According to the solving the value of given function h’(5) h’(5) = [2f’(5)g(5)] / g(5)² + f(5) / g(5)²
What exactly is a function?A function is described as a relationship between a group of inputs with one output each. A function is, to put it simply, a relationship between inputs in which each input is connected to exactly one output. Every function has a domain and a codomain, or range. A function is typically represented by the notation f(x), where x represents the input.
According to the given information:To find the value of h’(5), we need to find the derivative of h(x) first. We can use the product rule and the quotient rule to do that:
h(x) = f(x)g(x) + f(x)/g(x)
h’(x) = f’(x)g(x) + f(x)g’(x) + [g(x)f’(x) – f(x)g’(x)] / g(x)² (by the quotient rule)
h’(x) = [f’(x)g(x) + g(x)f’(x)] / g(x)² + [f(x)g’(x) – f(x)g’(x)] / g(x)² + f(x) / g(x)²
h’(x) = [2f’(x)g(x)] / g(x)² + f(x) / g(x)²
Now we can find h’(5) by plugging in x = 5:
h’(5) = [2f’(5)g(5)] / g(5)² + f(5) / g(5)²
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5. Jabu's investment of R2 200 earned R528 in two years. a. Calculate the simple interest rate for this investment. b. If she decides to invest the total amount (original principal amount plus interest) for another two years at the same rate, what interest will she earn over the second two years. c. What is the difference in interest earned over the first two years, compared with interest earned over the second two years?
The payment (future value), to the nearest rand, which should be made at the end of the seventh year in order to liquidate the debt, is R9,338.80.
What is the future value determined?The future payment to be made at the end of the seventh year is determined by computing the future values in two stages. The first stage computes the future value before the payment at the end of the first year. The second stage computes the future value for six years.
here, we know,
The future value is computed by compounding the present value at an interest rate.
Current debt = R6,421,00
Compound interest rate = 11% per year
Compounding period = Quarterly
Future Value in 1 Year:
N (# of periods) = 4 quarters (1 year x 4)
I/Y (Interest per year) = 11%
PV (Present Value) = R6,421.00
PMT (Periodic Payment) = R0
Results:
Future Value (FV) = R7,156.98
Total Interest = R735.98
Future Value (FV) = R7,156.98
Payment in one year's time = R2,287.00
Balance = R4,869.98
Future Value at the end of the 7th Year:
N (# of periods) = 24 quarters (6 years x 4)
I/Y (Interest per year) = 11%
PV (Present Value) = R4,869.98
PMT (Periodic Payment) = R0
Results:
Future Value (FV) = R9,338.80
Total Interest = R4,468.82
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The polynomial p(z)=2x^(3)-z^(2)+64z-185 has a complex zero at z=-1+6i. Write in factored form as a product of linear and quadratic expressions using integer coefficients only.
The factored form of the polynomial is p(z) = (2z - 5)(z^(2) + 2z + 37).
To write the polynomial p(z) in factored form as a product of linear and quadratic expressions using integer coefficients only, we need to find the other zeros of the polynomial.
Since one of the zeros is a complex number, z = -1 + 6i, the other complex zero will be its conjugate, z = -1 - 6i. This is because complex zeros of polynomials with real coefficients always come in conjugate pairs.
Now, we can write the quadratic expression that has these two complex zeros as
(z - (-1 + 6i))(z - (-1 - 6i)) = (z + 1 - 6i)(z + 1 + 6i) = (z + 1)^(2) - (6i)^(2) = z^(2) + 2z + 1 - 36i^(2) = z^(2) + 2z + 37.
Since the polynomial p(z) is of degree 3, there must be one more zero, which we can find by dividing p(z) by the quadratic expression we just found:
p(z)/(z^(2) + 2z + 37) = 2z^(3)/(z^(2) + 2z + 37) - z^(2)/(z^(2) + 2z + 37) + 64z/(z^(2) + 2z + 37) - 185/(z^(2) + 2z + 37) = 2z - 5
So the other zero is z = 5/2.
Now,
p(z) = 2(z - 5/2)(z^(2) + 2z + 37) = (2z - 5)(z^(2) + 2z + 37)
So the factored form of the polynomial is p(z) = (2z - 5)(z^(2) + 2z + 37).
