13. The reasonable domain for the function V(w) = (60-4w)w^2 is when the volume is greater than 0.
14. The graph will look like a parabola with a maximum point at w = 7.5.
15.The coordinates of the maximum point are (7.5, 506.25).
16. The width of the box that produces the maximum volume is 7.5 inches, as found in the previous question.
17. a. The expression for the height of the tube in terms of the radius is h = 60 - 2r.
This means that the values of w must be between 0 and 15, since the volume becomes negative when w is greater than 15. Therefore, the domain can be expressed as an inequality as 0 < w < 15, in interval notation as (0, 15), and in set notation as {w | 0 < w < 15}.
14. To sketch a graph of the function V(w) = (60-4w)w^2 over the domain (0, 15), we can plot points at different values of w and connect them with a smooth curve. The scale on each axis can be 1 unit per grid line.
15. Using a graphing calculator, we can find the coordinates of the maximum point of the function V(w) = (60-4w)w^2 by using the maximum function.
b. The expression for the volume of the tube in terms of the radius is V = πr^2h = πr^2(60 - 2r).
c. To find the radius that produces the maximum volume, we can take the derivative of the volume function and set it equal to 0. This gives us 2πr(60 - 4r) = 0. Solving for r, we get r = 7.5 cm.
d. The maximum volume of the tube is V = π(7.5)^2(60 - 2(7.5)) = 1265.49 cm^3.
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For what value of r is the statement an identity? (x^(2)-x-8)/(x-10)=x+9+(r)/(x-10) provided that x!=10
To find the value of r that makes the statement an identity, we need to set the two sides of the equation equal to each other and solve for r.
First, let's multiply both sides of the equation by (x-10) to eliminate the denominator:
(x^(2)-x-8) = (x-10)(x+9) + r
Next, let's expand the right side of the equation:
x^(2)-x-8 = x^(2) + 9x - 10x - 90 + r
Now, let's rearrange the equation and solve for r:
r = x^(2)-x-8 - x^(2) - 9x + 10x + 90
r = 90
Therefore, the value of r that makes the statement an identity is 90.
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rate of change is 1 and point is (-3,-3) what is the slope intercept
Answer:
0
Step-by-step explanation:
y = 1x + b
-3 = 1(-3) + b
-3 = -3 + b
+3 +3
0 = b
One edge of a painting is 6 in. longer than the other edge. The painting has a 2-inch-wide frame. The function f(x) = ×2 + 14x + 40 represents the total area of the painting and frame. Find the total area of the painting and the frame if the longer side of the frame is 14 inches long.
The total area of the painting and frame is 112 in²
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables. Equations can either be linear, quadratic, cubic and so on depending on the degree.
Let x represent the length of the smaller edge.
One edge of a painting is 6 in. longer than the other edge. Hence:
Longer edge = x + 6
The painting has a 2-inch-wide frame.
Smaller edge length with frame = x + 2 + 2 = x + 4
Longer edge length with frame = (x + 6) + 2 + 2 = x + 10
Total area = (x + 4)(x + 10)
Total area = x² + 14x + 40
The longer side of the frame is 14 inches long, hence:
x + 10 = 14
x = 4 in
Total area = x² + 14x + 40 = 4² + 14(4) + 40 = 112 in²
The total area is 112 in²
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u think u guys can help with this?
The IQR is the difference between Q3 and Q1: IQR = Q3 - Q1 = 4 - 1 = 3 hours.
The presented graph displays a line or dot plot of a data set that indicates the average daily activity time of a group of individuals over the previous week.
We must locate the data set's middle value in order to get the median. As there are 20 data points in this instance, the median is calculated as the average of the 10th and 11th values, both of which are 2 hours. As a result, 2 hours is the median amount of exercise per day.
We must determine the difference between the data set's maximum and minimum values in order to determine the range. The ranges are 0 hours for the minimum and 5 hours for the highest. The range is therefore 5 - 0 = 5 hours.
