Answers
Answer:
a. The x-coordinate of the minimum point of this parabola is halfway between the two roots (-1 and 2). So the x-coordinate here is 1/2, and we have:
[tex]f( \frac{1}{2} ) = { (\frac{1}{2} )}^{2} - \frac{1}{2} - 2 = - 2 \frac{1}{4} [/tex]
So the coordinate of the turning point is (1/2, -2 1/4), or (.5, -2.25).
b. The roots of this parabola are at (-1, 0) and (2, 0).
Related Questions
The graph of Y= | X |is transformed as shown in the graph below. Which equation represents the transformed function?
Answers
An equation that represents the transformed function include the following: A. y = |1/4x|.
What is a transformation?In Mathematics and Geometry, a transformation is the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed;
What is dilation?In Geometry, dilation can be defined as a type of transformation which typically changes the size of a geometric object, but not its shape.
Since the vertex of the two absolute value functions is located at points (0, 0), we would have to dilate the coordinates of the pre-image by using a scale factor of 1/4 centered at the origin as follows:
y = |1/4x|
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Create a rational inequality for which -2 < x < 1 is the solution.
(Use the chart to show work)
Answers
A rational inequality for which -2 < x < 1 is the solution is (3x - 1)/(x + 2) > 2.
What is a rational equation?An inequality that incorporates one or more rational expressions—that is, expressions that can be expressed as the quotient of two polynomial expressions—is referred to as a rational inequality. The same techniques used to resolve other kinds of inequalities, such as graphing, testing intervals, or algebraic manipulation, can also be used to resolve rational inequalities. The denominator of a rational expression cannot, however, equal zero, hence extra care must be made to identify and rule out any values of the variable that would do so, as these values would render the expression undefinable.
A rational inequality for which -2 < x < 1 is the solution is:
(3x - 1)/(x + 2) > 2
In the equation when,
For x -2, the inequality holds because both the denominator and the numerator are negative.
When -2 x 1, the inequality is true since both the denominator and the numerator are positive.
When x > 1, the denominator and numerator are both positive, but the expression's value is less than 2, rendering the inequality untrue.
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Can you anyone help me out with this?
I don’t know how to solve the equation and check for extraneous solutions!!!! PLEASEEEEEEEEE URGENT
Answers
The solution to the equation 1 is p = -7. and the only solution to the equation2 is k = 2.
what is equation?
An equation is a mathematical statement that asserts the equality of two expressions. It usually contains one or more variables (letters that represent unknown values) and an equal sign. The goal of solving an equation is to find the values of the variables that make the equation true.
In the given question,
Starting with -49 = -7√(p+8):
Dividing both sides by -7, we get:
7√(p+8) = 7
Dividing both sides by 7, we get:
√(p+8) = 1
Squaring both sides, we get:
p+8 = 1
Subtracting 8 from both sides, we get:
p = -7
Therefore, the solution to the equation is p = -7.
Starting with √(6-k) = k:
Squaring both sides, we get:
6-k = k²
Rearranging, we get:
k²+ k - 6 = 0
We can factor this quadratic equation as:
(k+3)(k-2) = 0
Therefore, the solutions to the equation are:
k+3 = 0 or k-2 = 0
Solving for k, we get:
k = -3 or k = 2
However, we need to check whether these solutions satisfy the original equation, since we squared both sides and introduced extraneous solutions:
When k = -3:
√(6-k) = √(6-(-3)) = √9 = 3
k = -3 is not a solution.
When k = 2:
√(6-k) = √(6-2) = √4 = 2
k = 2 is a solution.
Therefore, the only solution to the equation is k = 2.
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[TQ] is a tangent at A to a circle, centre O, /AB/ is a chord. if <BAT=y, show that <BOA=2y
Answers
HELP ASAP PLEASE I HAVE 12 HOURS:(
Answers
The value of angle EFG is 151°.
What is angle?
An angle is a measure of the amount of rotation or turn between two lines, rays, or planes that meet at a common point called the vertex. The unit of measurement for angles is degrees or radians.
