back again just multiplying the distance between the two places will give you the total distance travelled when travelling from one spot to another and returning to the beginning position.
The distance from frank's house to john's house = 4/10.
The distance back to Frank's house = 4/10.
Total distance = 4/10 + 4/10
Total distance = 8/10
The distance is thus,
1 - 8/10
Take the LCM:
(10 - 8) / 10
2/10 = 1/5
Hence, the distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.
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Mr. Nunez is tracking the progress of his plants' growth. Today the first plant is 5 cm high and the plant grows 2
cm per day. The second plant is 3 cm high and grows 3 cm per day.
A. Find the equations that represent the plants'
height after any given number of days.
B. Graph the system of equations
C. How tall is each plant after 15 days?
A. The equations representing plant height after a given number of days are:
h1(t) the height of the first plant in cm after t days and h2(t) the height of the second plant in cm after t days. Then:h1(t) = 2t + 5h2(t) = 3t + 3B. To graphically represent the system of equations, we can plot the points that correspond to the height of each plant after a given number of days. For example, after 5 days, the first plant is 15 cm tall (2(5) + 5 = 15) and the second plant is 18 cm tall (3(5) + 3 = 18).
You can draw this graph as a coordinate plane with the horizontal axis representing the number of days since the initial measurement and the vertical axis representing the height of the plants in centimeters.
C. To find the height of each plant after 15 days, we can plug t = 15 into the equations we encountered in letter A:
h1(15) = 2(15) + 5 = 35 cmh2(15) = 3(15) + 3 = 48 cmSo, after 15 days, the first plant is 35 cm tall and the second plant is 48 cm tall.What is a system of equations?It is a set of two or more equations that share common variables and are solved jointly.
Therefore, a system of equations is used when one wants to find the values of variables capable of satisfying all the equations of the system, whose solutions represent the points where the graphs of the equations in the Cartesian plane.
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Hector is playing a new video game. To get to the next level, he must capture the golden brick and then choose the correct door chance that he chooses the correct door is 40%. The probability that he dos the probability thatHector is playing a new video game. To get to the next level, he must capture the golden brick and then choose the correct door. The chance that he captures the golden brick is 70%. The chance that he chooses the correct door is 40%. The probability that he does both is 20%. Create a Venn Diagram and determine the probability that:
He captures the golden brick and chooses the correct door:
He only captures the golden brick:
He only chooses the correct door:
He doesn’t complete either task:
The probability that Hector captures the golden brick and chooses the correct door is approximately 0.2, or 20%.
What is probability ?
Probability can be defined as ratio number of favourable outcomes, total number of outcomes.
A represents the event that Hector captures the golden brick (with a probability of 0.7), B represents the event that Hector chooses the correct door (with a probability of 0.4), and C represents the intersection of A and B (the event that Hector captures the golden brick and chooses the correct door, with a probability of 0.2).
The probability of event C can be calculated using the formula:
P(C) = P(A and B) = P(A) * P(B|A)
where P(A) is the probability of event A (capturing the golden brick), and P(B|A) is the conditional probability of event B (choosing the correct door) given that event A has occurred (i.e., Hector has captured the golden brick). We are given that P(A) = 0.7, P(B) = 0.4, and P(A and B) = 0.2, so we can calculate:
P(B|A) = P(A and B) / P(A) = 0.2 / 0.7 ≈ 0.286
Therefore, the probability of event C (capturing the golden brick and choosing the correct door) is:
P(C) = P(A and B) = P(A) * P(B|A) = 0.7 * 0.286 ≈ 0.2
Therefore, the probability that Hector captures the golden brick and chooses the correct door is approximately 0.2, or 20%.
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'what are the equivalence classes of the equivalence relations in exercise 1?
Equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. → [x]R={y ∈ A ∣ xRy}.
What are equivalence relations?A binary relation which is symmetric, transitive and reflexive is called a equivalence relation.
Given is to find the equivalence classes of the equivalence relations.
If it is given that {R} is an equivalence relation on {A} and {x} ∈ A, then we can write the equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. →
[x]R={y ∈ A ∣ xRy}.
