The most general antiderivative of the given function g(t) is (8t + t³/3 + t²/2 + c).
The antiderivative of a function is the inverse function of a derivative.
This inverse function of the derivative is called integration.
Here the given function is: g(t) = 8 + t² + t
Therefore, the antiderivative of the given function is
∫g(t) dt
= ∫(8 + t² + t) dt
= ∫8 dt + ∫t² dt + ∫t dt
= [8t⁽⁰⁺¹⁾/(0+1) + t⁽²⁺¹⁾/(2+1) + t⁽¹⁺¹⁾/(1+1) + c]
= (8t + t³/3 + t²/2 + c)
Here 'c' is the constant.
Again, differentiating the result, we get:
d/dt(8t + t³/3 + t²/2 + c)
= [8 ˣ 1 ˣ t⁽¹⁻¹⁾ + 3 ˣ t⁽³⁻¹⁾/3 + 2 ˣ t⁽²⁻¹⁾/2 + 0]
= 8 + t² + t
= g(t)
The antiderivative of the given function g(t)is (8t + t³/3 + t²/2 + c).
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Suppose your parents decide to give you $10,000 to be put in a college trust fund that will be paid in equally quarterly installments over a 5 year period. If you deposit the money into an account paying 1.5% per quarter, how much are the quarterly payments (Assume the account will have a zero balance at the end of period.)
The quarterly annuity payment is $582.46.
What is an ordinary annuity?
An ordinary annuity in case of investment is the one that pays its end-of-the period cash flows rather than at the beginning as it is case of annuity due.
The applicable formula in this case is the present value formula of an ordinary annuity as shown below:
PV=quarterly payment*(1-(1+r)^-N/r
PV=initial investment=$10,000
quarterly payment=unknown(assume it is X)
r=quarterly interest rate=1.5%
N=number of quarterly payments in 5 years=5*4=20(there are 4 quarterly payments in a year)
$10,000=X*(1-(1+1.5%)^-20/1.5%
$10,000=X*(1-(1.015)^-20/0.015
$10,000=X*(1-0.742470418223773)/0.015
$10,000=X*0.257529581776227/0.015
X=$10,000/(0.257529581776227/0.015)
X=quarterly payment=$582.46
Note that the unknown in this case is the annuity payment, the present value can also be the unknown as well.
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In the expansion of (3a + 4b)8, which of the following are possible variable terms? Explain your reasoning.
a2b3; a7b; ab8; b8; a4b4; a6b2; a3b4; a8
pls help
The terms b⁸, a⁶ · b², a⁴ · b⁴, a⁷ · b and a⁸ are part of the expansion of (3 · a + 4 · b)⁸.
How to find the missing terms of a power polynomials
In this case, we have to expand a power binomial by means of the Pascal's triangle, which offers a useful and quick resource to expand expressions of this kind, now we proceed to present the result of this approach:
(3 · a + 4 · b)⁸ = 6 561 · a⁸ + 69 984 · a⁷ · b + 326 592 · a⁶ · b² + 870 912 · a⁵ · b³ + 1 451 520 · a⁴ · b⁴ + 1 548 288 · a³ · b⁵ + 1 032 192 · a² · b⁶ + 393 216 · a · b⁷ + 65 536 · b⁸
Such combinations of products between variables a and b are also supported by the theorem of the binomial. Finally, the terms b⁸, a⁶ · b², a⁴ · b⁴, a⁷ · b and a⁸ are part of the expansion of (3 · a + 4 · b)⁸.
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Hazing has been reported all across america. What percentage of students say they have witnessed hazing and did not report it?.
percentage of students say they have witnessed hazing and did not report it are = 95%
What is Hazing?No matter how voluntarily a person is to take part, hazing, initiation or deposition are all actions that are demanded of someone when they join or participate in a group and which humiliate, degrade, abuse, or imperil them.
involves humiliating a person or group. involves making fun of someone or something. involves or includes the deliberate destruction of or removal of private or public property for the purpose of affiliation with, initiation into, or as a requirement for continuing membership in an organization.