An angle is formed when two rays share a common endpoint, known as the vertex. The rays are often referred to as the arms or legs of the angle. The angle is measured in degrees, with a full circle measuring 360 degrees.
Angles can be classified according to their size. An acute angle measures less than 90 degrees, a right angle measures exactly 90 degrees, an obtuse angle measures more than 90 degrees but less than 180 degrees, and a straight angle measures exactly 180 degrees.
Angles can also be complementary or supplementary. Two angles are complementary if their sum is equal to 90 degrees, and they are supplementary if their sum is equal to 180 degrees.
Angles are used in many fields, including mathematics, physics, engineering, and architecture. They are used to measure distances, calculate trajectories, and design buildings and structures.
Here given,
[tex]\angle EFH = {44}^{o} [/tex] and
[tex] \angle \: HFG = {107}^{o} [/tex]
Here
[tex]\angle EFG = \angle EFH + \angle HFG \\ \angle EFG = {44}^{o} + {107}^{o} \\ = {151}^{o} [/tex]
So, required value of angle EFG is 151°.
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Find the difference.
Answers
Answer:
[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12
Hope this helps!
Step-by-step explanation:
[tex]t^{4}[/tex] - 1.5[tex]t^{2}[/tex] + t - 12 + 9.5[tex]t^{2}[/tex] + 7t : Combine like terms
[tex]t^{4}[/tex] + ( 9.5 - 1.5 )[tex]t^{2}[/tex] + ( 7t + t ) - 12
[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12
Audrey's mother bought a dozen muffins. There were
4 blueberry muffins, 4 chocolate chip muffins, and 4
banana nut muffins in the box. Audrey's mother ate a
chocolate chip muffin. What is the probability that
Audrey will randomly select a blueberry muffin if she
chooses after her mother?
A) 1/3
B) 4/11
C) 3/11
D) 1/12
Please answer the warm-up through the poll button.
Answers
There were originally 12 muffins in total, and Audrey’s mother ate one, resulting in 11 muffins now.
Aubrey wants a blueberry muffin, and since her mother didn’t east a blueberry muffin, there are still 4 muffins in total.
To get the probability, you need to divide how many of something there is divided by the total of things.
In this case, its 4 muffins divided by 11 muffins in total.
Therefore, your answer is B: 4/11
for a time series for simple exponential smoothing, a larger alpha is smoother than a shorter period. group of answer choices true false
Answers
The important to select the appropriate value of alpha based on the characteristics of the data and the goals of the forecasting model.
The statement "for a time series for simple exponential smoothing, a larger alpha is smoother than a shorter period" is true.
In simple exponential smoothing, the forecast for the next period is based on the previous forecast and the difference between the actual value and the previous forecast. The smoothing parameter alpha controls the weight given to the most recent observation compared to the previous forecasts.
A larger alpha places more weight on the most recent observation, resulting in a forecast that is more responsive to recent changes in the data. As a result, the forecast will exhibit less volatility or variability, making it smoother. This means that a larger alpha will lead to a more stable forecast with less fluctuation around the trend.
On the other hand, a smaller alpha places less weight on the most recent observation, resulting in a forecast that is less responsive to recent changes in the data.
As a result, the forecast will be more volatile or variable, making it less smooth. This means that a smaller alpha will lead to a forecast that is more susceptible to fluctuations around the trend.
In summary, the choice of alpha is a trade-off between stability and responsiveness to changes in the data. A larger alpha will lead to a smoother forecast, while a smaller alpha will lead to a more volatile forecast.
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delia bought a bedroom set for $3600 the sales tax on the purchase was 246.60 what was the sale tax rate 
Answers
Answer: 6.85%
Step-by-step explanation:
In a circle O shown, m
Answers
The sum of the angles in triangle MNP is 180, and subtracting 135 from both sides gives 45 degrees. With the angle sum property of triangle, ∠MNP = 121°.
What are angles?An angle is the result of the intersection of two lines.
