There are times when I use or calculate percentages in my personal life or at work. This is true.
The benefits that result from using percentages in this application is that it gives one an idea of the value to be paid as tax. The drawback is that it can be complicated for large number.
A word problem that can Illustrate percentage is that "An income is $1000 and there's a tax of 10%, how much will be paid?"
How to illustrate the percentage?A number or ratio expressed as a fraction of 100 is called a percentage. Divide the value by the total value to get the percentage, then multiply that number by 100.
For instance, many statistics in the media, bank interest rates, retail discounts, and inflation rates are all expressed as percentages. For understanding the financial aspects of daily life, percentages are crucial.
For example, an income is $1000 and there's a tax of 10%, the tax will be:
= 10% × $1000
= $100
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what is the answers to this question
By evaluating the quadratic equation, we will see that the values of the expressions are:
f(-1) + f(4) = 27
f(-1) - f(4) = -35
How to evaluate the given expressions?Here we need to work with the quadratic function:
f(x) = x^2 + 6x + 1
First, we want to find the value of the expression:
f(-1) + f(4)
To get that, we need to evaluate the quadratic equation:
f(-1) = (-1)^2 + 6*(-1) + 1
f(-1) = 1 - 6 + 1 = -4
f(4) = 4^2 + 6*4 + 1
f(4) = 16 + 24 + 1 = 31
Then the first expression is:
f(-1) + f(4) = -4 + 31 = 27
The second expression is:
f(-1) - f(4)
We already know these values, we can replace these to get:
f(-1) - f(4) = -4 - 31 = -35
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Write 3 numbers that are divisible by 2,
5, and 10.
Answer:
2,4,6,8,0
Step-by-step explanation:
A number is divisible by 2 if it ends in 2, 4, 6, 8 or 0. A number is divisible by 5 if it ends in 5 or 0. A number is divisible by 10 if it ends in a 0.
A capsule is in the form of a hollow cylinder surmounted by a hemispherical bowl of the same diameter, 12 cm on both ends. The total height of the capsule is 16 cm. Find the surface area of the capsule. п cm?
The surface area of the capsule is calculated by adding the surface area of the cylinder and the surface area of the hemispherical bowl.
What is the hemispherical bowl?A hemispherical bowl is a type of bowl with a rounded bottom that is shaped like half of a sphere. It is often used in laboratories, kitchens, and dining rooms. The bowl’s curved surface prevents the contents from spilling, which makes it an ideal choice for mixing, swirling, and scooping. Additionally, the bowl’s shape makes it look aesthetically pleasing, so it can be used as a decorative piece as well.
The surface area of the cylinder = 2πr2 + 2πrh
Where r is the radius of the cylinder and h is the height of the cylinder.
The radius of the cylinder = 12 cm
The height of the cylinder = 16 cm
Therefore, the surface area of the cylinder = 2π(12)2 + 2π(12)(16) = 1536π cm2
The surface area of the hemispherical bowl = 2πr2
Where r is the radius of the hemispherical bowl.
The radius of the hemispherical bowl = 12 cm
Therefore, the surface area of the hemispherical bowl = 2π(12)2 = 144π cm2
The total surface area of the capsule = 1536π cm2 + 144π cm2 = 1680π cm2
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A math club has 30 members. The number of girls is 2 less than 3 times the number of boys. How many members are boys? How many members are girls?
Answer:Well, we already see that the number of girls is greater than the number of boys, to find the result of girls, just do the division that is 30:2= 15 and boys is 30:3= 10
i hope i have helped you
Step-by-step explanation:
Answer:
8 boys, 22 girls
Algebraic ExpressionsTo solve, you must isolate the variable. This means moving the variable and it's coefficient to the other side of the equation with only the same variable on the same side with all the constants on the other side.
If you multiply, divide, add, subtract, square root, exponent both sides of the equation by the same value, that end value stays the same.
Q)
Lets set the amount of boys to x.