What is an example of hazing?Paddling, beating, or other types of attack. Branding. consumption of disgusting substances or mixtures under duress or force.
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Which algebraic expression represents the following: the sum of four times a number and six? (1 point) 4x + 6
Answer:
4x +6
Step-by-step explanation:
No explanation
Straight answer
Please explain to me how to do this
#43
1+2+3+..+200We may observe. 200+1=201,199+2=201,198+3=201So
200/2=100 pairs of 201
Sum
100(201)=20100#44
1+2+3+..+400400/2=200 pairs of 401
Sum
200+40180200#45
1+2+3+..+800800/2=400 pairs of 801
Sum
400(801)320400#46
1+2+3+..+20002000/2=1000 pairs of 2001
Sum
2001(1000)2001000Answer:
43. 20,100
44. 80,200
45. 320,400
46. 2,001,000
Step-by-step explanation:
Gauss's method
General formula for the sum of the first n integers:
[tex]1+2+3+ \dots +n=\dfrac{1}{2}n(n+1)[/tex]
Question 43
Given sum:
1 + 2 + 3 + ... + 200Therefore, n = 200.
Substitute the value of n into the formula:
[tex]\implies \dfrac{1}{2}(200)(200+1)=100 \cdot 201=20100[/tex]
Question 44
Given sum:
1 + 2 + 3 + ... + 400Therefore, n = 400.
Substitute the value of n into the formula:
[tex]\implies \dfrac{1}{2}(400)(400+1)=200 \cdot 401=80200[/tex]
Question 45
Given sum:
1 + 2 + 3 + ... + 800Therefore, n = 800.
Substitute the value of n into the formula:
[tex]\implies \dfrac{1}{2}(800)(800+1)=400 \cdot 801 =320400[/tex]
Question 46
Given sum:
1 + 2 + 3 + ... + 2000Therefore, n = 2000.
Substitute the value of n into the formula:
[tex]\implies \dfrac{1}{2}(2000)(2000+1)=1000 \cdot 2001=2001000[/tex]
Which of the following is an even function?
A) f(x) = (x – 1)^2
B) f(x) = 8x
C) f(x) = x^2 – x
D) f(x) = 7
Answer:
D) f(x) = 7Step-by-step explanation:
A function is even when f(-x) = f(x) for all x in the domain of f. We can say that the function is simetrical over x axis.
That's all i know, Hope it helps!
What is the factored form of 5x − 625x4
Options:
A. 5x(1 + 5x)(1 - 5x + 25x2)
B. 5x(1 - 5x)(1 + 5x + 25x2)
C. (1 - 5x)(1 + 5x + 25x2)
D. (1 + 5x)(1 - 5x + 25x2)
Answer:
B
Step-by-step explanation:
5x - 625[tex]x^{4}[/tex] ← factor out 5x from each term
= 5x(1 - 125x³) ← this factor is a difference of cubes and factors in general as
a³ - b³ = (a - b)(a² + ab + b²) , then
1 - 125x³
= 1³ - (5x)³
= (1 - 5x) (1² + 5x(1) + (5x)² )
= (1 - 5x) (1 + 5x + 25x²)
Then
5x - 625[tex]x^{4}[/tex] = 5x(1 + 5x + 25x²)
The equations of two lines are given below.
3 x minus y equals 5
3 y equals x minus 15
Are these lines parallel, perpendicular, or neither?
The lines are neither parallel nor perpendicular
How to determine the relationship between the lines?The equations of the lines are given as
3 x minus y equals 5
3 y equals x minus 15
Rewrite the equations properly
3x - y = 5
3y = x - 15
Make y the subject in both equations
In 3x - y = 5, we have
y = 3x + 5
In 3y = x - 15, we have
y = 1/3x - 5
The slopes of the two lines are
m1 = 3
m2 = 1/3
The above values are not equal, and they are not opposite reciprocal
This means that the lines are neither parallel nor perpendicular
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What is the maximum number of pants per employee that can be washed each week? (2 points)
6. What is the maximum number of chef coats per employee that can be washed
each week? (2 points)
7. What is the maximum total number of items per employee that can be washed in a week? (2 points)
The computation shows that the maximum number of pants per employee that can be washed each week is 6.