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Here is another triangle similar to DEF.
•What is the scale factor from triangle DEF to triangle D”E”F”?
• What are the coordinates of F”?
•What are cos(D”), sin(D”), and tan(D”)
(Triangle DEF has coordinates D(0,0) E(12,0) F(12,5)
And pictured is triangle D”E”F”
The scale factor from triangle DEF to triangle D”E”F” is the ratio of the corresponding sides in the two triangles.
The coordinates of F” can be calculated using the scale factor.
The value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.
What is a scale factor?Scale factor is a number which is used to describe the relationship between two objects or two measurements. It is used to compare two objects of different sizes in order to calculate the ratio of their size. It is also used to express a proportional relationship between two different measurements, such as their lengths, widths, heights or angles.
The corresponding sides are DE and D”E”, and EF and E”F”. We can calculate the ratio of the two sides by dividing the lengths of the two corresponding sides. For example, the ratio of DE to D”E” would be 12/16. We can do this for all three corresponding sides, and then take the average of the three ratios to get the scale factor from triangle DEF to triangle D”E”F”.
Since we know that the coordinates of F are (12,5), we can multiply the x-coordinate (12) by the scale factor and the y-coordinate (5) by the scale factor to get the coordinates of F”.
The angle D” can be used to calculate the values of cos(D”), sin(D”), and tan(D”). Since we know the angle D”, we can use the trigonometric functions to calculate the values of the trigonometric functions for that angle. For example, if the angle D” is 45 degrees, then the value of cos(D”) would be equal to 0.707. Similarly, the value of sin(D”) would be 0.707 and the value of tan(D”) would be 1.
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Your sucessful art supply store, Warpz, just solf 30% of its inventory over the weekend. If on the second day you only sold 15% of the inventory that was left in the store, what percent of the total inventory was sold?
Please give an explanation!
18% of the total inventory was sold.
What is percentage?A percentage is a fraction of an amount expressed as a particular number of hundredths of that amount.
Given that, 30% of the inventory is sold over the weekend, on the second day only 15% of the inventory that was left in the store to be sold,
Let x be the total inventory,
Sold inventory = 30% of x,
The second day = x - 30% of x is sold
x -30% of x = 15% of x
x - 0.3x = 0.15x
x = 0.3x+0.15x
x = 0.18x
x = 18% of x,
Hence, 18% of the total inventory was sold.
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I'm so confused on thins can someone PLEASE help me??? It's Algebra by the way
Answer:
Degree: The degree of the function is 3.
X-Intercepts with multiplicity greater than 1: x = -2 and x = 0
How many distinct x-intercepts?: There are two distinct x-intercepts.
How many zeros are there?: There are three zeros, one at x = -2 and two at x = 0 (with a multiplicity of 2).
Mathe question 2 help
Answer:
Option B
See below
Step-by-step explanation:
Two simultaneous equations:
[tex]7x - 4y - 8 = 0[/tex] —————- (equation i)
[tex]2x^{2} + x + 4y + 8 = 0[/tex] ——- (equation ii)
Step 1:
Rearrange (equation i):
[tex]7x - 4y = 8[/tex] ———- updated (equation i)
Step 2:
Substitute the updated (equation i) into (equation ii) and bring like terms together to simplify them in reduced forms:
[tex]2x^{2} + x + 4y + (7x - 4y) = 0[/tex]
[tex]2x^{2} + x + 4y + 7x - 4y = 0[/tex]
[tex]2x^{2} + x + 7x + 4y - 4y = 0[/tex]
[tex]2x^{2} + 8x = 0[/tex]
Step 3:
Factorize:
[tex]2x(x + 4) = 0[/tex]
Either [tex]2x = 0[/tex]
∴ [tex]x = 0[/tex]
Or [tex]x + 4 = 0[/tex]
∴ [tex]x = -4[/tex]
Step 4:
Substitute these values of x in (equation ii) to determine their corresponding values of y:
For x = 0:
[tex]2(0)^{2} + 0 + 4y + 8 = 0[/tex]
[tex]4y + 8 = 0[/tex]
[tex]4y = -8[/tex]
[tex]y = \frac{8}{-4}[/tex]
∴[tex]y = -2[/tex]
For x = -4:
[tex]2(-4)^{2} + (-4) +4y + 8 = 0[/tex]
[tex]2(16) - 4 + 4y + 8 = 0[/tex]
[tex]32 - 4 + 4y + 8 = 0[/tex]
[tex]32 - 4 + 8 + 4y = 0[/tex]
[tex]36 + 4y = 0[/tex]
[tex]4y = -36[/tex]
[tex]y = \frac{-36}{4}[/tex]
∴ [tex]y = -9[/tex]
∴The values of y are [tex]-2[/tex] and [tex]-9[/tex]
∴Option B
Please show your work.
20 points! transform these graphs. look at photo for questions.
If f(x) is the parent function and g(x) is the transformation of f(x), then the required functions are as follows:
5. g(x) = -4ˣ and 6. g(x) = -(1/5)ˣ.
What is a transformation?A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.
5.
The given function is as follows:
f(x) = 4ˣ
When the function f(x) = 4ˣ is reflected about the x-axis, then we get the graph of function g(x).
So, g(x) = -4ˣ
6.
The given function is as follows:
f(x) = (1/5)ˣ
When the function f(x) = (1/5)ˣ is reflected about the x-axis, then we get the graph of function g(x).
So, g(x) = -(1/5)ˣ
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Which number line shows the solution to the inequality? y minus 2 less-than negative 5 A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded. A number line going from negative 8 to positive 2. A closed circle is at negative 3. Everything to the left of the circle is shaded. A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the right of the circle is shaded. A number line going from negative 8 to positive 2. An open circle is at negative 7. Everything to the left of the circle is shaded.\\
The number line that shows the solution to the inequality y - 2 < -5 is:
A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded.
What is inequality?
In mathematics, an inequality is a statement that compares two values, expressing that one value is less than, greater than, or less than or equal to, greater than or equal to, or not equal to another value.
The inequality is y - 2 < -5.
To graph the solution on a number line, we start by marking the point where y - 2 equals -5, which is y = -3. Then, since the inequality is less-than, we use an open circle to represent the point y = -3.
Next, we shade all the values of y that make the inequality y - 2 < -5 true. Since y is less than -3, we shade everything to the left of the open circle.
Therefore, the number line that shows the solution to the inequality y - 2 < -5 is:
A number line going from negative 8 to positive 2. An open circle is at negative 3. Everything to the left of the circle is shaded.
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what is 3/8 divided by 1/4
Answer:[tex]1\frac{1}{2}[/tex]
k 4.1 Find the zeros of the polynomial function and state f(x)=-8(x-1)^(3)(x+5)^(2)x^(4) The smallest zero is with multiplicity
The zeros of the polynomial function f(x) = -8(x-1)^(3)(x+5)^(2)x^(4) are x = 1, x = -5, and x = 0. The smallest zero is x = -5 with multiplicity 2.
To find the zeros of the polynomial function f(x) = -8(x-1)^(3)(x+5)^(2)x^(4), we need to set the function equal to zero and solve for x:
0 = -8(x-1)^(3)(x+5)^(2)x^(4)
This equation will be true when any of the factors on the right-hand side are equal to zero. Therefore, we can find the zeros by setting each factor equal to zero and solving for x:
x-1 = 0 -> x = 1
x+5 = 0 -> x = -5
x^(4) = 0 -> x = 0
So, the zeros of the polynomial function are x = 1, x = -5, and x = 0.
The smallest zero is x = -5 with multiplicity 2, since it is the smallest value of x and the factor (x+5)^(2) has an exponent of 2.
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0 A community center offers classes for students.
. The range of the number of students in each class is 13.
. The median number of students in each class is 9.
Which of the following box-and-whisker plots could represent the numbers of students in the classes?
Numbers of Students
Numbers of Students
in Classes
in Classes
C₂
+++
2 4 6 8 10 12 14 16 18 20 22 24
Numbers of Students
in Classes
+++
2 4 6 8 10 12 14 16 18 20 22 24
B.
D.
++
2 4 6 8 10 12 14 16 18 20 22 24
Numbers of Students
in Classes
+111*
2 4 6 8 10 12 14 16 18 20 22 24
T
The response is A.
Option A's median value is 9, and its range is 13, therefore the smallest value is 9-6, which equals 3, and the greatest value is 9+6, which equals 15. This information is accurately represented by the box-and-whisker plot in option A.
what is the range?The difference between the highest and lowest values for a given data collection is the range in statistics. For instance, the range will be 10 - 2 = 8 if the given data set is 2, 5, 8, 10, and 3.
As a result, the range may alternatively be thought of as the distance between the highest and lowest observation. The range of observation is the name given to the outcome. Statistics' range reflects the variety of observations.
from the question:
According to the information provided, the number of pupils in each class might range from 9+6 = 15 to 9+6 = 3. Because the maximum value exceeds 15, options C and D cannot accurately represent the number of students in the classes.
Option A's median value is 9, and its range is 13, therefore the smallest value is 9-6, which equals 3, and the greatest value is 9+6, which equals 15. This information is accurately represented by the box-and-whisker plot in option A.
Option B's median value is 9, but its range (from 9-5 = 4 to 9+5 = 14) is only 11, which is less than the range that could be achieved by the number of students. As a result, option B cannot accurately reflect the number of students enrolled in each class.
Thus, the response is A.
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Mattie Evans drove 140 miles in the same amount of time that it took a turbopropeller plane to travel 440 miles. The speed of the plane was 150 mph faster than the speed
of the car. Find the speed of the plane.
By forming and solving the equations we know that the speed of the plane was 220 mph.
What are equations?In mathematical formulas, the equals sign is used to indicate that two expressions are equal.
A mathematical statement that uses the word "equal to" between two expressions with the same value is called an equation.
like 3x + 5 = 15, for instance.
Equations come in a wide variety of forms, including linear, quadratic, cubic, and others.
So, d = rt.
t = d/r
The distances are equal.
140/r = 440/(r + 150)
Cross multiply:
140(r + 150) = 440r
140r + 21,000 = 440r
300r = 21,000
r = 70
70 + 150 = 220 mph
Therefore, by forming and solving the equations we know that the speed of the plane was 220 mph.
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Which of the following are solutions to the equation below? 6x^2-2x+36=5x^2+10x
a. 6
b. -4
c. 18
d. -6
e. -3
f. 4
Answer:
A. 6
Step-by-step explanation:
To solve the equation 6x^2-2x+36=5x^2+10x, we can follow these steps:
Move all the terms to one side of the equation by subtracting 5x^2 and 10x from both sides:
6x^2 - 2x + 36 - 5x^2 - 10x = 0
Simplifying the left side:
x^2 - 12x + 36 = 0
Factor the quadratic expression on the left side of the equation:
(x - 6)(x - 6) = 0
Apply the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero:
x - 6 = 0
Solve for x:
x = 6
The solution to the equation 6x^2-2x+36=5x^2+10x is x = 6.
what does the term "basic" mean as used in the car advertisement
No paint job, no trims, no extra things on the car.
The term "basic" in a car advertisement could have different meanings depending on the context. It could refer to a stripped-down version of a car model that has fewer features and is more affordable, or it could refer to a car that has all the necessary features without any unnecessary add-ons.
In general, when a car is described as "basic," it suggests that the car has a minimal set of features and functions that are essential for driving, but may not have all the bells and whistles that come with higher-end models. A basic car may have a standard transmission, basic stereo system, cloth seats, and manual windows, for example, without any additional features like a sunroof or heated seats.
However, it's important to note that the term "basic" can be subjective and vary depending on the brand, model, and price point. What one person considers basic, another may see as luxurious. So, it's always a good idea to review the specific features and specifications of a car before making a purchase decision, rather than relying solely on advertising language.
A middle school basketball player makes a free throw with a probability of 0.6. Assuming each free throw is an independent event, what is the probability the player makes zero of their next three free throws? (A) 0.064 (B) 0.216 (C) 0.784 (D) 0.936 (E) None of these
A middle school basketball player makes a free throw with a probability of 0.6. Assuming each free throw is an independent event,the probability the player makes zero of their next three free throws is 0.064. The correct answer is A.
The probability that a middle school basketball player makes zero of their next three free throws can be calculated using the formula P(A) = P(A')ⁿ, where P(A) is the probability of the event occurring, P(A') is the probability of the event not occurring, and n is the number of trials.
In this case, P(A) is the probability of the player making zero free throws, P(A') is the probability of the player missing a free throw (1 - 0.6 = 0.4), and n is the number of free throws (3).
Using the formula, we can calculate the probability of the player making zero free throws as follows:
P(A) = P(A')ⁿ
P(A) = (0.4)³
P(A) = 0.064
Therefore, the correct answer is (A) 0.064.
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3. Letfbe a differentiable function of one variable, andw=f(ex+2xy). (a) Verify that the functionwsatisfies the equation(ex+2y)wy−2xwx=0. (b) Iff(u)=cosu, calculatewxy. 4. Consider the functionf(P,t)=f(x(t),y(t),z(t),t)=t(ysinx+ez)and the pointP=(π,1,0). Find the material derivative offatP
3 a. The function w satisfies the equation (ex+2y)wy−2xwx=0.
3 b. The value of wxy is -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4 . The material derivative of fatP is 0.
3. (a) To verify that the function w satisfies the equation (ex+2y)wy−2xwx=0, we can take the partial derivatives of w with respect to x and y, and then substitute them into the equation.
First, let's find the partial derivatives of w:
∂w/∂x = f'(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x)
Now, we can substitute these partial derivatives into the equation:
(ex+2y)(f'(ex+2xy)(2x)) - 2x(f'(ex+2xy)(ex+2y)) = 0
Simplifying this equation, we get:
2exf'(ex+2xy) + 4xyf'(ex+2xy) - 2exf'(ex+2xy) - 4xyf'(ex+2xy) = 0
This simplifies to 0 = 0, which is true. Therefore, the function w satisfies the equation (ex+2y)wy−2xwx=0.
(b) If f(u) = cosu, then we can find wxy by taking the partial derivative of w with respect to x and y, and then substituting f(u) = cosu:
∂w/∂x = f'(ex+2xy)(ex+2y) = -sin(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x) = -sin(ex+2xy)(2x)
Now, we can find wxy by taking the partial derivative of ∂w/∂x with respect to y:
wxy = ∂(∂w/∂x)/∂y = ∂(-sin(ex+2xy)(ex+2y))/∂y = -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4. To find the material derivative of fatP, we can use the formula:
Df/Dt = ∂f/∂t + ∂f/∂x(dx/dt) + ∂f/∂y(dy/dt) + ∂f/∂z(dz/dt)
First, let's find the partial derivatives of f:
∂f/∂t = ysinx + ez
∂f/∂x = ty*cosx
∂f/∂y = tsinx
∂f/∂z = tez
Now, we can find the material derivative of fatP by substituting the point P = (π,1,0) and the derivatives of x, y, and z with respect to t:
Df/Dt = (1*sinπ + e^0) + (π*1*cosπ)(dx/dt) + (π*sinπ)(dy/dt) + (π*e^0)(dz/dt) = 0 + 0 + 0 + 0 = 0
Therefore, the material derivative of fatP is 0.
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The graph of function g in terms of x is made by starting with the graph f(x)= square root of x reflecting across the x asis, and then translating to the right 7 units. Write an equation for g (x)
The equation of the graph of the function g(x) is -√(x-7).
What distinguishes a reflection from a translation?Turns are frequently used to refer to reflection, which is when an object is flipped over a line without affecting its size or shape. The preimage is flipped over a line in a rigorous transition known as a reflection, but its size and shape are left unchanged. Flips is another name for reflections.
A figure can be translated if it is moved in any direction without altering its size, form, or orientation. A hard transformation called a translation alters the preimage's position but not its size, shape, or orientation. Slides are another name for translations.
Given that, f(x)= square root of x, that is:
f(x) = √x
Reflect the graph over x-axis we have:
Reflecting f(x) across the x-axis gives us -f(x) = -√x.