Therefore, equivalence class of {x} is the set of all elements of the set {A} which are related to {x} i.e. → [x]R={y ∈ A ∣ xRy}.
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5a + 3b = 13
7a + 6b = 20
Answer:
5a + 3b = 13
7a + 6b = 20
And solve for the values of "a" and "b" using algebraic methods like substitution or elimination.
Step-by-step explanation:
Solving for a and b in the system of equations:
5a + 3b = 13
7a + 6b = 20
We can multiply the first equation by 2 to get:
10a + 6b = 26
Then, we can subtract the second equation from this to eliminate b:
(10a + 6b) - (7a + 6b) = 26 - 20
3a = 6
Solving for a, we get:
a = 2
Substituting this value into the first equation, we can solve for b:
5a + 3b = 13
5(2) + 3b = 13
10 + 3b = 13
3b = 3
b = 1
Therefore, the solution to the system of equations is:
a = 2
b = 1.
What is the area of the figure?
Solve the problem.
Four students used different methods to approximate the diagonal distance across the middle school gym. The students and their measurements are shown in the following table. Calculate the decimal equivalent for each measurement rounded to the nearest hundredth yard. Write the decimal next to each worker’s name in the tableStudent
Measurement (yards) Decimal Calculation
Arturo
6
√
28
Bonnie
8
√
17
Courtney
10
π
Dawson
3.1
√
100
The measurements of the diagonal distance across the middle school gym is given by the following persons and is rounded to the nearest hundredths
The calculation of Arturo = 31.75 yards
The calculation of Bonnie = 32.98 yards
The calculation of Courtney = 31.42 yards
The calculation of Dawson = 31 yards
What is rounding up numbers?There are basically two rules while rounding up numbers
If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down and if the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up.
Non-zero digits are always significant
Zeros between non-zero digits are always significant
Leading zeros are never significant
Trailing zeros are only significant if the number contains a decimal point
Given data ,
Let the measurements of the diagonal distance across the middle school gym be represented as A
Now , the measured values are
a)
The measurement of Arturo = 6√28 yards
The measurement of Arturo = 6 x 5.2915
On rounding , we get
The measurement of Arturo = 31.75 yards
b)
The measurement of Bonnie = 8√17 yards
The measurement of Bonnie = 8 x 4.123
On rounding , we get
The measurement of Bonnie = 32.98 yards
c)
The measurement of Courtney = 10π yards
The measurement of Courtney = 10 x 3.142
On rounding , we get
The measurement of Courtney = 31.42 yards
d)
The measurement of Dawson = 3.1√100 yards
The measurement of Dawson = 3.1 x 10
On rounding , we get
The measurement of Dawson = 31 yards
Hence , the measurements are rounded
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[tex]-3\sqrt{x} +2\ \textless \ 15[/tex]
Answer:
12
Step-by-step explanation:
solve; 12+|x+4|=10
(multiple choice)
No solution
x = -6
x= -2
x= -24
[tex] \: [/tex]
Given:-[tex] \rm \: 12 + |x + 4| = 10 ?[/tex][tex] \: [/tex]
Solution:-
[tex] \rm \: 12 + |x + 4| [/tex][tex] \: [/tex]
[tex] \rm \: 12 + ( - 6 + 4)[/tex][tex] \: [/tex]
[tex] \rm \: 12 + ( - 2)[/tex][tex] \: [/tex]
[tex] \underline{ \boxed{ \rm \green{ \: 10 \: }}}[/tex][tex] \: [/tex]
━━━━━━━━━━━━━━━━━━━━━━━━━━
hope it helps! :)
Please help me with this question! Explanation isn't necessary but would be helpful :)
Do know trig? you'll find out that AB is Hypotenuse and we also know the Adjacent side if you don't please take a look at the image below and the cos alpha is equal to Adjacent side/Hypotenuse = AC/AB so AB is Adjacent side/cos alpha
Need help fast 24 pts and the brainliest
Answer:
65
Step-by-step explanation:
All 4 sides of a polygon equals 360 degrees. We are given the equation of 2 sides to find the degree.
12x + 19 + 8x + 1 = 180
(We put it equal to 180 instead of 360 since we are only given 2 sides)
Subtract 19 from both sides
12x + 8x + 1 = 161
Subtract 1 from both sides
12x + 8x = 160
Add 12x + 8x
20x = 160
Divide 20x by 160
x = 8
Plug in x value to (8x + 1)
(8(8) + 1)
= 65
A line passes through the points (-1, 3) and (1, -5). Which points lie on the same line?
Select all that apply.
(0, -2)
(-3, 12)
(-5, 1)
(-2,7)
(4, 15)
(2, -9)
( 2,-9) points lie on the same line.
Line point intercept is defined.
Quantifying soil cover, including vegetation, litter, rocks, and biotic crusts, quickly and accurately can be done using the line-point intercept approach.
These metrics deal with the erosion caused by wind and water, water infiltration, and the site's capacity to withstand damage and recover from it.
k = -5 - 3/1 - (-1) = -8/2 = -4
the formula of the line is
y - 3 = -4(x+ 1)
when x = -3 , y = 11 ≠ 12 the points aren't on line.
when x = -5, y = 19 ≠ 1
when x = 4, y = -7 ≠ 15
when x = 2, y = -9 ≠ 7
when x = 0, y = -1 ≠ -2
( 2,-9) are on the same line.
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In the number 4,434, how does the value represented by the digit in the thousands place compare with the value represented by the same digit in the hundreds place? Math item stem image CLEAR CHECK It is 10 more. It is 10 times the value. It is 100 times the value. It is the same value.
When compared to the value denoted by the same digit with in hundreds place, the value in the thousands place is 10 times greater.
Explain about the place value chart?We can determine the place value from every digit by looking at a place value chart, which is a highly helpful table format.
In the place value chart, a digit's place value increases as we move left and drops as we move right by a factor of ten.We can also write numbers using their expanded form by using base-ten blocks to indicate the place values of the digits in the number.The digits in a decimal number to the left of that decimal point stand for a whole number. The parts are represented by the digits toward the right of the decimal.In the given number: 4,434.
thousands place : 4
hundreds place: 4
Thus, when compared to the value denoted by the same digit with in hundreds place, the value in the thousands place is 10 times greater.
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If a seed is planted, it has a 95% chance of growing into a healthy plant.
If 8 seeds are planted, what is the probability that exactly 3 don't grow?
The probability that exactly three plants don't grow is given as follows:
0.0054 = 0.54%.
What is the binomial distribution formula?The mass probability formula, giving the probability of x successes, is of:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are given by:
n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.The parameter values for this problem are given as follows:
n = 8, as there are 8 plants.p = 0.05, as 95% of the plants grow, hence 5% don't grow.The probability that exactly three don't grow is P(X = 3), hence:
P(X = 3) = C(8,3) x (0.05)^3 x (0.95)^5 = 0.0054.
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A new car is purchased for 27,200 dollars. The value of the car depreciates at a rate of
5% per year. Which equation represents the value of the car after 7 years?
The value of the car after 7 years is approximately $18,994.7.
What is the Linear equation?A linear equation is defined as an equation in which the highest power of the variable is always one.
Let V be the value of the car after 7 years.
In the first year, the value of the car depreciates by 5%, so the value after the first year is:
V1 = 0.95 × 27,200
In the second year, the value of the car depreciates again by 5%, so the value after the second year is:
V2 = 0.95 × V1
Similarly, the value of the car after the third year is:
V3 = 0.95 × V2
We can continue this process for each of the 7 years:
V4 = 0.95 × V3
V5 = 0.95 × V4
V6 = 0.95 × V5
V7 = 0.95 × V6
So, the equation that represents the value of the car after 7 years is:
V = 0.95⁷ × 27,200
Simplifying this expression, we get:
V ≈ 18,994.7
Therefore, the value of the car after 7 years is approximately $18,994.7.
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The length of Trevor’s pool is 5feet more than 3 times it’s width. The deck surrounding the pool is 2 feet in width. The area of the deck is 196 feet squared. What is the width of the pool.