How to compute the values?The maximum number of pants per employee that can be washed each week will be:
7 + 4.5x + 12 <= 46
4.5x <= 27
x <= 6
The maximum number is 6.
The maximum number of chef coats per employee that can be washed each week will be:
= 6 + 8
= 14
The maximum total number of items per employee that can be washed in a week will be:
= 6 + 14 = 20
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which of the tables represent a function??
please help me! asap!
5.
A section of a tessellated plane is shown. Which type of symmetry does the tessellated plane have?
The tessellated plane has translational symmetry option (A) translational symmetry is correct.
What is geometric transformation?It is defined as the change in coordinates and the shape of the geometrical body. It is also referred to as a two-dimensional transformation. In the geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, and glide translation.
A geometric figure is given:
As we know, from geometric transformation, geometric transformation, changes in the geometry can be possible by rotation, translation, reflection, etc.
Without any further changes, the tessellated plane is moved a few units to the right. If the translated plane is again moved left by the same amount of units, we will once more obtain the original plane. Translational symmetry is present as a result.
Thus, the tessellated plane has translational symmetry option (A) translational symmetry is correct.
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Use the laplace transform to solve the given initial-value problem. y'' + y = f(t), y(0) = 0, y'(0) = 1, where f(t) = 0, 0 ≤ t < 1, ≤ t < 2 0, t ≥ 2
In order to solve this IVP using Laplace transforms, we must first write f(t) in terms of the Heaviside function.
f(t)=0*(u(t)-u(t-Pi))+1*(u(t-Pi)-u(t-2Pi))+0*(u(t-2Pi))
f(t)=u(t-π)-u(t-2π)
So, the rewritten IVP is
y'' +y = u(t-π)-u(t-2π)y(0)=0, y'(0)=1
Taking the Laplace transform of both sides of the equation, we get:
s2L{y}-sy(0)-y'(0)+L{y}=(1/s)*e-πs-(1/s)*e-2πs
s2L{y}-1+L{y}=(1/s)*e-πs-(1/s)*e-2πs
(s2+1)L{y}=1+(1/s)*e-πs-(1/s)*e-2πs
L{y}=1/(s2+1)+(1/s(s2+1))e-πs-(1/s(s2+1))*e-2πs
Now, we must take the inverse transform of both sides to solve for y.
The first inverse transform is easy enough. By definition, it is sin(t).
The second two inverse transforms will be a little tougher, we will have to use partial fraction decomposition to break them down into terms that are easier to compute.
A/s+(Bs+C)/(s^2+1)=1/(s(s^2+1))
A(s^2+1)+(Bs+C)(s)=1
As^2+A+Bs^2+Cs=1
Rewriting this system in matrix form, we get:
1 1 0 A 0
0 0 1 * B = 0
1 0 0 C 1
Using row-reduction we find that A=1, B=-1, and C=0. So, our reduced inverse transforms are:
L-1{(e-πs)(1/s-s/(s2+1))}
and
L-1{(e-2πs)(1/s-s/(s2+1))}
Using the first and second shifting properties, these inverse transforms can be computed as.
L-1{(e-πs)(1/s-s/(s2+1))}=u(t-π)-cos(t-π)u(t-π)
L-1{(e-2πs)(1/s-s/(s2+1))}=u(t-2π)-cos(t-2π)u(t-2π)
Combining all of our inverses transforms, we get the solution the IVP as:
y=sin(t)+u(t-π)-cos(t-π)u(t-π)+u(t-2π)-cos(t-2π)u(t-2π)
In mathematics, the Laplace transform, named after its discoverer Pierre Simon Laplace (/ləˈplɑːs/), transforms a function of real variables (usually in the time domain) into a function of complex variables (in the time domain). is the integral transform that Complex frequency domain, also called S-area or S-plane).