Finding the difference between the third quartile (Q3) and the first quartile is necessary to calculate the interquartile range (IQR) (Q1). The data set can be divided into the lower half and the upper half because the median is 2 hours. 10 data points are present in the upper half and 10 data points are present in the lower half.
The median of the lower half of the data set must be located in order to determine Q1. The median is the average of the fifth and sixth values, both of which are one hour because there are ten data points in the lower half. Q1 is therefore 1 hour.
We must determine the median of the upper half of the data set in order to determine Q3. The median is the average of the fifth and sixth values, which are both 4 hours, as there are 10 data points in the upper half. Q3 is therefore 4 hours.
The IQR represents the variation between Q3 and Q1: IQR is equal to Q3 - Q1 (4 - 1 = 3 hours).
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The diameter of a circle is 10 ft. Find its area to the nearest whole number.
When the diameter οf the is 10 feet, then the area οf the circle is 78.54 ft².
What is circle?A circle is created in the plane by each pοint that is a specific distance frοm anοther pοint (center). Hence, it is a curve made up οf pοints that are separated frοm οne anοther by a defined distance in the plane. Mοreοver, it is rοtatiοnally symmetric abοut the centre at every angle. Every pair οf pοints in a circle's clοsed, twο-dimensiοnal plane are evenly spaced apart frοm the "centre." A circular symmetry line is made by drawing a line thrοugh the circle. Mοreοver, it is rοtatiοnally symmetric abοut the centre at every angle.
The circle's diameter is specified as 10 feet. Since we already knοw that the circle's diameter is twice its radius, we can calculate its radius as fοllοws:
diameter = radius / 2 = 10 / 2 = 5 feet
Area οf circle = πr²
π5²
= π5 × 5
= 78.54 ft²
The size οf the circle is 79 square feet when the answer is rοunded tο the next whοle number.
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The area of the circle to the nearest whole number is 79 square feet.
What is the diameter?
In geometry, the diameter of a circle is defined as the longest straight line segment that can be drawn between any two points on the circle, passing through the center of the circle. It is twice the length of the radius of the circle.
The formula for the area of a circle is A = πr^2, where r is the radius of the circle.
Given that the diameter of the circle is 10 feet, we can find the radius by dividing the diameter by 2:
radius = diameter / 2 = 10 ft / 2 = 5 ft
Now we can use the formula to find the area of the circle:
A = πr^2
= π(5 ft)^2
= 25π square feet
To get the answer to the nearest whole number, we can use the approximation π ≈ 3.14:
A ≈ 25 × 3.14
≈ 78.5
Therefore, the area of the circle to the nearest whole number is 79 square feet.
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Retention rates (the percentage of freshmen who return for their sophomore year) are of great interest to colleges and universities. One university's retention rate is typically about 80%. This year, they have 3,000 freshmen. How many of those do we expect to return for their sophomore year?
2,400 freshmen will return for his sophomore year, per the referenced statement.
What is a business interest?The cost a company pays a bank (creditor) for a loan is called interest. Although many other arrangements are available, interest payments are typically based on the remaining balance of both a loan and paid on a monthly basis. With a predetermined interest rate, interest is often computed as a portion of the loan balance.
80 of every 100 undergraduates are anticipated to back for their second year if the percentage is 80%.
We may multiply the overall amount of newcomers by a retention rate to get the number of undergraduates anticipated to return:
80 percent of 3,000 freshman equals 0.80 x 3,000, or 2,400.
So we may anticipate 2,400 freshman coming back for our sophomore year.
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Suppose the world's current oil reserves are 1820 billions barrels. If, on average, the total reserves is decreasing by 25 billion barrels of oil each year.
a. Give a linear equation for the remaining oil reserves, R, in terms of t, the number of years since now.
b. Seven Yeats from now, what will the oil reserves be?
c. If the rate of depletions isn't changed, when will the world's oil reserves be depleted?
a. R = 1820 - 25t
b. R = 1820 - 25(7) = 1345 billion barrels
c. The world's oil reserves will be depleted when R = 0, so 25t = 1820, t = 72.8 years
a. The linear equation for the remaining oil reserves, R, in terms of t, the number of years since now, is R = 1820 - 25t. This equation represents the starting amount of oil reserves (1820 billion barrels) and subtracts the amount that is depleted each year (25 billion barrels) multiplied by the number of years that have passed (t).