An "angle" is the length of the "opening" between these two beams.
Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.
In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.
The two primary components of an angle are the limbs and the vertex.
The joint vertex is the common terminal of the two beams.
According to the given question,
45° + 90° + ∠MNP = 180°
135° + ∠MNP = 180°
∠MNP = 45°
Hence, With the angle sum property of triangle, ∠MNP = 121°.
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If an angle is twice its supplement, the angle is ____ degrees.
Select one:
a. 30
b. 120
c. 60
d. 5
Answers
All the units of area
Answers
Answer:
Need to be more specific. What units?
Step-by-step explanation:
steve is saving up for a new bike, so he gets a summer job at the sugar scoops ice cream shop. all of the money he earns from tips goes toward the new bike. a customer leaves a tip 50% of the time, and there are 5 customers in line. how likely is it that all of the customers will give steve a tip? steve simulates the situation by rolling a six-sided die 5 times. each time an odd number appears, it represents a tip. this table shows the results of 100 trials: number of times an odd number appears 0 1 2 3 4 5 number of trials 2 15 33 31 16 3 based on steve's results, what is the probability that all of the customers will give him a tip?
Answers
The probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.
Based on the table presented, Steve simulated the situation by rolling a six-sided die 5 times. Each time an odd number appears, it represents a tip. This table shows the results of 100 trials: number of times an odd number appears 0 1 2 3 4 5 number of trials 2 15 33 31 16 3.
The customer leaves a tip 50% of the time, we can calculate the probability by multiplying the probability of a single tip (0.5) by itself 5 times.
Probability = (0.5) * (0.5) * (0.5) * (0.5) * (0.5)
Probability = 0.03125
From the table presented, it can be concluded that Steve got all of the five customers tipping him in 3 trials. Therefore, the probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.
The probability that an event will occur is the ratio of the number of ways that the event can occur to the number of total possible outcomes. In this scenario, the probability of a customer giving Steve a tip is 0.5.
Since the customer either gives a tip or doesn't.
In other words, there are two possible outcomes (gives a tip or doesn't give a tip) and only one outcome results in a tip (gives a tip).
Therefore, the probability of getting a tip from one customer is 1/2 or 0.5.
Since there are five customers, the probability of all five customers giving Steve a tip is calculated by multiplying the probability of each customer giving a tip.
This is because the occurrence of each event is independent of the others.
Therefore, the probability of all five customers giving Steve a tip is:
0.5 x 0.5 x 0.5 x 0.5 x 0.5 = 0.03125 or 3.125%.
From the table presented, it can be concluded that Steve got all of the five customers tipping him in 3 trials.
Therefore, the probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.
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Sams electric bill was $200 dollars for the month of June. The air conditional accounts for 1/2 of the bill and the water heater accounts for 1/5 of the bill. How much did it cost to one each appliance in June?
Answers
Answer:
$100 for air conditioning and $40 for water heater
Step-by-step explanation:
$200÷2= 100
200÷5= 40
What is the length of the missing leg? if necessary, round to the nearest tenth.
Answers
The length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.
What is the Pythagoras rule?The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Pythagoras rule we can evaluate for b as follows:
65² = 63² + b²
b = √(65² - 63²) {make b the subject}
b = √(4225 - 3969)
b = √256
b = 16
Therefore, the length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.
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Help Been Stuck on This
Answers
PEQ has a value of y of 60°.
The Sum of Angles property states that the sum of a polygon's internal angles is equal to (n - 2) times 180 degrees, where n is the number of sides.The polygon's s.
What is angle sum property?
The angle sum property in geometry asserts that the sum of a polygon's internal angles is equal to (n - 2) times 180 degrees, where n is the number of sides of the polygon.
In a triangle (n=3), for example, the total of the interior angles is (3 - 2) 180 = 1 180 = 180 degrees.
The sum of internal angles for a quadrilateral (n=4) equals (4 - 2) 180 = 2 180 = 360 degrees.
This is a fundamental geometric idea that can be used to calculate the value of an unknown interior angle in a polygon if the other interior angles are known. It can also be used to determine whether or not a given set of angles can create a valid polygon.
In PEQ, the angle sum attribute is used.
Hence, EPQ + EQP + PEQ = 180°
40° + 80° + y = 180°
120° + y = 180°
y = 180° - 120°
y = 60°
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heather and her two friends went to the nail salon to gte a manicure and a pedicure. manicures cost $18 and pedicures cost $30. heather upgraded her manicure to gel polish which costs an extra $10. The group paid 4% sales tax and left a tip that was about 5 times the ammount of tax they paid. How much did they spend in total including tax and tip.
Answers
they spent a total of $215.76 including tax and tip.
What is the arithmetic operation?
The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.
Therefore, the total cost per person without tax and tip would be:
$28 + $30 = $58
Since there were three people, the total cost without tax and tip would be:
$58 × 3 = $174
The sales tax would be 4% of the total cost:
$174 × 0.04 = $6.96
The tip left was about 5 times the amount of tax, so:
Tip = $6.96 × 5 = $34.80
Therefore, the total cost including tax and tip would be:
$174 + $6.96 + $34.80 = $215.76
So, they spent a total of $215.76 including tax and tip.
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kindly answer with working and i will mark you brainliest
Answers
The perimeter to the nearest tenth, we get: Perimeter ≈ 18.4 units.
What is perimeter?
We can start by using the Pythagorean theorem to find the length of AC:
AC² = AB² + BC²
AC² = 2² + 1.5²
AC² = 4.25
AC = √(4.25)
AC ≈ 2.06 units
Next, we can use the fact that AC is common to both triangles to find the length of CD:
ACD is a right triangle, so we can use the Pythagorean theorem:
CD² = AD² - AC²
CD² = 6.5² - 2.06²
CD² ≈ 39.86
CD ≈ 6.31 units
Now we can find the perimeter of the figure by adding the lengths of all three sides:
Perimeter = AB + BC + CD + DA + AC
Perimeter = 2 + 1.5 + 6.31 + 6.5 + 2.06
Perimeter ≈ 18.37 units
Therefore, none of the given options is the correct answer. However, if we round the perimeter to the nearest tenth, we get:
Perimeter ≈ 18.4 units
And the closest option to this value is option 3) 16 units, which is not very close. Therefore, we can conclude that none of the given options is a correct approximation of the perimeter of the figure.
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PLEASE HELP I NEED IT ASAP
Explain using the change of base formula and evaluating the change of base formula
Answers
The change of base formula is: log_b(x) = log_a(x) / log_a(b)
What do you mean by the term Change of base ?The term "change of base" refers to the process of converting a logarithm from one base to another. A logarithm is an exponent that indicates the power to which a base must be raised to obtain a given number. The most common logarithmic bases are 10, e (Euler's number), and 2, but logarithms can be taken to any base.
The change of base formula is a mathematical formula that allows you to convert logarithms from one base to another. It is useful when you need to calculate logarithms that cannot be easily evaluated on a calculator or when you need to work with logarithms in different bases.
The change of base formula is:
log_b(x) = log_a(x) / log_a(b)
Where:
log_b(x) is the logarithm of x to the base b
log_a(x) is the logarithm of x to the base a
log_a(b) is the logarithm of b to the base a
To use the change of base formula, you first need to identify the base of the logarithm that you want to convert and the base that you want to convert it to. Then, you can plug the values into the formula and simplify the expression.
For example, let's say you want to convert log_2(8) to base 10. Using the change of base formula, you can write:
log_2(8) = log_10(8) / log_10(2)
To evaluate this expression, you can use a calculator to find that:
log_10(8) = 0.9031
log_10(2) = 0.3010
Substituting these values into the formula gives:
log_2(8) = 0.9031 / 0.3010
Simplifying this expression gives:
log_2(8) = 3
Therefore, log_2(8) is equal to 3 when evaluated in base 10.