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PLS HELP ASAP! Ty
Let g(x)= 2x^2+3x-9 and h(x)=x^2+2x-6
Find (h-g)(2.1)
[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Let's solve ~
[tex]\qquad \sf \dashrightarrow \: (h - g)(x) [/tex]
[tex]\qquad \sf \dashrightarrow \: h(x) - g(x) [/tex]
[tex]\qquad \sf \dashrightarrow \: {x}^{2} + 2x - 6 - (2 {x}^{2} + 3x - 9)[/tex]
[tex]\qquad \sf \dashrightarrow \: {x}^{2} + 2x - 6 - 2 {x}^{2} - 3x + 9[/tex]
[tex]\qquad \sf \dashrightarrow \: {x}^{2} - 2 {x}^{2} + 2x - 3x - 6 + 9[/tex]
[tex]\qquad \sf \dashrightarrow \: - {x}^{2} - x + 3[/tex]
For, (h - g)(2.1), put x = 2.1
[tex]\qquad \sf \dashrightarrow \: - {(2.1)}^{2} - (2.1) + 3[/tex]
[tex]\qquad \sf \dashrightarrow \: - 4.41 - 2.1 + 3[/tex]
[tex]\qquad \sf \dashrightarrow \: - 6.51 + 3[/tex]
[tex]\qquad \sf \dashrightarrow \: - 3.51[/tex]
Pls help 15. Find the exact area of the shaded region of the circle shown in the diagram.
Answer:
The shaded region is 9.83 cm²Step-by-step explanation:
Refer to attached diagram with added details.
GivenCircle O with:
OA = OB = OD - radiusOC = OD = 2 cmTo findThe area of segment ADB.SolutionSince r = OC + CD, the radius is 4 cm.
Consider right triangles OAC or OBC:
They have one leg of 2 cm and hypotenuse of 4 cm, so the hypotenuse is twice the short leg.Recall the property of 30°x60°x90° triangle:
a : b : c = 1 : √3 : 2, where a- short leg, b- long leg, c- hypotenuse.It means OC: OA = 1 : 2, so angles AOC and BOC are both 60° as adjacent to short legs.
In order to find the shaded area we need to find the area of sector OADB and subtract the area of triangle OAB.
Area of sector:
A = π(θ/360)r², where θ- central angle,A = π*((mAOC + mBOC)/360)*r²,A = π*((60 + 60)/360))(4²) = 16.76 cm².Area of triangle AOB:
A = (1/2)*OC*(AC + BC), AC = BC = OC√3 according to the property of 30x60x90 triangle.A = (1/2)(2*2√3)*2 = 4√3 = 6.93 cm²The shaded area is:
A = 16.76 - 6.93 = 9.83 cm²Please help me fast
Determine an exact value for the expression and include a complete solution
(sin 225°) ² - cos 330° cos 240°
Answer:
1.187991
Step-by-step explanation:
sin^2(225)−(cos(330))(cos(240))
=(−0.930095)2−(cos(330))(cos(240))
=0.865076−(cos(330))(cos(240))
=0.865076−−0.991199(cos(240))
=0.865076−(−0.991199)(0.325781)
=0.865076−(−0.322914)
=1.187991
40 points please help
The athlete that has the shortest training ride is athlete A.
The athlete that has the shorter range is Athlete B.
The athlete that has the greater median is Athlete B.
The athlete that has the greater IQR is athlete A.
What are the summary statistics?A box plot is used to study the distribution and level of a set of scores. The box plot consists of two lines and a box. the two lines are known as whiskers. The first whisker represents the minimum number and the second whisker represents the maximum number. The difference between the whiskers is known as range.
Range of Athlete A = 36 - 13 = 13
Range of Athlete B = 30 - 19 = 11
Training ride of Athlete A = 16
Training ride of Athlete B = 21
On the box, the first line to the left represents the lower (first) quartile. The next line on the box represents the median. The third line on the box represents the upper (third) quartile.
Median of Athlete A = 20
Median of Athlete B = 25
IQR of Athlete A = 30 - 16 = 14
IQR of Athlete B = 28 - 21 = 7
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An empty box is shaped like a rectangular prism. the box has a base area of square foot and a height of foot. how much packing material is required to fill the box?
3 /10 foot³ packing material is required to fill the box.