The amount of girls is 3x-2 as the amount of boy is x.
Set up an equation.
x+3x-2 = 30
30 is the amount of members.
Add 2 on both sides.
x+3x+2-2=30+2
Simplify by combining like terms.
4x = 32
Divide 8 on both sides.
[tex]\frac{4x}{4} =\frac{32}{4}[/tex]
x = 8
There are 8 boys in the class.
Plug back into the equation or subtract from 30 to solve for the amount of girls.
30 - 8 = 22
8(3) - 2 = 22
solve the given differential equation by undetermined coefficients.
y"+3y=-48x2e3x
The given differential equation by undetermined coefficients are y(x)=c1cos(31/2 x)+c2sin(31/2x)+(-4x2+4x-4/3)e3x
A differential equation in mathematics is an equation that includes one or more functions and their derivatives. The rate of change of a function at a place is determined by the derivatives of the function. It is mostly employed in disciplines like physics, engineering, biology, and others. The study of solutions that satisfy the equations and the characteristics of the solutions is the main goal of the differential equation.
y'' + 3y=0
λ2+3 =0
λ 1=-i31/2,
λ 2=i31/2.
y0(x)=c1cos(31/2 x)+c2sin(31/2x),
yp(x)=(Ax2+Bx+C)e3x
yp'(x)=(3Ax2+(3B+2A)x+3C+B)e3x
yp''(x)=(9Ax2+(9B+12A)x+2A+6B+9C)e3x
x2e3x:3A+9A=-48,
xe3x:3B+9B+12A=0,
e3x:3C+2A+6B+9C=0.
=>A=-4,B=4,C=-4/3.
yp(x)=(-4x2+4x-4/3)e3x,
y(x)=y0(x)+yp(x)
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The dimensions of a box are 2x + 5 meters, x + 4 meters, and x + 2 meters. The volume of
the box is 2,520 m³. Find the dimensions of the box.
The rectangular box dimensions are 21m ,12m and 10m.
What is volume of the rectangular box?
The area of the base times the height of the rectangular prism equals the volume of the object. As a result, the formula for a rectangular prism's volume is given. A rectangular prism's volume is equal to its length, width, and height in cubic units.
Here the dimensions of a box is 2x+5 , x+4 and x+2 meters and volume of the box is 2520 [tex]m^3[/tex].
Volume of rectangular box = w×h×l [tex]unit^3[/tex]
=> 2520 = (2x+5)(x+4)(x+2)
=> 2520 = [tex](2x^2+13x+20)(x+2)[/tex]
=> 2520 = [tex]2x^3+4x^2+26x+13x^2+20x+40[/tex]
=> [tex]2x^3-16x^2+33x^2-264x+310x-2480=0[/tex]
=> [tex]2x^2(x-8)+33x(x-8)+310(x-8)=0[/tex]
=> [tex](x-8)(2x^2+33x+310)=0[/tex]
Here x-8=0 and [tex]2x^2+33x+310=0[/tex]
Then x=8 and x ∈ R.
Now the dimensions 2x+5=2(8)+5=16+5=21m.
=> x+4 = 8+4=12 m.
=> x+2= 8+2 = 10m.
Hence the rectangular box dimensions are 21m ,12m and 10m.
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Find two positive numbers whose product is 192 and whose sum is a minimum.
A. 3 and 64
B. 4[tex]\sqrt{3}[/tex] and 16[tex]\sqrt{3}[/tex]
C. 8 and 24
D. 8[tex]\sqrt{3}[/tex] and 8[tex]\sqrt{3}[/tex]
E. 12 and 16
Two positive numbers whose product is 192 and whose sum is a minimum are; 8√3 and 8√3.
The correct option is (D).
What is minima?Minima is the minimum value of a function in given domain.
First Order Derivative Test
Let f be the function defined in an open interval I and f be continuous at critical point c in I such that f’(c) = 0.
If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima. And the f(c) is the minimum value.
Given,
Product of two positive numbers = 192
Sum should be minimum
Let x be the first positive number and y be the other positive number. So, the equation would be
x . y = 192
⇒ y = 192/x --------(a)
Let S be the sum of the two positive numbers.
S = x + y
substituting value of y in terms of x from equation(a)
S = x + 192/x
⇒ S = x + 192 . x⁻¹
Differentiating with respect to x
S'(x) = 1 + (-1)192x⁻²
S'(x) = 1 - 192/x²
S'(x) = (x² - 192)/x²
To determine the minimum S, equating first derivative of S with respect to x to zero.
S'(x) = 0
(x²-192)/x² = 0
x² - 192 = 0
x² = 192
x = ±√192
Rejecting negative root, since the numbers are positive
x = √192
Substituting value of x in equation (a)
y = 192/√192
y = √192
x and y can be written as
x = √192 = 8√3
y = √192 = 8√3
Hence , 8√3 and 8√3 are the two positive numbers whose product is 192 and sum is a minimum.
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A system of equations is graphed on this coordinate grid.
Which ordered pair is the best estimate of the solution of the system of equations?
(-1, -2)is the best solution of the system of equations.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
A system of equations is graphed on this coordinate grid.
we need to find the ordered pair which is the best estimate of the solution of the system of equations.
(-1, -2) is the best solution of this graph because both the lines are intersecting at that point
Hence, (-1, -2)is the best solution of the system of equations.
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The City of Rock Hill has an approximate population of 70,000. An average of 100 people move in to Rock Hill every month and 150 people move out of Rock Hill each month. The City of Greenville has an approximate population of 61,000. An average of 200 people move into
Greenville every month. In how many months will the populations of
The number of months it will take for both cities to have the same population is; 36 months
How to solve Algebraic Word Problems?We are given;
Current population of city of rock hill = 70000
Rate at which people move into city of rock hill = 100 people per month
Rate at which people move out of city of rock hill = 150 people per month
Net average increase a month = 100 - 150 = -50 people per month
Current population of City of Greenville = 61000
Rate at which people move into City of Greenville = 200 people per month
If the number of months it will take for both cities to have the same population is x, then it means that;
70000 - 50x = 61000 + 200x
Rearrange to get;
200x + 50x = 70000 - 61000
250x = 9000
x = 9000/250
x = 36 months
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Answer to this math question
By means of relationship between powers and roots and algebra properties we conclude that radical expression ∛y · [2 · y · ∛(8 · y²) - ∛(y⁵) - 4∛(8 · y²)] is equivalent to 2 · y · ∛(8 · y⁵) - ∛(y⁶) - 4 ∛(8 · y³).
How to simplify a polynomial-like radical expression
In this problem we find a polynomial-like radical expression that must be simplified by algebra properties and by taking advantage of relationship between powers and roots. First, write the entire expression:
∛y · [2 · y · ∛(8 · y²) - ∛(y⁵) - 4∛(8 · y²)]
Second, apply distributive property:
2 · y · ∛(8 · y²) · ∛y - ∛(y⁵) · ∛y - 4∛(8 · y²) · ∛y
Third, use relationship between roots and powers:
2 · ∛(y³) · ∛(8 · y²) · ∛y - ∛(y⁵) · ∛y - 4∛(8 · y²) · ∛y
Fourth, apply multiplication of roots and multiplication of powers of equal base:
2 · y · ∛(8 · y⁵) - ∛(y⁶) - 4 ∛(8 · y³)
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Find the center and radius of the circle with the equation:
(x-3)² + (y-1)² = 16
center: (-3,-1)
radius: 4
a.
b. center: (-3,-1)
radius: 16
C.
center: (3, 1)
radius: 4
d. center: (3, 1)
radius: 16
On solving the provided question we can say that in the circle the radius, r = 4 and center, c = -3, -1, so area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.
