The Taylor series up to the degree 4 terms is:
e^8 + (x - 8)e^8 + (x - 8)^2/2! e^8 + (x - 8)^3/3! e^8 + (x - 8)^4/4! e^8
The Taylor series of a function f(x) at point a is an infinite sum of terms that approximates the value of the function near point a. The general form of the Taylor series of f(x) at a is:
f(x) = f(a) + (x - a)f'(a) + (x - a)^2/2! f''(a) + (x - a)^3/3! f'''(a) + ...
Where,
f'(a) is the first derivative of f(x) at a.f''(a) is the second derivative of f(x) at a.f'''(a) is the third derivative of f(x) at a.In the case of e^x, the function is infinitely differentiable, so all its derivatives are equal to itself. Hence the Taylor series at a=8 is:
e^8 + (x-8)e^8 + (x-8)^2/2! e^8 + (x-8)^3/3! e^8 + (x-8)^4/4! e^8 + ....
by taking the degree 4 terms we can find the Taylor series of e^x at a = 8 up to the degree 4 terms, which is
e^8 + (x-8)e^8 + (x-8)^2/2! e^8 + (x-8)^3/3! e^8 + (x-8)^4/4! e^8
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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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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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Merge onto Highway 40 and drive 3/5
mile. Stop and pay the toll. Then
continue on Highway 40 for twice this.
distance. How much longer will you be
on Highway 40 after you pay the foll?
Distance traveled after toll payment is 1.2 miles on highway 40.
What is Distance ?The distance may be calculated using a curved route. Displacement measurements can only be made along straight lines. Distance is path-dependent, meaning it varies depending on the direction followed. Displacement simply depends on the body's beginning and ending positions; it is independent of the route.
Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.
The size or extent of the displacement between two points is referred to as distance. Keep in mind that the distance between two points and the distance traveled between them are not the same. The entire length of the journey taken between two points is known as the distance traveled. Travel distance is not a vector.
Distance traveled before toll payment =3/5 miles on highway 40
Distance traveled after toll payment =2*3/5 = 6/5 =
1.2 miles on highway 40.
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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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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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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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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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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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p divided by 9=4 is what
Answer:
36
Step-by-step explanation:
p/9=4
p=4*9
p=36
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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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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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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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.
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
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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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.
Find the range of the given function y=3x-2 for the domain (-1,2,4)
The range of the given function y = 3x - 2 for the domain (-1, 2, 4) is
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.
Domain of the function is the values of x for which the function is defined.
Range of the function is the value of the function when we put the values in the domain.
Given function,
y = 3x - 2
Given the domain (-1, 2, 4)
x = -1, then y = (3 × -1) - 2 = -5
x = 2, then y = (3 × 2) - 2 = 4
x = 4, then y = (3 × 4) - 2 = 10
Range is (-5, 4, 10).
Hence the range of the function is (-5, 4, 10).
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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]
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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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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There are some counters in a bag.
The counters are red or white or blue or yellow.
Bob is going to take at random a counter from the bag.
The table shows each of the probabilities that the counter will be blue or will
be yellow.
Colour
Probability
red white blue yellow
0.45. 0.25
There are 18 blue counters in the bag.
The probability that the counter Bob takes will be red is twice the probability that the
counter will be white.
(a) Work out the number of red counters in the bag.
The number of red counters in the bag is given as follows:
8.
How to obtain a probability?A probability is obtained by the division of the number of desired outcomes by the number of total outcomes.
The probabilities for this problem are given as follows:
Red: 2x.White: x.Blue: 0.45.Yellow: 0.25.The sum of the probabilities is of one, hence the value of x is obtained as follows:
2x + x + 0.45 + 0.25 = 1
3x = 0.3
x = 0.1.
Hence the proportion of red is given as follows:
0.2.
There are 18 blue counters in the bag, which is of 0.45 of the total amount, hence the total amount is obtained as follows:
0.45n = 18
n = 18/0.45
n = 40.
Meaning that the number of red counters is given as follows:
0.2 x 40 = 8.
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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
The two-way frequency table contains data on the preference of two school subjects among a group of 100 students. English Science Total Grade 5 24 36 60 Grade 8 16 24 40 Total 40 60 100 Based on this data, are "Grade 5" and "Science" independent events? (1 point) No, P(Grade 5 ∩ Science) ≠ P(Grade 5) ⋅ P(Science) Yes, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) No, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) Yes, P(Grade 5 ∩ Science) ≠ P(Grade 5) ⋅ P(Science)
The events Grade 5 and Science" are independent events, P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) .Option B is the correct option.
What is frequency?
The frequency (f) of a specific value is the number of occurrences of the value in the data. The frequency distribution of a variable is the collection of all possible values and the frequencies associated with these values. Frequency distributions are represented graphically as frequency tables or charts.
Given table is
English Science Total
Grade 5 24 36 60
Grade 8 16 24 40
Total 40 60 100
Total students is 100.
Total student in grade 5 is 60.
The number of students who prefer science is 60.
The number of students who are in grade 5 and prefer science is 36.
The probability of grade 5 is P(grade 5) = 60/100 =0.6
The probability of science is P(science) = 60/100 = 0.6
The probability of Grade 5 ∩ Science is P(Grade 5 ∩ Science) = 36/100 =0.36.
Thus P(Grade 5 ∩ Science) =P(Grade 5) ⋅ P(Science).
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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).
Given the coordinates below, what is the scale factor of a dilation centered at the origin where "overline"(A'B') is the image of "overline"(A'B')
A(6,-4), B(2,-8)
A'(9,-6), B'(3, -12)
A.1/2
B.3/2
C.2/3
D. 3
Considering the coordinates of the preimage of line AB represented by A (6, -4), B (2, -8) and coordinates of the image of line A'B' represented by A' (9, -6), B'(3, -12) the dilation factor is 3/2
How to find the dilation factorThe transformation rule for dilation is as follows
(x, y) for a scale factor of r → (rx, ry)
following similar procedure for the given problem we have
A (6, -4) transformed to A' (9, -6)
B (2, -8) transformed to B' (3, -12)
hence 6r = 9, and -4r = -6
r = 9/6 = -6/-4
r = 3/2
The dilation factor of the transformation is 3/2
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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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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.
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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The sales tax in the town where Gina lives is 8%. Gina wants to buy a table saw that costs $300. How much sales tax will she pay?
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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