What is meant by volume of prism?
The volume of a prism is defined as the total space occupied by the three-dimensional object. Mathematically, it is defined as the product of the area of the base and the length. Therefore, The volume of a Prism = Base Area × Length.The formula for the volume of a prism is V=Bh , where B is the base area and h is the height.Volume = base area × height
= 9/10 × 1/3
= 3 /10 foot³
Therefore, 3/10 foot³ packing material is required to fill the box.
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The complete question is-
An empty box is shaped like a rectangular prism. The box has a base area of 9/10 square foot and a height of 1/3 foot. How much packing material is required to fill the box?
What is the domain of the function y = X+ 6 -7?
x>-7
x>-6
x>6
x>7
The domain of the function y = √(x + 6) - 7 is x > -6
How to determine the domain of the function?The equation of the function is given as
y = √(x + 6) - 7
Set the radical greater than 0
x + 6 > 0
Subtract 6 from both sides of the equation
x > -6
Hence, the domain of the function y = √(x + 6) - 7 is x > -6
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The domain of the function in discuss described as; y = √x+6 -7 is; x >= -6.
What is the domain of the function described as in the task content above?According to the task content, it follows that the domain of.the function can be evaluated by means of the characteristics associated with the square root.
The function given is; y = √x+6 -7
Since, the square root of a negative number renders a complex number as it's results, it follows that; x+6 >= 0.
Hence, x >= -6.
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(3-4^2)^2+8
Bdhdhdhsjsjsj
Answer: 177
Step-by-step explanation:
1. 4x4=16
(3-16)^2+8
2. 3-16=-13
(-13)^2+8
3. -13x-13=169
169+8
4.169+8=177
177
Answer:
177
Step-by-step explanation:
PEMDAS
The PEMDAS rule is an acronym representing the order of operations in math:
ParenthesesExponentsMultiplication and Division (from left to right)Addition and Subtraction (from left to right)Given expression:
[tex](3-4^2)^2+8[/tex]
Parentheses
Begin by calculating the part inside the parentheses.
Following PEMDAS, calculate the exponent first:
[tex]\implies (3-4^2)^2+8=(3-16)^2+8[/tex]
Then carry out the subtraction:
[tex]\implies (3-16)^2+8=(-13)^2+8[/tex]
Exponents
[tex]\textsf{Apply the exponent rule}: \quad (-a)^n=a^n, \quad \textsf{if }n \textsf{ is even}[/tex]
[tex]\implies (-13)^2+8=169+8[/tex]
Addition
[tex]\implies 169+8=177[/tex]
Therefore:
[tex]\implies (3-4^2)^2+8=177[/tex]
The ______ mean is the best estimate of next year's return. multiple choice question. the average of the arithmetic and geometric mean arithmetic geometric
The (B) arithmetic mean is the best estimate of next year's return.
What is the arithmetic mean?The arithmetic mean or arithmetic average, or simply the mean or the average (where the context is obvious), is the sum of a set of numbers divided by the number of numbers in the set. The collection is typically a set of results from an experiment or observational research, or a group of survey results. In some instances in mathematics and statistics, the word "arithmetic mean" is favored since it distinguishes it from other means such as the geometric mean and the harmonic mean. The arithmetic mean is the most accurate predictor of next year's return.Therefore, the (B) arithmetic mean is the best estimate of next year's return.
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The correct question is shown below:
The ______ mean is the best estimate of next year's return. multiple choice question.
(A) the average of the arithmetic and geometric mean
(B) arithmetic
(C) geometric
Geometry: complete this proof (ASAP! It’s urgent)
Answer:
1. <1 ~= <4 (Given)
2. <2 is supplementary to <1; <3 is supplementary to <4 (Because <1 & <2, <3 and <4 form linear pairs, they are supplementary by the linear pair postulate.)
3. <2 ~= <3 (By the definition of supplementary angles, m<1 + m<2 = 180 and m<3 + m<4 = 180. By definition of congruent angles, m<1 = m<4, therefore m<2 = m<3 by substitution property of equality.)