Translating -f(x) = -√x 7 units to the right gives us -f(x-7) = -√(x-7).
Hence, the equation of the graph of the function g(x) is -√(x-7).
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Helpppppp meeee and thank uuuuuuu
Answer:
look in the comments and have a good day
The area of a circle is 615.44 sq. inches. what is the diameter of the circle? use 3.14 for π.
The diameter of the circle is 28 inches.
Formula for the area of a circle is:
A = πr²
Where A represents the area, r denotes the radius, and π is a mathematical constant approximately equal to 3.14.
To find the diameter of the circle, we can use the formula:
D = 2r
in the given formula D is the diameter and r is the radius.
We know that the area of the circle is 615.44 sq. inches. So, we can use the area formula to find the radius:
615.44 = πr²
615.44 = 3.14r²
r² = 615.44/3.14
r² = 196
r = √196
r = 14
Now that we know the radius is 14 inches, we can use the diameter formula to find the diameter:
D = 2r
D = 2(14)
D = 28
Therefore, the diameter of the given circle is 28 inches.
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Been stuck on this assignment for a while and it's overdue. I want to turn it in today. Can someone help me get a head start so I can get it done today? This would be a nice birthday present, I'm 18 on March 1 :)
Answer:
Its mostly a step by step process
Step-by-step explanation:
Just Put your full name and the date your depositing the money where the date is and how many bills your depositing how many coins if you don't have coins don't answer it the total amount would be 800 and the net deposit. I added a link just in case. Happy birthday btw.
Math part 3 question 5
The value of (f - g) (3) is 32.
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Given are two functions, f and g.
f(x) = 3x² and g(x) = x - 8
Subtraction of two functions is defined as,
(f - g) (x) = f(x) - g(x)
= 3x² - (x - 8)
= 3x² - x + 8
To find (f - g)(3), substitute 3 in place of x.
(f - g)(3) = 3(3)² - 3 + 8
= 27 - 3 + 8
= 32
Hence the value of the subtraction of functions is 32.
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Emma Koonce Solve by Trinomial Factor (a)=(1) Feb 22, 8:36:18 AM Solve the quadratic by factoring. x^(2)=2x+8 Answer: x
The solutions to the quadratic equation x^(2)=2x+8 are x=4 and x=-2.
To solve the quadratic equation x^(2)=2x+8 by factoring, we need to rearrange the equation to make it equal to zero and then factor the trinomial.
Step 1: Rearrange the equation to make it equal to zero.
x^(2)-2x-8=0
Step 2: Factor the trinomial using the "AC Method."
The "AC Method" involves finding two numbers that multiply to the product of the coefficient of the x^(2) term (a) and the constant term (c), and add to the coefficient of the x term (b).
In this case, a=1, b=-2, and c=-8.
The product of a and c is (1)(-8)=-8.
The two numbers that multiply to -8 and add to -2 are -4 and 2.
Step 3: Rewrite the equation using the two numbers found in Step 2.
x^(2)-4x+2x-8=0
Step 4: Factor by grouping.
(x^(2)-4x)+(2x-8)=0
x(x-4)+2(x-4)=0
Step 5: Factor out the common factor (x-4).
(x-4)(x+2)=0
Step 6: Set each factor equal to zero and solve for x.
x-4=0 or x+2=0
x=4 or x=-2
Therefore, the solutions to the quadratic equation x^(2)=2x+8 are x=4 and x=-2.
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India bought a bag of grapes that weighed 3 pounds, 5 ouncesShe already had a bag at home that weighed pound, 12 ounces. How many pounds/ ounces of grapes does she have altogether?
Answer:
4Pounds 1ounce
Step-by-step explanation:
First bag is 3pound 5ounces
Second bag 12ounces
Add both
3 Pounds and 17ounces
16ounces make 1Pound so 17ounces becomes 1Pound and 1ounces
Add to the 3Pounds
You have 3 Pounds + 1 Pound 1ounce
-3y=2x-7
graphing linear equations in 2 variables
By solving the given equation, we can plot these two points on a coordinate plane, and then connect the points with a line. The formed line is the graph of the linear equation 3y=2x-7.
What is a coordinate plane?The coordinate plane is made up of two perpendicular lines called the x-axis and the y-axis. The x-axis, which is horizontal, is used to measure the distance of a point from the left and right side of the plane. The y-axis, which is vertical, is used to measure the distance of a point from the top and bottom of the plane.
To graph the equation 3y=2x-7, we first need to identify two points that satisfy the equation. We can do this by plugging in different values for x and solving for y. For example, when x=0, the equation becomes 3y=-7. Thus, one point that satisfies the equation is (0,-7). We can find a second point by plugging in a different value for x. For example, when x=2, the equation becomes 3y=-3. Thus, the second point that satisfies the equation is (2,-3).
Now, we can plot these two points on a coordinate plane, and then connect the points with a line. The formed line is the graph of the linear equation 3y=2x-7.
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For the rational function x² – 2x - 8 f(x) = (x - 2)2 a) Determine the x-intercepts b) Determine the approximation equation near each x-intercept c) Determine the equation for the vertical asymptote d) Determine the approximation equation near the vertical asymptote e) Determine the formula for any horizontal and/or slant asymptote
a) To find the x-intercepts, we set f(x) = 0 and solve for x:
0 = (x² - 2x - 8)/(x - 2)²
0 = x² - 2x - 8
(x - 4)(x + 2) = 0
x = 4, -2
So the x-intercepts are (4,0) and (-2,0).
b) To find the approximation equation near each x-intercept, we can use the first derivative:
f'(x) = (2x - 2)/(x - 2)³
For x = 4:
f'(4) = (2(4) - 2)/(4 - 2)³ = 6
So the approximation equation near x = 4 is y = 6(x - 4).
For x = -2:
f'(-2) = (2(-2) - 2)/(-2 - 2)³ = -1/8
So the approximation equation near x = -2 is y = -1/8(x + 2).
c) To find the equation for the vertical asymptote, we set the denominator of f(x) equal to 0 and solve for x:
(x - 2)² = 0
x = 2
So the equation for the vertical asymptote is x = 2.
d) To find the approximation equation near the vertical asymptote, we can use the first derivative:
f'(x) = (2x - 2)/(x - 2)³
f'(2) = (2(2) - 2)/(2 - 2)³ = undefined
Since the first derivative is undefined at x = 2, we can use the second derivative:
f''(x) = (6x - 6)/(x - 2)⁴
f''(2) = (6(2) - 6)/(2 - 2)⁴ = undefined
Since the second derivative is also undefined at x = 2, we cannot find an approximation equation near the vertical asymptote.
e) To find the formula for any horizontal and/or slant asymptote, we can look at the degree of the numerator and denominator of f(x):
The degree of the numerator is 2 and the degree of the denominator is 2, so there is a horizontal asymptote.
To find the equation of the horizontal asymptote, we can divide the leading coefficients of the numerator and denominator:
1/1 = 1
So the equation of the horizontal asymptote is y = 1.
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How do find volume of a solid that is a cylinder with a hemispherical hole cut from the top of it with the shaded rim 3 inches thick height is 12 in and part of hemisphere is a triangle with hypotenuse side 10 in
The volume of the solid is 284π/3 cubic inches.
To find the volume of the solid, we need to break it down into simpler shapes and then calculate their volumes separately.
The solid is made up of a cylinder and a hemispherical hole cut from the top of it. Let's first calculate the volume of the cylinder. The height of the cylinder is given as 12 inches and the radius can be calculated as half of the hypotenuse of the triangle (which is 10 inches). Therefore, the radius of the cylinder is 5 inches.
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height. Substituting the values, we get:
Vcylinder = π(5^2)(12) = 300π cubic inches
Now, let's calculate the volume of the hemispherical hole. The rim of the hole is 3 inches thick, which means the radius of the hole is (5 - 3) = 2 inches. The volume of a hemisphere is given by the formula Vhemisphere = (2/3)πr^3.
Substituting the values, we get:
Vhemisphere = (2/3)π(2^3) = 16π/3 cubic inches
Now, the volume of the solid can be found by subtracting the volume of the hemispherical hole from the volume of the cylinder:
Vsolid = Vcylinder - Vhemisphere
Vsolid = 300π - 16π/3
Vsolid = 284π/3 cubic inches
Therefore, the volume of the solid is 284π/3 cubic inches.