b. Seven years from now, the oil reserves will be R = 1820 - 25(7) = 1820 - 175 = 1645 billion barrels.
c. To find when the world's oil reserves will be depleted, we need to solve for t when R = 0. So we have:
0 = 1820 - 25t
25t = 1820
t = 1820/25
t = 72.8
So, if the rate of depletion isn't changed, the world's oil reserves will be depleted in about 72.8 years.
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Draw the image of the following figure after a dilation centered at the origin with a scale factor of 3/2
The image of the figure under a dilation with scale factor 3/2 is attached
How to determine the image of the figureFrom the question, we have the following parameters that can be used in our computation:
Center = (0, 0).
Point = (6, 12), (8, 12) and (8, 8)
Scale factor = 3
A dilation with center (0, 0) and scale factor k multiplies the coordinates of a point by k.
So, to find the image of the point under the dilation with scale factor 3/2, we multiply each coordinate by 3/2:
Image = [(6, 12), (8, 12) and (8, 8)] * 3/2
Image = [(9, 18), (12, 18) and (12, 12)]
Hence, the image of the figure is attached
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a) Find the definite integral L (x + 1)(x - 1)dx Answer: I= b) Find the indefinite integral (x– 1)dx Answer: I =
c) Calculate the integral \2cos(t)dt Answer: I= =
a) To find the definite integral of L (x + 1)(x - 1)dx, we first need to expand the expression and then integrate it.
L (x + 1)(x - 1)dx = L (x^2 - 1)dx
Now we can integrate this expression:
I = ∫(x^2 - 1)dx = (x^3/3) - x + C
Since we are looking for the definite integral, we need to evaluate this expression at the limits of integration.
I = [(b^3/3) - b] - [(a^3/3) - a]
b) To find the indefinite integral of (x - 1)dx, we simply need to integrate the expression and add a constant of integration.
I = ∫(x - 1)dx = (x^2/2) - x + C
c) To calculate the integral of 2cos(t)dt, we simply need to integrate the expression and add a constant of integration.
I = ∫2cos(t)dt = 2sin(t) + C
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Evaluate the function f(x)=2x^2-3x for the given values of x.
(a) f(-1)
(b) f(0)
(c) f(1)
(d) f(2)
(e) f(3)
The evaluated values of the function for the given values of x are:
f(-1) = 5
f(0) = 0
f(1) = -1
f(2) = 2
f(3) = 9
To evaluate the function f(x)=2x^2-3x for the given values of x, we simply plug in the given value of x into the function and solve for f(x).
(a) f(-1) = 2(-1)^2 - 3(-1) = 2(1) + 3 = 5
(b) f(0) = 2(0)^2 - 3(0) = 0 - 0 = 0
(c) f(1) = 2(1)^2 - 3(1) = 2 - 3 = -1
(d) f(2) = 2(2)^2 - 3(2) = 8 - 6 = 2
(e) f(3) = 2(3)^2 - 3(3) = 18 - 9 = 9
Therefore, the evaluated values of the function for the given values of x are:
f(-1) = 5
f(0) = 0
f(1) = -1
f(2) = 2
f(3) = 9
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Probability For the error e in a distance observable y, with y=x+e and x the unknown true distance, it is given that it is distributed as e ~ N(0,02). What value for k has to be taken, to ensure that the probability, that - when we take a distance measurement y in practice - this distance measurement lies inside the interval [x – ko, x + ko), equals 99%? The value for k is
The value for k has to be taken, to ensure that the probability, that - when we take a distance measurement y in practice - this distance measurement lies inside the interval [x – ko, x + ko), equals 99% is 0.516.
To find the value of k that ensures that the probability that the distance measurement lies inside the interval [x – ko, x + ko) equals 99%, we can use the standard normal distribution table.