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If an exteiror angle measure of a regular polygon is 120 how many sides does the polygon have
Answers
The regular polygon with an exterior angle measuring 120 will have 3 equal sides. Thus, an equilateral triangle have three equal sidea and angles is a regular polygon.
What do you mean by regular polygon?Regular polygon: A regular polygon is both "equiangular" and "equilateral," meaning it has all sides that are the same length and interior angles that are the same size.
The following algorithm determines an ordinary polygon's exterior angle:
Exterior Angle = [tex]360 Degrees / Sides[/tex]
We can therefore make up an equation if the exterior angle measurement is [tex]120[/tex] degrees:
[tex]120 = 360/n[/tex] the number of sides being n
[tex]n (sides) = 360/120 = 3[/tex]
Three sides [tex]=[/tex] triangles are polygons with three sides.
An equilateral triangle is a regular three-sided polygon.
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Pls help ill offer 13 points
Answers
Answer:
Point K
Step-by-step explanation:
Which Point represents the ordered pair
( -4[tex]\frac{1}{2}[/tex], -1)
Left -4[tex]\frac{1}{2}[/tex] we have points J and K
Down -1 we have point K
So, the answer is point K
The diameter of the circle above is 6cm what is the circumference of the circle?
Answers
Answer:
C ≈ 18.84 cm
Step-by-step explanation:
circumference = π · diameter
We can plug the given diameter into this formula, approximating π as 3.14.
C = 3.14 · 6 cm
C ≈ 18.84 cm
We know that,
[tex] \bf \leadsto Radius = Diameter / 2[/tex]
So,
[tex] \rightarrow \: \sf \: Radius = \frac{6}{3} \\ \\ \rightarrow \: \sf \: Radius = 3[/tex]
[tex] \rule{140pt}{5pt}[/tex]
Now,
[tex] \bf \bigstar \: Circumference \: of \: circle = 2πr[/tex]
[tex]→ \sf \: 2 × 3.14 × 3 \\ \\ → \sf \: 6 × 3.14[/tex]
[tex] \bf \: 18.84cm[/tex]
Option a is correct
[tex] \underline{ \rule{190pt}{6pt}}[/tex]
The given table of values shows a linear relationship.
x 1 3 5 7 9
y 8 14 20 26 32
What is the rate of change for this relationship?
A.
3
B.
-3
C.
D.
Answers
The rate of change (slope) for a linear relationship can be found by taking the difference in the y-values divided by the difference in the corresponding x-values.
In this case, the change in y from x=1 to x=9 is 32-8=24, and the change in x is 9-1=8. Therefore, the rate of change is:
rate of change = (change in y) / (change in x) = 24/8 = 3
So the answer is A. 3.
the wall street journal reported that of taxpayers with adjusted gross incomes between and itemized deductions on their federal income tax return. the mean amount of deductions for this population of taxpayers was . assume that the standard deviation is . use z-table. a. what is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within of the population mean for each of the following sample sizes: , , , and ? round your answers to four decimals. b. what is the advantage of a larger sample size when attempting to estimate the population mean? round your answers to four decimals. a larger sample - select your answer - the probability that the sample mean will be within a specified distance of the population mean. in this instance, the probability of being within of ranges from for a sample of size to for a sample of size .
Answers
a. The z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999
b. The probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.
a. We can use the formula for the standard error of the mean:
SE = σ/√n
where σ is the population standard deviation, n is the sample size, and SE is the standard error.
For a sample size of n = 25:
SE = 0.0588/√25 = 0.01176
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.01176 < Z < 0.02/0.01176) = P(-1.70 < Z < 1.70) = 0.9045
For a sample size of n = 50:
SE = 0.0588/√50 = 0.00832
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00832 < Z < 0.02/0.00832) = P(-2.41 < Z < 2.41) = 0.9872
For a sample size of n = 100:
SE = 0.0588/√100 = 0.00588
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00588 < Z < 0.02/0.00588) = P(-3.40 < Z < 3.40) = 0.9984
For a sample size of n = 200:
SE = 0.0588/√200 = 0.00416
Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:
P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999
b. The advantage of a larger sample size is that it leads to a smaller standard error of the mean, which means that the distribution of sample means is narrower and more closely centered around the population mean. As a result, the probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.
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Which statement describes the end behavior of this absolute value function?
Answers
As x approaches positive infinity, f(x) approaches positive infinity.
What is the absolute value function?In algebra, an absolute value function is a function where the variable is inside the bars denoting absolute values. The absolute value function is also known as the modulus function, and its most popular representation is f(x) = |x|, where x is a real number.
Here,
We have to find the end behavior of this absolute value function.
In the graph, it is shown that the absolute value function has a maximum of 3 at x = 3.
The function f(x) reaches positive infinity in both cases when x reaches either minimum or maximum value.
Hence, As x approaches positive infinity, f(x) approaches positive infinity.
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Alex bought a new car for his daughter. He knows the value of the car will decrease at a constant rate. After 3 years, the value of the car is $15,000. After 5 years, the value of the car of $11,000. Write and solve a linear equation to find the value of the car after 8 years. (Drop down and select)
y-______=______(x-____)
The value of the car after 8 years will be ______.
Answers
Answer: To find the value of the car after 8 years, we can use the given information to create a linear equation in the form:
y - y1 = m(x - x1)
Where:
y is the value of the car after 8 years (what we want to find)
y1 is the value of the car after a given time (either 3 or 5 years)
x is the given time (either 3 or 5 years)
x1 is the starting time (when the car was purchased)
m is the rate of decrease (slope of the line)
We can use the two given data points to find the slope of the line:
m = (y2 - y1) / (x2 - x1)
m = (11,000 - 15,000) / (5 - 3)
m = -2,000 per year
Now, we can use one of the data points to solve for the y-intercept of the line:
y - y1 = m(x - x1)
y - 15,000 = (-2,000)(3 - x1)
y - 15,000 = (-2,000)(3 - x1)
y - 15,000 = (-6,000 + 2,000x1)
y = 2,000x1 - 6,000 + 15,000
y = 2,000x1 + 9,000
So the linear equation that represents the value of the car after x years is:
y - 15,000 = (-2,000)(x - 3)
Simplifying this equation gives:
y - 15,000 = -2,000x + 6,000
Now, we can use x = 8 to find the value of y:
y - 15,000 = -2,000(8) + 6,000
y - 15,000 = -2,000
y = $13,000
Therefore, the value of the car after 8 years will be $13,000.
So the completed expression is:
y - 15,000 = -2,000(x - 3)
The value of the car after 8 years will be $13,000.
Step-by-step explanation:
1) One week the price of gas dropped by $0.05 per gallon. Steven travels 37 miles each way to work and her car travels 30 miles on each gallon of gas. What were his total savings over a 5 day work week.
Answers
37 miles each way * 2 = 74 miles round trip
74 miles / 30 miles per gallon = 2.47 gallons
This means that Steven needs approximately 2.47 gallons of gas each day to commute to work.
Now, let's calculate how much Steven saved on gas during the week when the price dropped by $0.05 per gallon:
2.47 gallons per day * $0.05 per gallon = $0.1235 per day
So Steven saved approximately $0.12 per day on gas during that week.
To calculate his total savings over the 5-day work week, we just need to multiply his daily savings by the number of days:
$0.12 per day * 5 days = $0.60
Therefore, Steven saved about $0.60 on gas during that week.
is it possible to find a set of bij and cij such that the state-space equation is controllable? observable?
Answers
The controllability and observability matrices of the given state-space equation were calculated, but no set of bij and Cij was found to make the system controllable or observable. Therefore, the system is not controllable or observable.
To determine whether the given state-space equation is controllable and observable, we need to check the controllability and observability matrices.
The controllability matrix is given by:
[ B AB A^2B A^3B A^4B ]
where A and B are the state and input matrices, respectively.
The observability matrix is given by:
[ C ]
[ CA ]
[ CA^2 ]
[ CA^3 ]
[ CA^4 ]
where C is the output matrix.
Using the given state-space equation, we can calculate the controllability and observability matrices as follows:
A = [ 1 0 0 0 0 ]
[ L0 1 0 0 1 ]
[ b12 1 0 0 0 ]
[ 0 0 1 0 0 ]
[ b21 bzz 0 1 1-L ]
B = [ 0 ]
[ 0 ]
[ 0 ]
[ 0 ]
[ 1 ]
C = [ 1 0 0 0 0 ]
[ 0 1 0 0 0 ]
[ 0 0 1 0 0 ]
Controllability Matrix:
[ B AB A^2B A^3B A^4B ] =
[ 0 0 0 0 1 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ] [ 0 0 0 0 0 ]
[ 0 0 0 0 0 ]
[ 0 bzz b21bzz 0 Lb51 ] [ 0 L0 b12bzz Lb51 0 ] [ L0 b12bzz L0b21bzz 0 0 ]
[ b12bzz 0 L0b21bzz 0 0 ] [ 0 0 L0b12bzz 0 0 ]
Observability Matrix:
[ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ] [ 1 0 0 0 0 ]
[ 0 1 0 0 0 ] [ 0 L0 1 b12 1 ] [ L0 b12bzz L0b21bzz 0 0 ] [ b12bzz 0 L0bb21bzz 0 0 ]
To elaborate on the solution, the controllability matrix shows that some of the columns are linearly dependent on the others, which means that it is not possible to find a set of inputs that can steer the system from any initial state to any desired state in finite time. Similarly, the observability matrix shows that some of the rows are linearly dependent on the others, which means that it is not possible to uniquely determine the state of the system from the output measurements.
Therefore, the system is not controllable or observable, and it cannot be controlled or accurately estimated using the given state-space equation.
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--The given question is incomplete, the complete question is given
" Is it possible to find a set of bij and a set of Cij such that the following state-space equation is controllable? Observable? (If the answer is yes, find a set) i(t) = 1 0 0 0 LO 1 0 0 01 5611 b12 1 0 0 0 |b21 bzz 0 1 1 0 X(t) + b31 b32|u(t) 0 0 1 0 | b41 buz 0 0 0 1- Lb51 652 [C11 y(t) = C21 C31 C12 C22 C32 C13 C23 C33 C14 C24 C34 C15 C25 x(t) C35
A jogger running around a rectangular park takes a shortcut back to his car by running 91 yards from one corner to the opposite corner. If the park is 35 yards long, what is the width?
Answers
Step-by-step explanation:
The length and the width of the rectangle are LEGS of a RIGHT triangle
the 91 yards is the hypotenuse .....so you can use the Pythagorean Theorem for right triangles
hyp^2 = leg1^2 + leg2^2
91^2 = 35^2 + width^2
91^2 - 35^2 = width^2
7056 = width^2
then width = sqrt (7056) = 84 yards
3200 dollars is placed in an account with an annual interest rate of 8%. To the nearest tenth of a year, how long will it take for the account value to reach 9000 dollars?
Answers
It will take approximately 13.6 years for the account value to reach $9000.
What is an account value?
We can use the formula for compound interest to solve this problem:
A = P*[tex](1 + r/n)^{nt}[/tex]
where A is the final amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.
We want to find t, so we can rearrange the formula as:
t = (1/n) × log(A/P) / log(1 + r/n)
Substituting the given values:
P = $3200
r = 8% = 0.08
A = $9000
n = 1 (interest is compounded annually)
Using a calculator:
t = (1/1) × log(9000/3200) / log(1 + 0.08/1)
t ≈ 13.6
Rounding to the nearest tenth of a year, it will take approximately 13.6 years for the account value to reach $9000.
What is an interest ?
Interest is the cost of borrowing money, or the compensation paid by a borrower to a lender for the use of money. When you borrow money, you are usually required to pay back the amount borrowed (the principal) plus an additional amount of money, which is calculated as a percentage of the principal and represents the interest.
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