4. AB ~= BC (Converse of Isosceles Triangles Theorem)
5. Triangle ABC is isosceles (Definition of Isosceles Triangles)
A parabola can be represented by the equation y2 = 12x. which equation represents the directrix?
The equation of the directrix is x = -3
What is Parabola?A mirror-symmetrical, roughly U-shaped plane curve known as a parabola. It corresponds to a number of mathematical models that appear to be superficially dissimilar but yet all define the exact same curves. A point and a line are one way to describe a parabola.
According to the given information:Given the general equation of a parabola expressed as
(y-y0)² = 4a(x-x0) ... 1
Equation of the directrix will be expressed as x = x0 - a
From the equation given;
y² = 12x ... 2
Comparing 1 and 2;
x - x0 = x
-x0 = x - x
-x0 = 0
x0 = 0
Similarly;
y-y0 = y
-y0 = y - y
-y0 = 0
y0 = 0
Also;
4a = 12
a = 12/4
a = 3
Substituting x0 =0, y0 = 0 and a = 3 into the equation of the directrix above, we will have;
x = x0 - a
x = 0 - 3
x = -3
Hence,
the equation of the directrix is x = -3
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S=1 - 1/4 + 1/6 - 1/9 + 1/11 - 1/14..........infinity. Find the sum of series.
[tex]1+\frac{1}{6}+\frac{1}{11}+\cdots=\sum^{\infty}_{n=1} \frac{1}{5n-4} \\ \\ \frac{1}{4}+\frac{1}{9}+\cdots=\sum^{\infty}_{n=1} \frac{1}{5n-1} \\ \\ S=\sum^{\infty}_{n=1} \left(\frac{1}{5n-4} -\frac{1}{5n-1} \right) \\ \\ =\frac{\pi}{5} \sqrt{\frac{1}{5} (5+2\sqrt{5})}[/tex]
When the points in a scatter plot cluster closely around the regression line, the correlation can be said to be?
The points in a scatter plot cluster closely around the regression line, the correlation can be said to be the proportion of variance in Y that is determined or explained by X.
It is critical to understand that a correlation coefficient of zero indicates that there is no linear courting, but there can also nonetheless be a strong relationship among the 2 variables.
If one point of a scatter diagram is farther from the regression line than a few other factor, then the scatter diagram has as a minimum one outlier. If two or extra points are the identical farthest distance from the regression line not a commonplace prevalence), then every of those points is an outlier.
When the y variable has a tendency to increase because the x variable increases, we say there may be a high-quality correlation between the variables. when the y variable has a tendency to lower as the x variable increases, we are saying there's a bad correlation between the variables.
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Please Help!
A dog is attached to a 35-foot rope fastened to the outside corner of a fenced-in garden that measures 28 feet by 36 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander.
The dog can wander an area of ___ square feet
(exact answer, using the pi symbol)
The area that the dog can wander based on the information provided will be 931π.
How to calculate the area?From the information, we are told that the dog dog is attached to a 35-foot rope fastened to the outside corner of a fenced-in garden that measures 28 feet by 36 feet.
In this case, the dog is in an outside corner. The dog can trace 3/4 of a circle when it starts walking in a circle away form the wall.
The area that the dog can wander will be:
= 3/4(π35²) + 1/4(π(35 - 28)²)
= 3/4π(35)² + 1/4π(7)²
= (3/4 × 1225)π + (1/4 × 49)π
= 918.75π + 12.25π
= 931π
Therefore, the area that the dog can wander based on the information provided will be 931π.
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olives and pineapple pizza delivered two large pizzas cut into eight slices each. james and his sister both ate 5/8 of the pizza together. how many slices are left?
Find the surface area of each figure. Round your answers to the nearest tenth, if necessary.
The surface area of the pyramid is 34.8 in²
How to find the surface area of the figure?Since the figure is a square pyramid, its surface area, A = 4 × area of triangular face + area of square base.