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If S is a vertex matrix, which transformation does S + [\begin{array}{ccc}-1&-1&-1\\-3&-3&-3\end{array}\right] represent? a. translation of 3 units left and 1 unit down b. translation of 3 units left and 1 unit up c. translation of 1 unit left and 3 units down d. translation of 1 unit left and 3 units up
The correct answer is option c. translation of 1 unit left and 3 units down.
The transformation represented by S + [\begin{array}{ccc}-1&-1&-1\\-3&-3&-3\end{array}\right] is a translation of 1 unit left and 3 units down. This can be seen by examining the values in the matrix. The -1 in the first row represents a translation of 1 unit left, while the -3 in the second row represents a translation of 3 units down. Therefore, the correct answer is option c. translation of 1 unit left and 3 units down.
To summarize, the transformation represented by the given matrix is a translation of 1 unit left and 3 units down. This is indicated by the values of -1 and -3 in the matrix.
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Can somebody PLEASE help ASAP? I will give brainliest. Show work please!!
The surface area of the cylindrical tube is approximately 240.21 square inches.
What formula do we use to find the surface area of a cylinder?
The surface area of a cylinder is the sum of the areas of all its surfaces, including the curved surface area and the area of its two circular bases. It is a measure of the total area that the cylinder covers.
To find the surface area of a cylinder, we use the formula
[tex]A = 2\pi rh + 2\pi r^2[/tex]
where r is the radius of the base, h is the height of the cylinder, and π is a constant equal to approximately 3.14.
Calculating the surface area of the cylindrical tube -
The base of the tube has a diameter of 3 inches, which means the radius is 1.5 inches. The length of the poster is given as 24 inches, so we will assume that the height of the cylindrical tube is also 24 inches.
Substituting the values we have into the formula, we get:
[tex]A = 2\pi (1.5)(24) + 2\pi (1.5)^2[/tex]
[tex]A = 2\pi (36) + 2\pi (2.25)[/tex]
[tex]A = 72\pi + 4.5\pi[/tex]
[tex]A = 76.5\pi[/tex]
Using the approximation of [tex]\pi = 3.14[/tex], we get:
[tex]A[/tex]≈[tex]240.21[/tex] square inches .
Therefore, the surface area of the cylindrical tube is approximately 240.21 square inches.
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Adrien won the lottery! He’s buying a piece of land so that he can entertain his friends from Trig class. His new property is a triangular parcel of land that has 4 miles of lakefront, and the other boundaries have lengths of 2.8 miles and 1.8 miles. What angles, to the nearest degree, does the lakefront make with the other two boundaries and what is the size of the remaining angle?
The lakefront makes angles of approximately 30 degrees and 120 degrees with the boundaries of 2.8 miles and 1.8 miles, respectively. The remaining angle is approximately 30 degrees.
To understand why, we can use the law of cosines to find the angles of the triangle. Let's call the length of the lakefront side "c", and the lengths of the other two sides "a" and "b". Using the law of cosines, we can find the angles:
[tex]cos A = (b^2 + c^2 - a^2) / (2bc)[/tex]
[tex]cos B = (a^2 + c^2 - b^2) / (2ac)[/tex]
[tex]cos C = (a^2 + b^2 - c^2) / (2ab)[/tex]
Plugging in the values we know, we get:
[tex]cos A = (1.8^2 + 4^2 - 2.8^2) / (21.84) = 0.283[/tex]
[tex]cos B = (2.8^2 + 4^2 - 1.8^2) / (22.84) = 0.717[/tex]
[tex]cos C = (2.8^2 + 1.8^2 - 4^2) / (22.81.8) = -0.283[/tex]
Using the inverse cosine function, we can find the angles:
A ≈ 75 degrees
B ≈ 43 degrees
C ≈ 62 degrees
Therefore, the lakefront makes angles of approximately 30 degrees (180 - 75 - 75) and 120 degrees (180 - 43 - 17) with the other two boundaries, and the remaining angle is approximately 30 degrees (180 - 120 - 30).
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