What is circle?Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. At every angle, it is also rotationally symmetric about the center. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.
here,
in the circle
the radius, r = 4
and center, c = -3, -1
area of the circle will be = [tex]\pi r^2[/tex] = 3.14*4*4 = 50.24 cm sq.
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AB=16 and BC = 22 what does AC equal
Answer:AC is equal to 88
Step-by-step explanation:
C=11
A=8
B=2
At Park Junior High, 10%, or 160, of the students, play a musical instrument. How many students attend the school?
A tape diagram. StartFraction part Over whole EndFraction = StartFraction 10 Over 100 EndFraction = StartFraction 160 Over question mark EndFraction
Which statements are correct? Check all that apply.
The total number of students is < 160.
The total number of students is > 160.
The percent as a part-to-whole ratio is StartFraction 10 Over 100 EndFraction.
The percent as a part-to-whole ratio is StartFraction 160 Over 100 EndFraction
There are 1,600 students in the school.
There are 250 students in the school.
The statements that are correct include the following:
B. The total number of students is > 160.
C. The percent as a part-to-whole ratio is 16/100.
E. There are 1600 students in the school.
What is a percentage?In Mathematics, a percentage can be defined as any number that is expressed as a fraction of hundred (100). This ultimately implies that, a percentage indicates the hundredth parts of any given number.
For the total number of students, we have:
Total number of students = 10/100 × T = 160
0.1T = 160
T = 160/0.1
T = 1,600 students.
Note: Percentage as a part-to-whole ratio is 10/100.
In conclusion, we can reasonably infer and logically deduce that the total number of students at Park Junior High is equal to 1,600 students.
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Complete Question:
At park junior high 10% or 160 of the students play a musical instrument how many students attended the school?
Which statements are correct? check all that apply.
A.the total number of students is 160
B. The total number of students is > 160.
C.The percent as a part-to-whole ratio is 10/100
D.the percent as a part to whole ratio is 160/100
E.there are 1600 students in the school
F.there are 250 students in the school.
Exhibit 2-3
The number of sick days taken (per month) by 200 factory workers is summarized below.
Number of Days Frequency
0 − 5 120
6 − 10 65
11 − 15 14
16 − 20 1
The number of workers who took more than 10 sick days per month is _____.
A summary of 200 factory workers' monthly sick days is provided. From this, we can tell that about 15 workers took more than 10 sick days per month.
The number of times an event or observation occurred during an experiment or research is referred to as its frequency in statistics. It can alternatively be described as a straightforward count of a specific occurrence. Relative frequency and cumulative frequency are the two main types of frequency seen in statistics.
The given table contains the number of sick days per month and the frequency of factory workers on sick days. Then, the number or frequency of workers who took more than 10 sick days per month is,
number of workers = frequency for 11-15 days + frequency for 16-20 days
=14+1
=15
The required answer is 15 workers.
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How do we solve this?
Correct option is E, The roots of Auxiliary equation are √6 and -√6.
The solution of an nth-order differential equation or difference equation depends on an algebraic equation of degree n called the characteristic equation (or auxiliary equation) in mathematics.
The auxiliary equation, however, lacks true roots. Consider the simple harmonic equation y + y = 0, which has solutions and was given that name because of how it relates to the vibration of a musical tone. Y2(t) = cos t and Y1(t) = sin t. r2 + 1 = 0 is the auxiliary equation for the straightforward harmonic equation.
The Auxiliary form of the given equation is m² - 6 = 0
Using identity a² - b² = (a - b) (a + b),
⇒ The equation becomes (m - √6)(m+√6) = 0
⇒ either m = √6 or m = -√6
Hence, The roots of Auxiliary equation are √6 and -√6.
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Please help me with this!
The question is in the screenshot below. TIA
Answer: 3c+3
Step-by-step explanation:
You throw a ball straight up from a rooftop 70 feet high with an initial speed of 80
feet per second. The functions s(t)=â16t2+80t+70models theâ ball's height above theâ ground,s(t)
inâ feet, t seconds after it was thrown. During which time period will theâ ball's height exceed that of theâ rooftop?
When will the ball exceed the height of theâ rooftop?
On solving the provided question we can say that the value of equation here will be as duration is 0<t<5 seconds.
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions
Information given in the query:
Rooftop height is 70 feet.
Speed at start: 80 feet per second
s(t ) = - 16t² + 80t + 70
s(t) in feet here
now,
for the balls to touch the rooftop
[tex]s(t ) = - 16t^2+ 80t + 70 > 70\\ - 16t^2 + 80t + 70 > 70\\ - 16t^2 + 80t > 70 - 70\\ - 16t^2 + 80t > 0\\ 16t^2 - 80t < 0\\t( 16t - 80 ) < 0\\t < 0 and 16t - 80 < 0 or 16t < 80 or t < 5\\[/tex]
Hence,
since time cannot be negative, the time period is 0 t 5 seconds.
0<t<5
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p divided by 9=4 is what
Answer:
36
Step-by-step explanation:
p/9=4
p=4*9
p=36
hello, i need help with my math money management math homework but no one wants to help me please help me with my math homework!!
a) The total cost of purchasing the car with the harmonized sales tax (HST) is $36,160.
b) If the customer leases the car instead of buying it, they will save $14,760.
c) After the customer returns the leased car, their lease options include:
Lease buyoutExtending the leaseSigning a new lease agreementBuying out the car and then reselling it.What is a lease?A lease is a financing arrangement for capital assets, that enables the lessee to use the asset for a determined period while making periodic payments to the lessor.
Leases are classified into operating and finance (capital) leases.
Value of the car = $32,000
Harmonized Sales Tax (HST) = 13%
Cost of the car with HST = $36,160 ($32,000 x 1.13)
Under Lease:Down payment = $1,000
Financing part = $35,160 ($36,160 - $1,000)
Installment payments = 48
Periodic payments = $425
Total lease cost = $21,400 ($1,000 + 48 x $425)
Savings from leasing = $14,760 ($36,160 - $21,400)
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The graph of linear equation is always a straight line ?
Yes, the graph of linear equation is always a straight line.
What is a linear equation?Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.
Consider the equation y = mx + c to define a linear function with a straight line as its graph.
We are aware that the rate of change, m, for a linear function is constant. On the graph, shifting by 1 always causes you to go up by m as shown in attached image.
The graph thus resembles a staircase. It is a straight line because it always rises in equal-sized steps.
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Kiyo invites 5 friends to a sporting event.
Tickets cost $40 each. He knows at least one
friend will choose to attend the event.
Answer: 40 dollars
Step-by-step explanation:
i hope the explanation is right:)
If he invites 5 friends for 40 bucks each then one friend that he knows will go is 40 dollars. So the only thing you have to do for this problem is multiply 40 by the number of friends which in this case i believe is 1.
if AT= 4x-7 and CT = -x + 13, solv e for x
PLEASE HELP, I ATTACHED AN IMAGE OF THE PROBLEM!!!
The correct rule of the transformation is,
⇒ Dilation of 0.5.
What is Translation?A transformation that occurs when a figure is moved from one location to another location without changing its size or shape is called translation.
We have to given that;
The coordinate of CDE are,
⇒ C (4, - 5), D (3, - 1), E (5, - 3)
And, After transformation the coordinate of image C'D'E' are,
⇒ C' (2, - 2.5) , D' (1.5, - 0.5), E' (2.5, - 1.5)
Now, We have;
⇒ C (4, - 5) = C' (2, - 2.5)
Let a dilation of the points = k
So, We can formulate;
⇒ 4 × k = 2
⇒ k = 2/4
⇒ k = 1/2
⇒k = 0.5
Hence, The correct rule of the transformation is,
⇒ Dilation of 0.5.
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You have a cone whose height and base radius are equal and, inside it, you put the largest possible sphere that will completely fit inside. What is the exact fraction of the volume of the cone that is occupied by the sphere?
The exact ratio of of the volume of the cone that is occupied by the sphere is 4/(1+√2)³.
What is a cone?
A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (typically circular base) to a point called the apex or vertex (which creates an axis to the centre of base).
Given that the height and radius of the cone is the same.
Assume that the radius of the cone is x.
Thus the slant height of the cone is √(x² + x²) = x√2.
Assume the radius of the sphere is r.
△OEC and △BDC are similar triangles according to AAA rule.
Thus, BD/OE =BC/OC
x/r = x√2/(x -r)
Cross multiply:
x(x - r) = xr√2
x² - rx = xr√2
x² = xr√2 + rx
rx( 1+√2) = x²
r = x²/[x( 1+√2)]
r = x/ (1+√2)
The volume of the sphere is 4/3 ∏ r³ = 4/3 ∏ [ x/ (1+√2)]³
The volume of the cone is 1/3 ∏r²h = 1/3 ∏x³
The ratio of the volume of the sphere to the cone is
4/3 ∏ [ x/ (1+√2)]³ : 1/3 ∏r³
= 4 : (1+√2)³
= 4/(1+√2)³
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What is the vertex of y = |x-9| +7?
Answer:
Step-by-step explanation:
The vertex (h, k) is always in the format of "x-" so if we have a positive h value for the vertex, it would be written as "x-(h)" and if we have a negative h value for the vertex, it would be written as "x-(-h)" which is the same as "x+h". Since our sign inside the absolute value symbols is negative, the h value for the the vertex is 9. The k value is always what's outside the absolute value symbols, including the sign. Ours is a positive 7. Thus the vertex is (9, 7).
what is the average (mean) value of 3t^3-t^2 over the interval -1<=t<=2?
The average value of 3t^3-t^2 over the interval -1<=t<=2 is 11/4
Mean Value Theorem states if a function f(x) is continuous on the interval [a,b] and differentiable on (a,b), there is at least one value c in the interval (a<c<b) such that:
f '(c)=f(b)-f(a)/b-a
Average Value:
The average value of a function of a variable over a range represents the height of a rectangle of the same area as that defined by the function under the curve. This value can be calculated using a formula:
favg=1/b−a∫f(t)dt
consider f(x)=3t³-t² and limits are -1 to 2
Now, substitute these values in the above formula:
favg=1/2-(1)∫3t³-t²dt
favg=1/3∫(3/4t⁴-1/3t³)|₋₁²
favg=(12-8/3)-(3/4+1/3)
By solving the equation we get
favg=11/4
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NO LINKS!!
a. Identify each sequence as arithmetic, geometric, or neither
b. If it is arithmetic or geometric, describe the sequence generator
n t(n)
0 16
1 9
2 4
3 1
4 0
5 1
6 4
Answer:
a) Neither,b) t(n) = (n - 4)²--------------------------------------
We observe that, the sequence is symmetric and all the terms are perfect squares:
16, 9, 4, 1, 0, 1, 4 ⇒ 4², 3², 2², 1², 0², 1², 2²This is neither arithmetic nor geometric.
The zero term is 16 and the fourth term is 0 so the nth term would be:
t(n) = (n - 4)²Answer:
a) Neither
b) t(n) = n² - 8n + 16
Step-by-step explanation:
An Arithmetic Sequence has a constant difference between each consecutive term.
A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.
Part (a)As the sequence has neither a constant difference or a constant ratio, the sequence is neither arithmetic or geometric.
Part (b)Work out the differences between the terms until the differences are the same:
First differences
[tex]16 \underset{-7}{\longrightarrow} 9 \underset{-5}{\longrightarrow} 4 \underset{-3}{\longrightarrow} 1 \underset{-1}{\longrightarrow} 0 \underset{+1}{\longrightarrow} 1 \underset{+3}{\longrightarrow} 4[/tex]
Second differences
[tex]-7 \underset{+2}{\longrightarrow} -5 \underset{+2}{\longrightarrow} -3\underset{+2}{\longrightarrow} -1\underset{+2}{\longrightarrow} 1\underset{+2}{\longrightarrow} 3[/tex]
As the second differences are the same, the sequence is quadratic and will contain an n² term. The coefficient of n² is always half of the second difference. Therefore, the coefficient of n² = 1.
Write out the numbers in the sequence n² and determine the operation that takes n² to the given sequence:
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 &0& 1& 4& 9&16 & 25&36 \\\cline{1-8} \sf operation&+16& +8&+0&-8&-16&-24&-32\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]
As the operation is not constant, work out the differences between the operations:
[tex]16\underset{-8}{\longrightarrow} 8\underset{-8}{\longrightarrow} 0\underset{-8}{\longrightarrow} -8\underset{-8}{\longrightarrow} -16\underset{-8}{\longrightarrow} -24\underset{-8}{\longrightarrow} -32[/tex]
As the differences are the same, the second operation in the sequence is -8n. Write out the numbers in the sequence with both operations and and determine the operation that takes (n² - 8n) to the given sequence:
[tex]\begin{array}{|c|c|c|c|c|c|c|c|}\cline{1-8} n&0& 1& 2&3 &4 &5 &6 \\\cline{1-8}n^2 -8n&-0&-7&-12&-15&-16&-15&-12\\\cline{1-8}\sf operation &+16&+16&+16&+16&+16&+16&+16\\\cline{1-8} \sf sequence & 16&9 &4 & 1& 0& 1& 4\\\cline{1-8}\end{array}[/tex]
As the operation is constant, the final operation in the sequence is +16.
So the equation for the nth term is:
[tex]\implies t(n)=n^2-8n+16[/tex]
A rectangular piece of metal is 15 in longer than it is wide. Squares with 3 in long are cut from the 4 corners and the flap are folded upward to form an open box. If the volume of the box is 858in^3, what were the original dimensions of the piece of metal?
Note that in the above scenario, the original dimensions of the piece of metal that is rectangular are:
Length: 32 InchesWidth: 15 InchesWhat is the rationale for the above solution?Let x equal the width of the box and x + 15 equal the length of the box.
Subtract 6 (the amount cut out) from both of these to get the following terms: (x+9) and (x-6).
These are the length and width of the box respectively. We also know that the height of the box is 3. Plugging these into the formula for the volume of a box gives you:
3(x+9)(x-6)=858
If the above is expanded, we have:
3x²+9x-162=858
Subtract 858 from both sides to make the quadratic equation equal zero. Plug the a, b and c coefficients into the quadratic formula (or factor) to find x.
3x²+9x-162- 858 = 0
⇒ 3x²+9x -1020= 0
At the stage we solve for x:
3x2+9x−1020 = 0
Factor left side of the equation.
3(x−17)(x+20)=0
Set factors equal to 0.
x−17=0 or x+20=0
x=17 or x=−20
We can't get a negative side, so the answer that makes sense is x =17 recall that the metal is 15 inches longer than its width.
So if the Width is x and the Lenght is x +15
Then Width is 17 while the length is 17 + 15 = 32inces.
Thus the original dimensions of the metal are:
Length: 32 Inches
Width: 15 Inches
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Which of the following are true statements about waves?
A. All waves carry matter but not energy.
B. All waves have both an amplitude and frequency.
C. All waves have a repeating pattern.
Answer:
B,C
Step-by-step explanation:
B. All waves have both an amplitude and frequency.
C. All waves have a repeating pattern.
A statement A is incorrect, Waves are a disturbance that transfer energy through space without transferring matter.
B is true, All types of waves have an amplitude, which is the measure of the maximum displacement of a point on the wave from its rest position and a frequency, which is the number of complete oscillations of a point on the wave per unit of time.
C is also true, All types of waves have a repeating pattern, which is known as a waveform. The shape of the waveform determines the type of wave. For example, sine waves have a smooth, curved shape, while square waves have sharp, angular edges.