The area of triangular face
Area of triangular face A' = 1/2bh where
h = height = 4.3 in and b = base = 3 in.So, A' = 1/2 × 4.3 in × 3 in
A' = 1/2 × 12.9 in²
A' = 6.45 in²
The area of square base
Area of square base A" = L² where L = length of base = 3 in
So, A" = (3 in)²
= 9 in²
The Surface area of pyramid.
Surface area of pyramid, A = 4A' + A"
= 4 × 6.45 in² + 9 in²
= 25.8 in² + 9 in²
= 34.8 in²
So, the surface area of the pyramid is 34.8 in²
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Use of identity for sum of cubes:
x³ + y³ = (x + y)(x² - xy + y²)Change it as:
x³ + y³ = (x + y)(x² + 2xy + y² - 3xy) = (x + y)[(x + y)² - 3xy)]It is easy to notice that:
64a³ = (4a)³ and125b³ = (5b)³.So the sum of cubes for the given expression will be:
64a³ + 125b³ = (4a + 5b)[(4a + 5b)² - 3(4a*5b)] =(4a + 5b)[(4a + 5b)² - 60ab]Now substitute the values and calculate each of these:
(a) 4a + 5b = 5, ab = -1.
(4a + 5b)[(4a + 5b)² - 60ab] = 5[5² - 60(-1)] = 5(25 + 60) = 5(85) = 425(b) 4a + 5b = -2, ab = 1/15.
(4a + 5b)[(4a + 5b)² - 60ab] = -2[(-2)² - 60(1/15)] =-2(4 - 4) = -2(0) = 0(c) 4a + 5b = 1/3, ab = -1/9.
(4a + 5b)[(4a + 5b)² - 60ab] = (1/3)[(1/3)² - 60(-1/9)] =(1/3)(1/9 + 60/9) = (1/3)(61/9) = 61/27 = 2 7/27(d) 4a + 5b = -1, ab = 1.
(4a + 5b)[(4a + 5b)² - 60ab] = -1[(-1)² - 60(1)] = -1(1 - 60) = -1(-59) = 59Identify the side lengths of the triangle.
The figure shows an isosceles triangle. One of the congruent sides has a length of 3 x plus 4 units. The other congruent side has a length of x plus 12 units. The third side has a length of 4 x minus 10 units.
The side lengths of the isosceles triangle are:
AO = 16 units
AB = 16 units
OB = 6 units.
What is an Isosceles Triangle?An isosceles triangle is a type of triangles that has two sides that are congruent to each other, and two base angles that are opposite these two sides that are also congruent to each other.
How to Identify the Side lengths of the Triangle?Referring to the isosceles triangle in the image attached below, we have the following:
One of the congruent sides = AO = 3x + 4 units
The other congruent side = AB = x + 12 units
The third side = OB = 4x - 10 units
Applying the definition of an isosceles triangle, we can create an equation as shown below to find x:
AO = AB [congruent sides]
Substitute
3x + 4 = x + 12
3x - x = - 4 + 12
2x = 8
2x/2 = 8/2
x = 4
Find the side lengths of the isosceles triangle by plugging in the value of x:
One of the congruent sides = AO = 3x + 4 = 3(4) + 4 = 16 units
The other congruent side = AB = x + 12 = 4 + 12 = 16 units
The third side = OB = 4x - 10 = 4(4) - 10 = 6 units.
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The graph of the step function g(x) = –⌊x⌋ + 3 is shown.
On a coordinate plane, a step graph has horizontal segments that are each 1 unit long. The left end of each segment is a closed circle. The right end of each segment is an open circle. The left-most segment goes from (negative 2, 5) to (negative 1, 5). Each segment is 1 unit lower and 1 unit farther to the right than the previous segment. The right-most segment goes from (4, negative 1) to (5, negative 1).
What is the domain of g(x)?
{x| x is a real number}
{x| x is an integer}
{x| –2 ≤ x < 5}
{x| –1 ≤ x ≤ 5
Using it's concept, the domain of the function is given as follows:
{x| –2 ≤ x <= 5}
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function.
The function described is defined for x between -2 and 5, hence the domain is given by the following interval:
{x| –2 ≤ x <= 5}
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Answer:
{x| –2 ≤ x < 5}
Step-by-step explanation:
