There are 17 one-dollar coins and 19 two-dollar coins in the machine.
How did we arrive at these values?4.3.1 Model the statement into two linear equations by letting the number of one-dollar coins to be x and the number of two-dollar coins to be y:
Let x be the number of one-dollar coins and y be the number of two-dollar coins in the machine. Then we can form the following two linear equations:
Equation 1: x + y = 36 (total number of coins)
Equation 2: 1x + 2y = 55 (total value of coins)
4.3.2 Solve the two equations you formed in 4.3.1 using Cramer’s rule to find the number of coins of each value:
To solve the two equations using Cramer's rule, we need to write them in matrix form:
| 1 1 | | x | | 36 |
| | x | | |
| 1 2 | | y | | 55 |
Using Cramer's rule, we first calculate the determinant of the coefficient matrix:
D = | 1 1 |
| 1 2 |
D = (1 x 2) - (1 x 1) = 1
Then we replace the first column of the coefficient matrix with the constants from the right-hand side of the equations:
Dx = | 36 1 |
| 55 2 |
And calculate the determinant of this matrix:
Dx = (36 x 2) - (55 x 1) = 17
Similarly, we replace the second column of the coefficient matrix with the constants from the right-hand side of the equations:
Dy = | 1 36 |
| 1 55 |
And calculate the determinant of this matrix:
Dy = (1 x 55) - (1 x 36) = 19
Finally, we can calculate the values of x and y using the formulas:
x = Dx / D = 17 / 1 = 17
y = Dy / D = 19 / 1 = 19
Therefore, there are 17 one-dollar coins and 19 two-dollar coins in the machine.
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You notice that a stairwell railing is a parallelogram. In parallelogram RALS: ALsegment A L = 26.5 feet RIsegment R I = 5.3 feet ARsegment A R = 6.2 feet What is the area of parallelogram RALS? Use the given information to complete the worksheet.
HELPP I MEED IT
Answer:
To find the area of a parallelogram, we need to multiply the base by the height. In this case, we need to find the height of the parallelogram.
We can see that segment AL and segment RI are both perpendicular to segment AR. Therefore, we can use either of them to find the height.
Let's use segment AL as our height. To find the length of segment LS, we can use the Pythagorean theorem:
LS² = AR² - RI²
LS² = 6.2² - 5.3²
LS² = 16.21
LS = √16.21
LS ≈ 4.02 feet
Now we can find the area of the parallelogram:
Area = base x height
Area = AL x LS
Area = 26.5 x 4.02
Area ≈ 106.23 square feet
Therefore, the area of parallelogram RALS is approximately 106.23 square feet.
Step-by-step explanation:
Analyze the diagram below and complete the instructions that follow.Find m
The measure of angle A is equal to 41.11 rounded to two decimal place, using trigonometric ratio for the cosine of the angle A. The correct option is B.
What is trigonometric ratios?The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.
The basic trigonometric ratios includes;
sine, cosine and tangent.
cos A = 55/73 {adjacent/hypotenuse}
A= cos⁻¹(55/73) {cross multiplication}
A = 41.1121
Therefore, the measure of angle A is equal to 41.11 rounded to two decimal place, using trigonometric ratio for the cosine of the angle A
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i really need help ple
Answer:
-6.99 , -4 , -6.9 , -2 , -6.999 , -6 , 1
Step-by-step explanation:
Any number to the right of -7 is greater than -7.
Malaya has 5different shapes of blocks: rectangle, rhombus, square, trapezoid, and triangle. She has 10 of each shape. She selects a block at random.
Using probability, we can find that the probability of getting a triangle shape block when selected at random is 1/5.
Define probability?The chance of an event can be calculated using the probability formula by only dividing the favourable number of possibilities by the total number of potential outcomes.
The likelihood of an event occurring can be anything between 0 and 1, as the favourable number of outcomes can never exceed the total number of outcomes. Therefore, the percentage of successful results cannot be zero.
In the question,
Malaya has 5 different shapes of blocks.
Malaya has 10 blocks of each shape.
So, the total no. of blocks = 5 × 10 = 50.
Now, Malaya has 10 blocks of triangular shape.
probability of getting a triangular shape is:
P = 10/50
= 1/5
Therefore, the probability of getting a triangle shape block when selected at random is 1/5.
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The complete question is:
Malaya has 5different shapes of blocks: rectangle, rhombus, square, trapezoid, and triangle. She has 10 of each shape. She selects a block at random. What is the probability of getting a triangle shape block when selected at random?
an inner city revitalization zone is a rectangle that is twice as long as it is wide. the width of the region is growing at a rate of 31 m per year at a time when the region is 450 m wide. how fast is the area changing at that point in time?
The area of the rectangle is increasing at a rate of 27,900,000 meters³ per year when the width of the region is 450 meters.
The inner-city revitalization region is a rectangle that is twice as long as it is wide. At a time when the area is 450 meters wide, the width of the region is increasing at a rate of 31 meters per year. The aim is to find out how quickly the area is changing at this point in time. The area of the rectangle is given by the formula,
Area = length × width
Given that the width of the rectangle is increasing at a rate of 31 meters per year. Let's say the width of the rectangle is w meters, and the rate of change in width is dw/dt meters per year. Then, we can say, dw/dt = 31 meters per year
Width of the rectangle is w meters
Length of the rectangle is twice the width, or 2w meters
Area of the rectangle is given by the formula, A = lw= 2w × w= 2w²
Now, we need to find dA/dt, the rate at which the area of the rectangle is changing with respect to time (t).We can find it using the formula,
dA/dt = dA/dw × dw/dt
dA/dw = 4w (differentiate 2w² with respect to w)
dw/dt = 31 meters per yeard
A/dt = dA/dw × dw/dt= 4w × 31= 124w meters² per year
From the given information, the width of the rectangle is 450 meters wide. So, the width is w = 450 meters.
dA/dt = 124w meters² per year
= 124 × 450 meters² per year
= 27,900,000 meters³ per year
Which is the increasing rate of the rectangle.
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Required information Note this is a multi-part problem. That means the parts must be worked in order (a first, then b, then c, and so on). Furthermore, if you check your answer on one part, it will count as a check for all parts of the problem. Therefore, work all parts of the problem (in order!), then check your answer. Rework all missed parts before using your next "check my answer". Let X=5 L 40' and Y = 20 L -30° The polar form of the quantity (X + Y)X* is Oy Please report your answer so the magnitude is positive and all angles are in the range of negative 180 degrees to positive 180 degrees.
The polar form of the quantity will be 100.83 L 76.47°.
Given X = 5 L 40'andY = 20 L -30°(X + Y)X* .The expression (X + Y)X* can be simplified as: X * X + Y * X* =X² + Y * X* .Now, we need to find the polar form of the quantity (X + Y)X*.That is, we need to representX² + Y * X* in polar form. By using rectangular to polar conversion, we can convert the above expression to polar form.
Using the formula:
r² = x² + y²... Equation 1
θ = tan⁻¹(y / x)... Eq 2 (Where,x = real part of the expression)
x = X² = 5² = 25... Eq 3 (y = imaginary part of the expression)
y = Y * X*... Eq 4
Therefore, Y
Y * X* = 20 L -30° * 5 L 40'20 L -30° * 5 L 40' = 100 L 10°... Eq5
y = 100 L 10°.
Substituting the values of x and y in equations 1 and 2, we get:r² = x² + y²r² = 25² + 100²r² = 10025 + 100²r = 100.83θ = tan⁻¹(y / x)θ = tan⁻¹(100 L 10° / 25)θ = tan⁻¹(4 L 10°)θ = 76.47°
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how do you find the dimensions of the parallelogram
To find the dimensions of a parallelogram, you need to know the length of its base and the perpendicular distance from the base to the opposite side, which is also called the height or altitude of the parallelogram.
What is a parallelogram?A quadrilateral (a four-sided polygon) with two pairs of parallel sides is referred to as a parallelogram. This means that the opposite sides of the parallelogram are parallel and have the same length, while the adjacent sides are unequal in length.
The properties of a parallelogram include:
The opposing sides seem to be congruent and parallel (equal in length)
Congruent opposite angles (equal in measure)
The angle adjacent to each other are supplementary (their sum is 180 degrees)
Diagonals bisect each other (the diagonals divide each other into two equal parts)
If you have the measurements of the sides and angles of the parallelogram, you can use trigonometry and geometry to find the length of the base and the height. For example, if you have the lengths of two adjacent sides and the angle between them, you can use the sine function to find the height. The base can be found by using the formula for the area of a parallelogram, which is base times height.
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high voltage, inc., a light bulb manufacturer, wants to know if there's a difference in mean life span between its product and that of a competitor. they collect paired data and calculate the difference in each pair to create one set of numbers that represents the differences within each pair. what notation should high voltage use to construct its null and alternative hypotheses?
High Voltage, Inc. should use the following notation to construct its null and alternative hypotheses: Null Hypothesis: H0: μ1 - μ2 = 0; Alternative Hypothesis: H1: μ1 - μ2 ≠ 0; where μ1 represents the mean life span of High Voltage's product and μ2 represents the mean life span of the competitor's product.
The notation High Voltage should use to construct its null and alternative hypotheses is:
(i) Null Hypothesis: H0: μ1−μ2=0
where μ1 represents the mean life span of High Voltage's product and μ2 represents the mean life span of the competitor's product.
(ii) Alternative Hypothesis: Ha: μ1−μ2≠0
Where,μ1 and μ2 are the mean lifespan of the light bulbs of High Voltage and its competitor, respectively.
In this problem, paired samples t-test should be used to test the hypotheses.
Paired sample t-test: It is a statistical procedure used to compare two population means with two samples of dependent observations, where the dependent variable is the same or the same subject measured under different conditions. It is used when the observations are paired, i.e., each pair of observations consists of two measurements made on a single subject or item.
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if the actual length is 28.3m and the scale is 1cm to 2m, what is the length on drawing
The length on drawing with the given scale is 14.15 cm.
How to determine the length on drawingUsing the given scale of 1cm to 2m, we can set up the following proportion to find the length on the drawing:
1 cm : 2 m = x cm : 28.3 m
where x is the length on the drawing that we want to find.
To solve for x, we can cross-multiply and simplify:
1 cm * 28.3 m = 2 m * x cm
Evaluate the products and remove the units
28.3 = 2x
Divide both sides by 2 to solve for x
x = 28.3 / 2
Evaluate
x = 14.15
Therefore, the length on the drawing is 14.15 cm.
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Pls help due tomorrow
Determine the values of p such that the rank of A=[[1,1,-1,0],[4,4,-3,1],[p,2,2,2],[9,9,p,3]] is 3 .
The values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.
Given, A = [[1,1,-1,0],[4,4,-3,1],[p,2,2,2],[9,9,p,3]] To find the value of p, such that the rank of A is 3.
Rank of a matrix is defined as the maximum number of linearly independent row vectors or column vectors. It is denoted by R(A).When a matrix is in echelon form, its rank is equal to the number of pivots.The rank of a matrix is equal to the maximum number of linearly independent rows or columns in the matrix.Here, A is a matrix,The rank of A is 3.Thus, we can say that there will be 3 linearly independent rows or columns in the matrix.
The augmented matrix [A|0] should have 1 pivot for each linearly independent rows.Therefore, [A|0] will have 3 pivots.Then, the last row of [A|0] should be a linear combination of the first three rows.To find the value of p,Let the matrix A is in echelon form [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]Let's analyze the matrix A for rank 3If we swap R2 and R3,R2 <-> R3 [1,1,-1,0][p,2,2,2][4,4,-3,1][9,9,p,3]Then, the matrix in echelon form is [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]We can see that the third row is not a linear combination of the first two rows. Therefore, the first three rows of A are linearly independent.Then, A has rank 3 if we can get rid of the fourth row using linear combinations of the first three rows.
9R1 + (-9)R2 + (-p)R3 = 0Thus, 9 - 9p - p² = 0p² + 9p - 9 = 0(p + 3)(p - 3) = 0p = -3 or 3So, the values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.
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The list of ordered pairs below represents a function.
{(10,−2),(−9,6),(5,−8),(2,−4)}
Find the range of the function.
10,-2
Step-by-step explanation: 10-2, -9,6, 5-8, 2,-4
Domain: 10,-9, 5, 2,
Range -8, 6,-4, -2
A large rectangular swimming pool is 10,000 feet long, 100 feet wide, and 10 feet deep. The pool is filled to the top with water.
1. What is the area of the surface of the water in the pool? ______ square feet
2. How much water does the pool hold? _______ cubic feet
1. The surface area of the pool is given as follows: 2,202,000 square feet.
2. The amount of water that the pool holds is of: 10,000,000 cubic feet.
What is the surface area of a rectangular prism?The surface area of a rectangular prism of height h, width w and length l is given by:
S = 2(hw + lw + lh).
This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.
The dimensions for this problem are given as follows:
l = 10000, w = 100, h = 10.
Hence the surface area is given as follows:
S = 2 x (10000 x 100 + 10000 x 10 + 100 x 10)
S = 2,202,000 square feet.
What is the volume?The volume of a rectangular prism is given by the multiplication of it's dimensions, hence:
V = 10000 x 100 x 10
V = 10,000,000 cubic feet.
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Michelle needs to rent storage space for some of her belongings. She paid a one-time original storage fee of $50.00, and now pays $15.00 each month,
. Which answer choice shows an expression that represents the total amount Michelle has paid after a certain number of months,
?
An SRS of 1000 voters finds that 57% believe that competence is more important than character in voting for President of the United States.
(A) Determine a 95% confidence interval estimate for the percentage of all voters who believe competence is more important than character.
(B) Based on the interval you found in part (a), is it reasonable to assume that a majority of voters believe that competence is more important than character. Explain why or why not.
(C) Explain what is meant by 95% confidence in this situation
(A) A 95% confidence interval estimate for the percentage of all voters who believe competence is more important than character is (0.538, 0.602).
(B) This is due to the interval's lower bound's value of 0.538, which is higher than 0.5.
(C) In this instance, 95% confidence indicates that if the sampling procedure were repeated numerous times and a 95% confidence interval estimate was calculated for each sample
(A) Using the method below, we can calculate an estimate with a 95% confidence interval for the proportion of voters who think competence is more significant than character:
CI equals p z*(sqrt(p*(1-p)/n)/p)).
where: p = the percentage of voters who say character is more essential than competence = 0.57; n = the sample size; 1000; and z = the z-score for a 95% confidence level; 1.96.
By entering the numbers, we obtain:
CI = 0.57 1.96*(sqrt(0.57*(1-0.57)/1000)), which equals 0.57 0.032. (0.538, 0.602)
the 95% confidence interval estimate for the proportion of respondents overall who think competence is more significant than character is (0.538, 0.602).
(B) Based on the range discovered in part, it is reasonable to infer that the majority of voters value competence over character (a). This is due to the interval's lower bound's value of 0.538, which is higher than 0.5. Therefore, we can state with 95% certainty that the actual proportion of voters who think competence is more crucial than character is at least 53.8%, which is a majority.
(C) In this instance, 95% confidence indicates that if the sampling procedure were repeated numerous times and a 95% confidence interval estimate was calculated for each sample, then roughly 95% of those intervals would contain the actual proportion of voters who believe competence is more significant than character. That is to say, we have a 95% confidence interval (0.538, 0.602) that contains the actual proportion of voters who think competence is more significant than character.
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The distance between the points (10,4) and (1,-8)
Round decimals to the nearest tenth
the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.
To find the distance between two points in a coordinate plane, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:
d = √[(x2 - x1)² + (y2 - y1)²]
Where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.
In this case, the coordinates of the two points are (10, 4) and (1, -8). Substituting these values into the distance formula, we get:
d = √[(1 - 10)² + (-8 - 4)²]
= √[(-9)² + (-12)²]
= √(81 + 144)
= √225
= 15
Therefore, the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.
In summary, to find the distance between two points in a coordinate plane, we can use the distance formula. In this case, we found that the distance between the points (10, 4) and (1, -8) is 15 units, rounded to the nearest tenth as 15.0 units.
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Part C
What is the probability that Geraldo, the 25-year-old you're considering for a 30-year policy, lives to be 55 years old?
Remember that Geraldo is a Hispanic male.
B
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Characters used: 0/ 15000
Part D
The probability that client C, the 25-year-old you’re considering for a 30-year policy, lives to be 55 years old is 1.
The potential is implied by the word "probability". This area of mathematics studies how random events occur. The range is the set of values between 0 and 1. Probability has been incorporated into the mathematics to forecast the likelihood of various events. In essence, probability refers to the degree to which something is likely to occur. You can appreciate the possible results of a random experiment using this fundamental probability theory, which is also applied to a probability distribution. Before assessing how likely an event is to occur, we need to know how many possible outcomes there are.
Gerald age = 25 years
Policy considered = 30 years
Age at the end of the policy = 55 years
The probability that Geraldo lives to be 55 years will be:
= P(A + B)
= (25÷55) + (30÷55)
= 55÷55
= 1
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Data was collected on the amount of time that a random sample of 8 students spent studying for a test and the grades they earned on the test. a scatter plot and line of fit were created for the data. scatter plot titled students' data, with points plotted at 1 comma 60, 2 comma 60, 2 comma 70, 2 comma 80, 3 comma 80, 3 comma 90, 4 comma 95, and 4 comma 100, and a line of fit drawn passing through the points 0 comma 40 and 2 comma 70. find the y-intercept of the line of fit and explain its meaning in the context of the data.a. 30; for each additional hour a student studies, their grade is predicted to increase by 30% on the testb. 15; for each additional hour a student studies, their grade is predicted to increase by 15% on the test c. 70; a student who studies for 0 hours is predicted to earn 70% on the test d. 40; a student who studies for 0 hours is predicted to earn 40% on the test
Answer: The y-intercept of the line of fit can be found by looking at the point where the line crosses the y-axis. From the given information, we know that the line passes through the points (0, 40) and (2, 70). To find the equation of the line, we can use the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. We can find the slope of the line by using the two points:
m = (y2 - y1) / (x2 - x1)
m = (70 - 40) / (2 - 0)
m = 30
So the equation of the line is:
y = 30x + b
To find the y-intercept, we can plug in one of the points and solve for b. Let's use the point (0, 40):
40 = 30(0) + b
b = 40
Therefore, the y-intercept of the line of fit is 40. In the context of the data, this means that a student who did not study at all (0 hours) is predicted to earn a grade of 40 on the test. However, it's important to note that this prediction is based on the data collected from the sample of 8 students and the line of fit may not accurately predict the grades of all students who did not study.
Your welcome.
Can sum1 help please
You can use the desmos graphing calculator
According to the given information, Mack should make 4 necklaces and 8 wristbands to maximize his profit.
What are linear programming problems?
LPP stands for Linear Programming Problems. Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a set of linear constraints. An LPP involves identifying a set of decision variables, an objective function that depends linearly on these variables, and a set of constraints that specify linear relationships between the variables. The objective is to find the values of the decision variables that optimize the objective function while satisfying all the constraints.
To solve this problem, we need to use a system of linear inequalities to represent all the constraints. Let's start by defining the variables:
x = number of necklaces
y = number of wristbands
Now we can write the constraints:
Time constraint: Mack has 360 minutes to make the necklaces and wristbands.
40x + 25y ≤ 360
Production constraint: Mack wants to make no more than 12 items.
x + y ≤ 12
Non-negative constraint: Mack cannot make a negative number of necklaces or wristbands.
x ≥ 0, y ≥ 0
To maximize Mack's profit, we need to define the objective function as the total profit:
P = 3x + 2y
Now we can graph these constraints and find the feasible region:
40x + 25y ≤ 360
x + y ≤ 12
x ≥ 0, y ≥ 0
As shown in attachment 1.
The feasible region is the shaded polygon bounded by the lines: x=0, y=0, x + y ≤ 12, and 40x+25y=360.
To find the optimal solution, we need to evaluate the objective function at each corner point of the feasible region:
Corner point A: (0,0)
P = 3(0) + 2(0) = $0
Corner point B: (0,12)
P = 3(0) + 2(12) = $24
Corner point C: (4,8)
P = 3(4) + 2(8) = $28
Corner point D: (9,0)
P = 3(9) + 2(0) = $27
The maximum profit is $28, which occurs when Mack makes 4necklaces and 8 wristbands. Therefore, Mack should make 4 necklaces and 8 wristbands to maximize his profit.
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Show that x^5-5x^3+5x2+-1=0 has 3 equal roots and find that root
solve y = -1/4 x and x + 2y = 4
Answer:
8
Step-by-step explanation:
Not sure this is correct
substitute y in the second equation
X+2(-1/4x)=4
X-2/4x=4
4/4x-2/4x=2/4x
2/4x=4
4 /2/4
4•4/2
16/2
8
The solid below is dilated by a scale factor of 3 3. Find the volume of the solid created upon dilation.
Use the drawing tools to form the correct answer on the graph.
Plot function h on the graph.
The graph of the piecewise function is shown in the image attached below.
How to plot a piecewise function
In this problem we need to graph a piecewise function formed by two linear equations, a horizontal line and an oblique line. According to Euclidean geometry, a line can be formed from two distinct points set on Cartesian plane. The procedure is summarized below:
Plot the points (-5, - 4) and (- 4, - 4) of function f(x) = - 4.Generate the line of function f(x) for x < - 3.Plot the points (0, 5) and (5, 10) of function g(x) = x + 5.Generate the line of function f(x) for x ≥ - 3.Lastly, the piecewise function is shown in the image attached below.
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Graph the inequality in the coordinate plane. x < -7
For the given inequality x < -7 we draw a straight line at the point -7. The inequality is not strict, hence the line is dotted.
What is inequality?A mathematical statement called an inequality compares two numbers or expressions, typically by employing a symbol. Inequalities show that two quantities are not equal and that one is more than or less than the other.
Number lines can be used to visualise inequality conditions. Depending on whether the endpoint is part of the solution set or not, numbers that meet the inequality are shaded in or denoted by an open or closed circle. A number line would, for instance, show the inequality x > 2 with an open circle at 2 and shading to the right to show all values higher than 2.
In many areas of mathematics as well as in other disciplines like economics and social sciences, inequality is used.
For the given inequality x < -7 we draw a straight line at the point -7.
The inequality is not strict, hence the line is dotted.
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if a = -1/2 is a root of the quadratic equation 8x²-bx-3 . find the value of b, the other root, and (1/a - 1/b)²
Answer:
If a = -1/2 is a root of the quadratic equation 8x² - bx - 3, then we know that when x = -1/2, the equation is equal to 0. We can use this information to solve for b.
Substituting x = -1/2 into the equation, we get:
8(-1/2)² - b(-1/2) - 3 = 0
Simplifying and solving for b, we get:
2 - (b/2) - 3 = 0
b/2 = -1
b = -2
Therefore, b = -2 is the value we are looking for.
To find the other root, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. In this case, the constant term is -3 and the leading coefficient is 8. Therefore, the product of the roots is:
(-1/2) times the other root = -3/8
Solving for the other root, we get:
(-1/2) times the other root = -3/8
other root = (-3/8) / (-1/2)
other root = (3/8) * 2
other root = 3/4
Therefore, the other root is 3/4.
Finally, to find (1/a - 1/b)², we can substitute a = -1/2 and b = -2 into the expression:
(1/a - 1/b)² = (1/(-1/2) - 1/(-2))²
= (-2 - 1/2)²
= (-5/2)²
= 25/4
Therefore, (1/a - 1/b)² is equal to 25/4.
Answer:
[tex]b=2[/tex]
[tex]\textsf{Other root} = \dfrac{3}{4}[/tex]
[tex]\left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2=\dfrac{25}{4}[/tex]
Step-by-step explanation:
Roots are also called x-intercepts or zeros. They are the x-values of the points at which the function crosses the x-axis, so the values of x when f(x) = 0.
If x = α is a root of a polynomial f(x), then f(α) = 0.
Therefore, given that a = -1/2 is a root of the quadratic equation 8x² - bx - 3, substitute x = -1/2 into the equation and set it to zero:
[tex]\implies 8\left(-\dfrac{1}{2}\right)^2-b\left(-\dfrac{1}{2}\right)-3=0[/tex]
Solve for b:
[tex]\implies 8\left(\dfrac{1}{4}\right)+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies \dfrac{8}{4}+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies 2+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies \dfrac{1}{2}b-1=0[/tex]
[tex]\implies \dfrac{1}{2}b=1[/tex]
[tex]\implies b=2[/tex]
Therefore, the quadratic equation is:
[tex]\boxed{8x^2-2x-3}[/tex]
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
The constant term of the quadratic equation is -3 and the leading coefficient is 8. Let the other root be "r". Therefore:
[tex]\implies a \cdot r=\dfrac{-3}{8}[/tex]
Substitute the known value of a = -1/2 and solve for r:
[tex]\implies -\dfrac{1}{2} \cdot r=\dfrac{-3}{8}[/tex]
[tex]\implies r=\dfrac{3}{4}[/tex]
Therefore, the other root of the quadratic equation is 3/4.
To find the value of (1/a - 1/b)², substitute the given value of a and the found value of b into the equation and solve:
[tex]\implies \left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2[/tex]
[tex]\implies \left(\dfrac{1}{-\frac{1}{2}}-\dfrac{1}{2}\right)^2[/tex]
[tex]\implies \left(-2-\dfrac{1}{2}\right)^2[/tex]
[tex]\implies \left(-\dfrac{5}{2}\right)^2[/tex]
[tex]\implies \dfrac{25}{4}[/tex]
A popular video game retailer develops apps. The total revenue, in dollars per day, after x days can be modeled by the function R(x) = 2x3 + 11x2 – 23x – 14. The total cost, in dollars per day, after x days, can be modeled by the function C(x) = 13x2 + 11x + 16. After how many days does the retailer start making money?
1
2
5
7
After 5 days, the retailer starts making the money. The solution has been obtained by using the concept of break - even.
What is break - even?
The balance point known as "break-even" results in neither a profit nor a loss. Every value below the break-even mark represents a loss, whereas any value above it represents a gain.
We are given revenue function as R(x) = [tex]2x^3 + 11x^2 - 23x - 14[/tex].
The cost function is given as C(x) = [tex]13x^2 + 11x + 16[/tex].
The retailer will start earning only when the cost will be covered. This means that he will earn after when the R(x) = C(x).
So,
⇒ R(x) = C(x)
⇒ [tex]2x^3 + 11x^2 - 23x - 14[/tex] = [tex]13x^2 + 11x + 16[/tex]
⇒ [tex]2x^3[/tex] = [tex]2x^2 + 34x + 30[/tex]
⇒ [tex]x^3[/tex] = [tex]x^2 + 17x + 15[/tex]
⇒ [tex]x^3[/tex] - [tex]x^2 - 17x - 15[/tex] = 0
⇒ (x - 5) (x + 1) (x + 3) = 0
⇒ x = 5, -1, -3
Since, the days cannot be negative so, x = 5.
Hence, after 5 days, the retailer starts making the money.