First, we need to find the z-score that corresponds to the 99% probability. This is found by looking at the standard normal distribution table and finding the z-score that corresponds to a cumulative probability of 0.995 (since the probability is split between both sides of the mean, we need to look for 0.995 instead of 0.99).
The z-score that corresponds to 0.995 is approximately 2.58.
Now, we can use the formula for the z-score to find the value of k:
z = (y - x)/o
Since we are looking for the value of k that corresponds to the interval [x – ko, x + ko), we can plug in the values for the z-score and the standard deviation:
2.58 = (x + ko - x)/0.2
Solving for k, we get:
k = 2.58 * 0.2
k = 0.516
Therefore, the value for k that ensures that the probability that the distance measurement lies inside the interval [x – ko, x + ko) equals 99% is 0.516.
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A polynomial has one root that equals 5 - 7i. Name one other root of this
polynomial.
The other root of the polynomial is 5 + 7i.
What is a Polynomial?Polynomial, a mathematical expression made comprised of numbers and variables arranged in specific patterns. Polynomials are sums of monomials of the form ax^n, where n (the degree) must be a whole number and a (the coefficient) can be any real number. The greatest degree monomial in a polynomial determines the polynomial's degree. Polynomials can be prime or factorable into products of primes, just like whole numbers. As long as each variable's power is a non negative integer, they can include any number of variables. They serve as the foundation for solving algebraic equations.
As per the given data:
One root of a polynomial is 5 - 7i
For finding the other root:
Consider the quadratic formula:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
From here, we can observe that imaginary roots will always occur in a pair.
So if one root is a + ib the other will be a - ib.
Similarly, for 5 - 7i the other root will be 5 + 7i.
Hence, the other root of the polynomial is 5 + 7i.
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(1.8x3)x(2.1x7)=(3x7)x(1.8x2.1) true or false?
The given equation is true. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?It is believed that the four fundamental operations, often referred to as "arithmetic operations", can explain all real numbers. The four mathematical operations that produce the quotient, product, sum, and difference are divide, multiply, add, and subtract.
We are given an equation as (1.8 x 3) x (2.1 x 7) = (3 x 7) x (1.8 x 2.1)
In order to see whether it is true or false, we will solve both the sides.
So, first solving L.H.S., we get
⇒(1.8 x 3) x (2.1 x 7)
⇒5.4 x 14.7
⇒79.38
Now, solving R.H.S., we get
⇒(3 x 7) x (1.8 x 2.1)
⇒21 x 3.78
⇒79.38
Since, L.H.S. = R.H.S., so the given equation is true.
Hence, the given equation is true.
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Points A, B, C, and D are consecutive points on circle W. Given that m
According to the figure the congruent angles are: ∠BAC = ∠CDB, ∠ABD = ∠ACD, ∠ACB = ∠ADB and ∠CBD = ∠CAD
What is a congruent angles?Congruent angles may be define as if two or more angles measures the same value. In other words, if two angles are congruent, they will have the same degree measure, and they will look identical when placed on top of each other. This is similar to the concept of congruent shapes or figures, where two shapes have the same size and shape.
According to the figure the points A, B, C, and D are consecutive points on Circle W (center) as shown in figure.
We need to find out the angles must be congruent to the angles of ABD, BAC, ACB and CBD.
according to the figure we have to get some values of congruent angles are:
∠BAC = ∠CDB,
∠ABD = ∠ACD,
∠ACB = ∠ADB and
∠CBD = ∠CAD
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What is 92199+20923+29290+83292+2819+99279+38471+378144
Answer:
744417
Step-by-step explanation:
Answer:
744,417
Step-by-step explanation:
Add the terms together
Please help with question
Answer:
The answer is 145°
Step-by-step explanation:
Angle 3 and 4 make a right angle (90°) Subtract 55 from 90.
90-55=35
Angle 3 and Angle 1 are vertical angles, hence they have the same measurement.
Angle 1 is 35°
Angle 1 and 2 make a 180° angle. Subtract 35 from 180.
180-35=145
The measurement of angle 2 is 145°
Answer:
Step-by-step explanation:
∠5 + ∠4 = 90 + 55 = 145
∠2 = ∠5 + ∠4 (vertically opposite angles)
∴ ∠2 = 145
The general solution to the second-order differential equation y′′+4y=0 is in the form y(x)=c1cosβx+c2sinβx. Find the value of β, where β>0
The general solution to the differential equation y''+4y=0 is y(x) = c₁ cos ₂x + c₂ sin ₂x and the value of β is 2.
To find the value of β, we substitute the general solution into the differential equation and solve for β. We start by finding the first and second derivatives of y(x):
y'(x) = -c₁β sin βx + c₂β cos βx
y''(x) = -c₁β² cos βx - c₂β² sin βx
Substituting these expressions into the differential equation, we get:
-c₁β² cos βx - c₂β² sin βx + 4(c₁ cos βx + c₂ sin βx) = 0
Simplifying this equation, we get:
(c₁β² + 4c₁) cos βx + (c₂β² + 4c₂) sin βx = 0
This equation must hold for all values of x, which means that the coefficients of cos βx and sin βx must both be zero. Therefore, we have the following system of equations:
c₁β² + 4c₁ = 0
c₂β² + 4c₂ = 0
We can solve for β by dividing the second equation by c₂ and substituting c₁ = -4β²/c₂ from the first equation:
β² = -4c₁/c₂ = 4
Since β>0, we take the positive square root of 4, which gives β=2.
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75. Linear Speed at the Equator. The earth has a 4000-mi radius and rotates one revolution every 24 hr. What is the linear speed of a point on the equator, in miles per hour? 76. Linear Speed of the Earth. The earth is about 93,000,000 mi from the sun and traverses its orbit, which is nearly circular, every 365.25 days. What is the linear velocity of the earth in its orbit, in miles per hour?
75. To find the linear speed of a point on the equator, we need to use the formula for the circumference of a circle: C = 2πr. The radius of the earth is 4000 mi, so the circumference of the earth at the equator is C = 2π(4000) = 8000π mi. The earth rotates one revolution every 24 hours, so the linear speed of a point on the equator is 8000π/24 = 333.33π mi/hr ≈ 1047.2 mi/hr.
76. To find the linear speed of the earth in its orbit, we need to use the same formula for the circumference of a circle, but this time the radius is the distance from the earth to the sun, which is 93,000,000 mi. The circumference of the earth's orbit is C = 2π(93,000,000) = 186,000,000π mi. The earth traverses its orbit every 365.25 days, or 8766 hours. So the linear speed of the earth in its orbit is 186,000,000π/8766 = 21,174.6π mi/hr ≈ 66,555.8 mi/hr.
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Un recipiente de aluminio tiene una capacidad de 6 litros a 28 grados celcius.
Determina el volumen del recipiente cuando este se caliento a 100 grados celcius
Answer: I don't speak Spanish but the answer might be 21.4285714286 liters
Step-by-step explanation:
"An aluminum container has a capacity of 6 liters at 28 degrees Celcius. Determine the volume of the container when it is heated to 100 degrees Celsius."
Tamera Purchased 31 Cases of tile to use on her bathroom floors. She used 17 cases in her upstairs bathroom and the rest in her downstairs bathroom, whic equation expresses how many cases. She used in her downstairs bathroom?
The equation that expresses how many cases Tamera used in her downstairs bathroom is: Downstairs cases = Total cases - Upstairs cases
We know that Tamera purchased a total of 31 cases of tile, and used 17 cases in her upstairs bathroom. Therefore, the number of cases she used in her downstairs bathroom can be found by subtracting the number of cases used in her upstairs bathroom from the total number of cases:
Downstairs cases = Total cases - Upstairs cases
Downstairs cases = 31 - 17
Downstairs cases = 14
Therefore, Tamera used 14 cases of tile in her downstairs bathroom.
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A dilation has center (0, 0, 0). Find the image of the point (-1, -2, 0) for the scale factor of 3.
The image of the point (-1, -2, 0) under a dilation with center (0, 0, 0) and scale factor 3 is (-3, -6, 0).
How to determine the image of the pointFrom the question, we have the following parameters that can be used in our computation:
Center = (0, 0, 0).
Point = (-1, -2, 0)
Scale factor = 3
A dilation with center (0, 0, 0) and scale factor k multiplies the coordinates of a point by k.
So, to find the image of the point (-1, -2, 0) under a dilation with scale factor 3, we multiply each coordinate by 3:
(-1, -2, 0) --> (3(-1), 3(-2), 3(0))
Image = (-3, -6, 0)
Hence, the image of the point (-1, -2, 0)is (-3, -6, 0).
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The measure of an angle is 12.1°. What is the measure of its complementary angle?
complementary mean they both add to 90 degrees
90 - 12.1 = 77.9
the other angle is 77.9 degrees
Answer: 77.9°
Step-by-step explanation:
Complementary angles are angles that, when added together, equal 90°.
So to find the complementary angle of an angle that measures 12.1°, you subtract:
90-12.1 = 77.9°
If a board is 4/4 x 12" x 12" how many board ft is it?
The board has a length of one board foot.
How long is an example?A thing's length has been its breadth or size calculated from the ends to the ends. In other words, it is the larger of an object's higher or lower geometric dimensions. For instance, a rectangle's dimensions are given by its length and breadth.
How long is a size?When describing an organism's size or the distance between two locations, the word "length" is often employed. For instance, the length of the a ruler is revealed in the table below.
Given that the board is [tex]4/4*12*12[/tex]
BF = (thickness in inches x width in inches x length in inches) / [tex]144[/tex]
BF = (1 inch x 12 inches x 12 inches) / 144
BF = 144 cubic inches / 144
BF = 1 board foot
Therefore, the board is 1 board foot in size.
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please help me out please
The slope of the line as shown in the image attached below is: 2/3.
How to Find the Slope of a Line Using the Rise and Run?The slope of a line represents the ratio of the change in y (vertical) to the change in x (horizontal). It is a measure of the steepness of the line and tells us how much the y-value changes for each unit change in the x-value.
It is given as:
Slope (m) = rise/run.
The rise (blue line) = 2 units
The run (red line) = 3 units
Slope (m) = -2/3
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Find the value of x.
BP and DP are straight lines
The value of the variable 'x' using the external angle theorem will be 84°.
What is the triangle?The polygonal form of a triangle has a number of flanks and three independent variables. Angles in the triangle add up to 180°.
The exterior angle of a triangle is practically always equivalent to the accumulation of the interior and opposing interior angles. The term "external angle property" refers to this segment.
The graph is completed and given below.
By the external angle theorem, the equation is given as,
x + 180° - 154° + 180° - 110° = 180°
x + 26° + 70° = 180°
x + 96° = 180°
x = 84°
The value of the variable 'x' using the external angle theorem will be 84°.
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Matrix Multiplication Non-Commutativity ( 2 by 2 ) Feb 20, 6:12:06 PM Watch help video Given the matrices A=[[1,-4],[-2,4]] and B=[[-4,0],[4,3]], find the product AB as well as the product BA. AB=[[1,
The product AB is not equal to the product BA. This demonstrates the non-commutativity of matrix multiplication.
Matrix multiplication is non-commutative, which means that the product of two matrices can be different depending on the order in which they are multiplied. In other words, AB is not always equal to BA. Let's find the product of the given matrices A and B in both orders to demonstrate this.
First, let's find the AB Product:
AB = [[1,-4],[-2,4]] * [[-4,0],[4,3]]
= [(1 * -4) + (-4 * 4), (1 * 0) + (-4 * 3), (-2 * -4) + (4 * 4), (-2 * 0) + (4 * 3)]
= [[-20, -12], [16, 12]]
Now, let's find the product BA:
BA = [[-4,0],[4,3]] * [[1,-4],[-2,4]]
= [(-4 * 1) + (0 * -2), (-4 * -4) + (0 * 4), (4 * 1) + (3 * -2), (4 * -4) + (3 * 4)]
= [[-4, 16], [2, 4]]
As we can see, the product AB is not equal to the product BA. This demonstrates the non-commutativity of matrix multiplication.
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Consider functions fand g.
f(x) = -23
g(x) = |x − 1
What is the value of (go f)(4)?
A. 9
B. - 1/8
C. -9
D. 1/8
The value of (gof)(4) is 9 if the function f(x) is f(x) =[tex]-x^3[/tex], and function g(x) is g(x) = |1/8x-1| option (A) is correct.
What is a function?It is described as a particular kind of relationship, and each value in the domain is associated with exactly one value in the range according to the function. They have a predefined domain and range.
from the question:
We have a function:
f(x) = -x³
Plug x = 4 in the f(x)
f(4) = -4³ = -64
Plug the above value in the g(x)
g(f(4)) = |-8-1| = |-9| = 9
Thus, the value of (gof)(4) is 9 if the function f(x) is f(x) = -x³, and function g(x) is g(x) = |1/8x-1| option (a) is correct.
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complete and correct questions
f(x) = -x3
g(x) = |1/8x-1|
What is the value of (gof)(4)?.
A. 9
B. - 1/8
C. -9
D. 1/8
Calculate the base length of the isosceles triangular pieces
The base length of the isosceles triangular pieces with height 8 cm, side length 4cm and area of triangle is 64 cm² is equals to the 16 cm.
Isoceles triangle: The two sides of triangle which are equal in length are called "legs". In case of isosceles triangle, ABC (present above), AB and AC are the called "legs".
The third side of an isosceles triangle is called "base" of it. In above triangle ABC, BC is the base of the isosceles triangle.Steps to calculate the base of the triangle :
The area of the triangle multiplied by 2. Measure the height of the triangle.Divide the result of step 1, by the height.The new result is the base of the triangle.Now we have an isosceles triangle,
height of isosceles triangle, h = 8 cm
area of isosceles triangle, A = 64 cm²
we know that area of isosceles triangle
= ½ x base x height
=> base triangle = (2 x area of triangle)/ height
Substituting all known values,
=> base length of triangle = 2 x 64/8
=> base length = 16 cm
Hence, required base length is 16 cm.
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Complete question:
Calculate the base length of the isosceles triangular pieces with height 8 cm, side length 4cm and area of triangle is 64 cm².
Show the family of conics with the same focus
x^2/a^2+C + y^2/b^2+C = 1
is its own orthogonal family of curves.
The original equation and the orthogonal equation are the same, we can conclude that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves.
To show that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves, we need to take the derivative of the equation and set it equal to -1/b^2, the slope of the orthogonal line.
First, we take the derivative of the equation with respect to x:
2x/a^2 = -2y/b^2 * dy/dx
Simplifying, we get:
dy/dx = -b^2*x/a^2*y
Now, we set this equal to -1/b^2:
-b^2*x/a^2*y = -1/b^2
Cross-multiplying and simplifying, we get:
x/a^2*y = 1/b^2
Finally, we can rearrange this equation to get:
y = b^2*x/a^2
This equation represents the orthogonal family of curves to the original family of conics. Since the original equation and the orthogonal equation are the same, we can conclude that the family of conics with the same focus x^2/a^2+C + y^2/b^2+C = 1 is its own orthogonal family of curves.
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At an ice cream shop, the cost of 4 milkshakes and 2 ice cream sundaes is $23.50. The cost of 8 milkshakes and 6 ice cream sundaes is $56.50.
What is the price of a milkshake?
What is the price of an ice cream sundae?
Answer:
If I had to take a guess I'd say the price of the milkshakes is $4.50 and the price of the sundaes are $2.75 but I'm not 100% sure.
Answer:
Step-by-step explanation:
8m + 6i = 56.50
4m + 2i = 23.50
8m + 6i = 56.50
-8m - 4i =-47.00
2i = 9.50
i = $4.75 ice cream sundae
4m + 2(4.75) = 23.50
4m + 9.50 = 23.50
4m = 14
m = 3.50 milkshake