Answer:
Let's start by defining some variables to represent the width and length of the pool:
Let w be the width of the pool (in feet).
Let l be the length of the pool (in feet).
We know from the problem statement that the length of the pool is 5 feet more than 3 times its width:
l = 3w + 5
We also know that the deck surrounding the pool is 2 feet wide. This means that the overall width of the pool and the deck combined is:
w_total = w + 2 + 2 = w + 4
Similarly, the overall length of the pool and the deck combined is:
l_total = l + 2 + 2 = l + 4
We are given that the area of the deck is 196 square feet. The area of the deck can be calculated by subtracting the area of the pool from the area of the pool and deck combined:
Area of deck = Area of pool and deck - Area of pool
196 = w_total * l_total - w * l
We can substitute the expressions for w_total and l_total in terms of w and l:
196 = (w + 4) * (l + 4) - w * l
Simplifying this expression gives:
196 = wl + 4w + 4l + 16
We can substitute the expression for l in terms of w:
196 = w(3w + 5) + 4w + 4(3w + 5) + 16
Simplifying this expression gives:
196 = 13w + 36
Solving for w, we get:
w = 14
Therefore, the width of the pool is 14 feet.
On certain lakes, the state of Minnesota imposesnexperimental fishing regulations designed to increase fish populations by protecting certain sizes of a species.The fishing regulations for Crane Lake state that all walleye less than 13 inches long or larger than 17 inches must be released, except one fish over 23 inches may be kept as part of the daily limit of six walleye.Draw a graph on a number line to show the sizes of walleye that may be kept.
On the number line, we can represent the different size ranges of walleye that may be kept as shown.
What is Number Line?In math, a number line can be defined as a straight line with numbers arranged at equal segments or intervals throughout. A number line is typically shown horizontally and can be extended indefinitely in any direction.
The line segment from 13 to 17 inches represents the protected size range, where all walleye caught must be released. The line segments from 0 to 13 inches and from 17 to 23 inches represent the size range of walleye that can be caught but must be released.
The line segment from 23 to 26 inches represents the size range of walleye that can be caught and kept, but only one fish per day is allowed. The vertical arrows show the minimum and maximum sizes allowed for keeping walleye, as specified in the fishing regulations for Crane Lake in Minnesota.
Thus, on the number line, we can represent the different size ranges of walleye that may be kept as shown.
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The probability of a passenger missing a flight is 0.0993. Airlines sometimes overbook their flights so that they can be sure that there are no empty seats. Suppose that a particular plane has the capacity for 250 passengers. If 270 tickets are sold, what is the probability that there are not enough seats on the flight?
The probability that there are not enough seats on the flight is approximately 0.9735 or 97.35%.
The probability of a passenger missing a flight is 0.0993. Therefore, the probability that a passenger does not miss the flight is:
P(not missing the flight) = 1 - 0.0993 = 0.9007
The plane has a capacity for 250 passengers, but 270 tickets are sold. Assuming that the probability of a passenger missing a flight is independent of other passengers, we can model the number of passengers who show up for the flight as a binomial distribution with n = 270 and p = 0.9007.
Let X be the number of passengers who show up for the flight. We want to find the probability that there are not enough seats on the flight, which means that X > 250. We can use the complement rule to find this probability:
P(X > 250) = 1 - P(X ≤ 250)
To find P(X ≤ 250), we can use a binomial probability table or a calculator with a binomial probability function. Using a calculator, we get:
P(X ≤ 250) = binomcdf(270, 0.9007, 250) ≈ 0.0265
Substituting this into the complement rule formula, we get:
P(X > 250) = 1 - 0.0265 ≈ 0.9735
Therefore, the probability that there are not enough seats on the flight is approximately 0.9735 or 97.35%.
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What is the area of a cross section parallel to the base of a cylinder that has a height of 12 inches and a radius of 4 inches? Use 3.14 for π.
Answer:
The cross section parallel to the base of a cylinder is a circle. The area of a circle is given by the formula:
A = πr^2
where A is the area and r is the radius.
In this case, the radius of the circle is 4 inches, so we have:
A = π(4)^2
A = 16π
Using 3.14 for π, we can simplify:
A = 16(3.14)
A = 50.24
Therefore, the area of the cross section parallel to the base of the cylinder is 50.24 square inches.
Step-by-step explanation:
Answer:
Step-by-step explanation:
8
the length of a rectangle poster is nine more inches than half it’s with. The area of the poster is 140 square inches. Solve for the dimensions of the poster.
The dimensions of the poster are 10 inches by 14 inches.
What is Area of Rectangle?The area of Rectangle is length times of width.
Given that length of a rectangle poster is nine more inches than half it’s width.
Length = 1/2 width +9
The area of the poster is 140 square inches.
Area=Length×Width
140=(W/2+9)W
140=W²/2 +9W
W²+18W-240=0
(w + 28)(w - 10) = 0
W=-28 and w=10
We take only positive value so width is 10
Length=10/2+9
=5+9=14 inches
Hence, the dimensions of the poster are 10 inches by 14 inches.
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There are 1 3/4 cookies. If there is 1/8 of a cookie in each bag, how many bags are there
Answer: 8
Step-by-step explanation:
Since 1 batch uses 1/8 of a bag of flour, I would use equivalent fractions to find how many 8ths are in 3/4. 3/4=?/8
4 times 2 equals 8, so I would multiply 3 by 2 also. 3/4=6/8. If 1 batch takes 1/8, then 6 batches would use 6/8 or 3/4 of a bag of flour.
3x-1>-4 and -2x-3<1 in interval notation
This means that the solution is any x value that is less than -1 AND greater than -2.
What kind of notation is it?A good example of notation is musical notation. This kind of notation system can be used by a composer to instruct performers and listeners on how to play and hear their music. There are many different types of symbols and pictures in it.
To solve the inequality 3x - 1 > -4, we add 1 to both sides:
3x - 1 + 1 > -4 + 1
3x > -3
Then, we divide both sides by 3 (remembering to flip the inequality sign since we're dividing by a negative number):
x < -1
So the solution to 3x - 1 > -4 is x < -1.
To solve the inequality -2x - 3 < 1, we add 3 to both sides:
-2x - 3 + 3 < 1 + 3
-2x < 4
Then, we divide both sides by -2 (remembering to flip the inequality sign again):
x > -2
So the solution to -2x - 3 < 1 is x > -2.
Putting these two solutions together in interval notation, we get:
(-∞, -1) ∩ (-2, ∞)
This means that the solution is any x value that is less than -1 AND greater than -2.
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What are the domain and range of the function shown below? write the answers as a list using commas. Enter values in order from least to greatest.
The domine of function = {0, 1, 2, 3}
The range of function = {-3, -1, 1, 3}
Domain:The domain of a function is the set of all possible input values for which the function is defined. It can be restricted by certain conditions, such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers.
Range:The range of a function is the set of all possible output values that the function can produce. It depends on the nature of the function and the restrictions on the domain.
Here we have a table
x - 0 1 2 3
f(x) -3 -1 1 3
Here x values are indicates input values for the function f(x)
Hence, the domine of function = {0, 1, 2, 3}
The values of f(x) represent the output values for each input value of 'x'
Hence, the range of function = {-3, -1, 1, 3}
Therefore,
The domine of function = {0, 1, 2, 3}
The range of function = {-3, -1, 1, 3}
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the muscular muscle is 75kg and the muscle is pulling on the bone at an angle of 15 degrees. what are the vertical and horizontal components of this muscular force?
a. which of these forces would be responsible for the rotation of this bone around the joint
b. which of these forces would cause a compressive force at the joint?
please show work and draw a diagram and label
Answer:
Step-by-step explanation:
To find the vertical and horizontal components of the muscular force, we can use trigonometry. Let F be the force exerted by the muscle, and θ be the angle it makes with the horizontal.
Horizontal component of the force Fx = F cos θ
Vertical component of the force Fy = F sin θ
Given that the muscular force is 75kg and the angle of pull is 15 degrees, we have:
Fx = 75kg x cos(15) = 72.62 kg (approx.)
Fy = 75kg x sin(15) = 19.34 kg (approx.)
To determine which force would be responsible for the rotation of the bone around the joint, we need to consider the torque or moment created by the force. The torque is given by the force times the lever arm, which is the perpendicular distance from the line of action of the force to the axis of rotation. In this case, the axis of rotation is the joint, and the lever arm is the distance from the joint to the point where the muscle attaches to the bone.
The torque due to the horizontal component of the force is zero, since it acts perpendicular to the lever arm. The torque due to the vertical component of the force is:
Torque = Fy x d
where d is the lever arm. If we assume that the lever arm is 5 cm, we have:
Torque = 19.34 kg x 0.05 m = 0.967 Nm
Therefore, the vertical component of the force would be responsible for the rotation of the bone around the joint.
To determine which force would cause a compressive force at the joint, we need to consider the direction of the force. A force that acts perpendicular to the bone would cause compression at the joint. In this case, the vertical component of the force is perpendicular to the bone, while the horizontal component is parallel to it. Therefore, the vertical component of the force would cause a compressive force at the joint.
Here is a diagram of the situation:
|\
Fy | \ /|
| \ / |
| \ / |
| \ / |
| \ / |
| \/ |
| /\ |
| / \ |
| / \ |
| / \ |
| / \ |
| / \|
|/______\
Fx
In the diagram, Fx is the horizontal component of the force, Fy is the vertical component of the force, and the bone is represented by the vertical line. The angle θ is the angle between the force and the horizontal. The lever arm d is the distance between the joint and the point where the muscle attaches to the bone.
solve for x: (-x+18)(x+4)
(-x+18)(x+4)
-x(x+4) + 18 (x+4)
-(x^2+4x) + 18(x+4)
-x^2-4x+18x+72
−x^2+14x+72
I hope this helps! :)
Figure ABC has vertices A(-3,3), B (1,-1), and C (0,5). Sketch ABC and draw its image after the translation (x,y) (x+4, y + 2).
Sketch of ABC and drawn image after the translation (x,y) (x+4, y + 2) is mentioned below.
Describe Translation?In mathematics, translation refers to the process of moving a geometric shape or object from one location to another without changing its size, shape, or orientation. This movement is typically described in terms of the distance and direction of the translation.
For example, a point on a coordinate plane can be translated by adding or subtracting a constant value to its x and y coordinates. Similarly, a line segment or polygon can be translated by moving each of its vertices by the same distance and in the same direction.
Mathematical translations can be represented using transformation matrices or vectors. A translation matrix is a square matrix that describes the movement of a point or object in a specific direction and distance. A translation vector is a one-dimensional vector that describes the direction and distance of the translation.
To sketch the triangle ABC, we first plot the three given points on a coordinate plane:
We connect the three points to form triangle ABC:
|
6 | C(0,5)
| /\
3 | / \
| / \
| / \
| / \
| /__________\
|A(-3,3) B(1,-1)
|
|
|
|
|
|
-3 -1 1 3
To find the image of the triangle under the translation (x,y) → (x+4, y+2), we add 4 to the x-coordinates and 2 to the y-coordinates of each point. This gives us the following new coordinates:
A(-3,3) → A'(-3+4, 3+2) = A'(1,5)
B(1,-1) → B'(1+4, -1+2) = B'(5,1)
C(0,5) → C'(0+4, 5+2) = C'(4,7)
We can now plot these new points and connect them to form the image of triangle ABC under the given translation:
|
9 | C'(4,7)
| / \
7 | / \
| / \
5 | / \
| / \
| /__________\
3 |A'(1,5) B'(5,1)
|
|
|
1 |
|
|
-3 -1 1 3 5
This is the final image of triangle ABC after the translation.
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The vertices of the translated figure are A'(1, 5), B'(5, 1), and C'(4, 7).
What is translation?
A translation in mathematics does not turn a shape; instead, it moves it left, right, up, or down. They are congruent if the translated shapes (or the image) seem to be the same size as the original shapes. They have only changed their direction or directions.
To translate the figure by (x+4, y+2), we add 4 to the x-coordinate and 2 to the y-coordinate of each vertex.
So the new coordinates of A' would be -
x-coordinate: -3 + 4 = 1
y-coordinate: 3 + 2 = 5
Therefore, A' has coordinates (1, 5).
The new coordinates of B' would be -
x-coordinate: 1 + 4 = 5
y-coordinate: -1 + 2 = 1
Therefore, B' has coordinates (5, 1).
The new coordinates of C' would be -
x-coordinate: 0 + 4 = 4
y-coordinate: 5 + 2 = 7
Therefore, C' has coordinates (4, 7).
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Calculus 1. Can someone please help me? Thanks.
For the differential equation y = [(x² - 1) sin x] / (sin x + 1), the derivative is dy/dx = [(x² + 2x) sin² x] / (sin x + 1)².
What is differential equation?
Any equation with one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation.
To find dy/dx, we need to use the quotient rule of differentiation, which states that -
d/dx [f(x)/g(x)] = [g(x) × f'(x) - f(x) × g'(x)] / [g(x)]²
Applying this rule to the given function, we get -
y = [(x² - 1) sin x] / (sin x + 1)
y' = [(sin x + 1) d/dx[(x² - 1) sin x] - (x² - 1) sin x d/dx[sin x + 1]] / (sin x + 1)²
Now we need to find the derivatives of the two terms in the numerator separately -
d/dx[(x² - 1) sin x] = (2x sin x + (x² - 1) cos x) sin x
d/dx[sin x + 1] = cos x
Substituting these values in the formula for y', we get -
y' = [(sin x + 1) * (2x sin x + (x² - 1) cos x) - (x² - 1) sin x * cos x] / (sin x + 1)^2
Simplifying this expression, we get -
y' = [(x² + 1) sin² x + 2x sin x cos x - (x² - 1) sin x cos x - (x² - 1) sin x cos x] / (sin x + 1)²
y' = [(x² + 2x) sin² x] / (sin x + 1)²
Therefore, the derivative of y with respect to x (dy/dx) is -
dy/dx = [(x² + 2x) sin² x] / (sin x + 1)²
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Please help me this id due today
Based on the information, I can notice that mathematics is used in various situations in daily life. Additionally, I wonder how to apply algebra in daily life.
What do we notice about mathematics?Based on what we have learned about mathematics, I have noticed that it is a very broad area that covers various types of operations and mathematical problems that allow us to solve various problems. For example: It helps us to know how to distribute a budget in our daily life, it helps us to divide the birthday cake in equal parts and to find the area of our house and our room.
What do we ask ourselves about mathematics?What we ask ourselves about mathematics is how we can apply the problems and formulas of algebra and other areas derived from mathematics in daily life. Sometimes it seems that these complex problems do not have an application in daily life.
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Write the equation of the circle in standard form
X^2+y^2-8x-6y+16=0
Show work
The equation of the circle x² + y² - 8x - 6y + 16 = 0 in standard form is (x - 4)² + (y - 3)² = 3².
What is the equation of the circle in standard form?Given the equation of the circle ;
x² + y² - 8x - 6y + 16 = 0
To write the equation of a circle in standard form, we need to express it as:
(x - h)² + (y - k)² = r²
where (h, k) is the center of the circle and r is the radius.
To do this, we need to complete the square for both x and y terms in the given equation.
Starting with the x terms:
x² - 8x = x² - 8x + 16 - 16
= (x - 4)² - 16
Similarly, for the y terms:
y² - 6y = y² - 6y + 9 - 9
= (y - 3)² - 9
Substituting these results into the original equation, we get:
x² + y² - 8x - 6y + 16 = 0
(x - 4)² - 16 + (y - 3)² - 9 + 16 = 0
Simplifying, we get:
(x - 4)² + (y - 3)² - 9 = 0
(x - 4)² + (y - 3)² = 9
So the equation of the circle in standard form is:
(x - 4)² + (y - 3)² = 3²
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