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Question:
A popular video game retailer develops apps. The total revenue, in dollars per day, after x days can be modeled by the function R(x) = [tex]2x^3 + 11x^2 - 23x - 14[/tex]. The total cost, in dollars per day, after x days, can be modeled by the function C(x) = [tex]13x^2 + 11x + 16[/tex]. After how many days does the retailer start making money?
a. 1
b. 2
c. 5
d. 7
Use the Pythagorean Theorem to find the missing side of this right triangle. Estimate with a calculator (to one decimal place) if the answer doesn't simplify to a whole number.
Answer:
13.2
Step-by-step explanation:
using Pythagorean theorem, create the equation for the unknown side, x.
x^2+9^2=16^2
subtract 9^2
x^2=16^2-9^2
Use difference of squares.
x^2=(16-9)*(16+9)
Solve
x^2=7*25
x^2=175
Take the square root of both sides
x=sqrt175
x=13.2
If y =27 when x=9, determine y when x=11
Answer:
[tex]The \: problem \: probably \\ \: assumes \: direct \: variation \: \\ \\
y \: = \: k \: x \\ \\
IF \: so, \: then \: plug \: in \: the \: \\ values \: and solve \: for \: k \\ \\
27 \: = \: k(8) \\ \\
k \: = \: \frac{27}{8}
\\ \\
y \: = \: ( \frac{27}{8} )x. \\ \\ Now \: let \: x \: = \: 11 \\ \\
y \: = \: ( \frac{27}{8} )11 = \\ \\ \frac{27(11)}{8} \\ \\ = 37.125 = \\ \\ \frac{371}{8y} \\ \\
y \: = \frac{371}{8}
[/tex]
Erica has 2. 2 pounds of uncooked hamburger meat. A hamburger patty weighs about 6 ounces before it is cooked. Which is the maximum number of hamburger patties that Erica can make? i need help!!
Erica can make a maximum of 5 hamburger patties with the amount of meat she has.
What role do unit conversions have in mathematical and scientific calculations?In scientific and mathematical computations, unit conversions are crucial because they guarantee that data are stated in the proper units, enabling consistency and correctness. To combine or compare numbers and to understand the findings meaningfully, values must be converted to a standard unit of measurement. Calculation mistakes and erroneous conclusions can result from improper unit conversion.
Given that, a hamburger patty weighs about 6 ounces before it is cooked.
Thus,
2.2 pounds = 35.2 ounces (1 pound = 16 ounces)
35.2 ounces / 6 ounces per patty = 5.87
Erica can make a maximum of 5 hamburger patties with the amount of meat she has.
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Let θ be an angle in standard position, with its terminal side in quadrant IV such that tanθ = -7/9. Find the exact values of sinθ and cosθ.
The value of sin θ is -[tex]\frac{7\sqrt{130} }{130 }[/tex] and cos θ is [tex]\frac{9\sqrt{130} }{130 }[/tex] . The solution has been obtained by using trigonometry.
What is trigonometry?
The study of right-angled triangles, including their sides, angles, and connections, is referred to as trigonometry.
We are given that tan θ is -7/9. The minus sign is there because it lies in the fourth quadrant.
This means that the perpendicular is 7 and the base is 9.
Let the hypotenuse be x.
Now, by using Pythagoras theorem, we get
⇒ [tex]7^{2}[/tex] + [tex]9^{2}[/tex] = [tex]x^{2}[/tex]
⇒ 49 + 81 = [tex]x^{2}[/tex]
⇒ [tex]x^{2}[/tex] = 130
⇒ x = √130
By trigonometry,
⇒ Sin θ = -[tex]\frac{7}{\sqrt{130} }[/tex]
⇒ Sin θ = -[tex]\frac{7\sqrt{130} }{130 }[/tex]
Similarly,
⇒ Cos θ = [tex]\frac{9}{\sqrt{130} }[/tex]
⇒ Cos θ = [tex]\frac{9\sqrt{130} }{130 }[/tex]
Hence, the values for sin θ and cos θ have been obtained.
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