The differential equation is [tex]y=\frac{-7t^2+22t-7}{7t-22}[/tex]
We are given the Initial value problem:
y'=(t=y)²-1, y(3)=4
Substitute the value z=t+y
When t=3 and y=4 then z=3+4=7
y'=z²+1
Differentiate z w.r.t t
[tex]\frac{dz}{dt} =1+y'[/tex]
Then, we get [tex]z'=1+z'-1=z^2[/tex]
z⁻²dz=dt
Integrate on both sides:
-1/zdz=t+c
z=-1/t=c
Substitute t=3 and z=7
Then, we get
7=-1/3+c
21+7c=-1
7c=-1-21=-22
c=-22/7
Substitute the value of C then we get:
z=-1/t-22/7
z=-7/7t-22
y=z-t
y=-7/7t-22-t
y=-7-7t²+22t/7t-22
y=-7t²+22t-7/7t-22.
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g(x)
10
8
6
4
2
fo
-10-8-64-22
-4
-6
-8
-10
f(x)
2 4 6
8 10 x
What is the equation of the translated function, g(x), i
f(x) = x²?
g(x) = (x-4)² +6
Og(x) = (x + 6)²-4
g(x) = (x-6)²-4
Og(x) = (x+4)² + 6
O
Therefore , the solution of the given problem of function comes out to be g(x) is given by g(x) = (x-4)² + 6
What does the term function refer to?The study of numbers, their variations, mathematics and the regional tissue, structures, and both their actual and imagined locations, is known as mathematics. A function is a mathematical representation of the relationship between a set of inputs, each of which has an associated output. A function is a group of inputs that, when combined, result in a single, recognized output for each input. Each function is given a city, town, or scope, which is frequently referred to as a realm.
Here,
Given the graph of the original function f(x), whose vertex is at (0, 0), and indeed the graph of the translating function g(x), whose vertex is at (0, 0), (5, 2).
Be aware that the translated function's vertex is 2 units higher and 5 units to the right of the f graph's graph (x).
The function f(x) is then translated 5 units toward the right и 2 units up to arrive at g(x).
Recall that provided a function, f(x), is characterized as a parabola with a vertex at (h, k).
=> f(x) = (x-h)² + k
The outcome of translating a unit to the right with b unit up is
=>f'(x) = (x-h-a)² + k + b
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Two angles are complementary. The larger angle is 50∘ more than the smaller angle. Find the measure of the larger angle.
The measure of the smaller complementary angles is 40°
What are complementary anglesWhen two angles add up to 90°, we say they complement each other in terms of mathematics. Hence in geometry, complementary angles are defined as two angles whose sum is 90 degrees.
The larger angle = 50°
we shall represent the smaller angle with the letter x so that;
x + 50° = 90°
subtract 50° from both sides of the equation
x + 50° - 50° = 90° - 50°
x = 40°
Therefore, the smaller complementary angle has a measure of 40°.
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I NEED HELP ASAP WITH THIS MATH HOMEWORK PLEASE
Answer: 1. 7a 2. 5r 3. 8y+4 4. 12m-7 5. 7u+5 6. 8q+6 7. 3w+5x 8. 4j + 4k 9. 5d+6
Step-by-step explanation: Add the coefficients of the terms with same variables. ex. 5a+4a+7... 5a and 4a have same variable which is a so add the 5 and 4 and you get 9a and the 7 doesn't match so leave it out and you get 9a + 7
Answer:
1) 7a
2) 5r
3) 8y +4
4) 12m-7
5) 7u+5
6) 8q+6
7) 3w+5x
8) 4j +4k
9) 5d+6
Step-by-step explanation:
you just add or subtract the number with the same letter after it (if there is no letter just leave it as is, unless there are two or more numbers with no letter after, then add or subtract as normal)
Evaluate the following integral: ∫20(45t3−34t2+25t)dt
The integral[tex]∫20(45t^3-34t^2+25t)dt[/tex] can be evaluated by using the power rule of integration, which states that the derivative of x^n is [tex]nx^(n-1)[/tex]. This rule can be applied to each term of the integral separately.
The first step is to integrate [tex]∫20(45t^3-34t^2+25t)dt[/tex] with respect to t. By using the power rule, we can find the antiderivative of each term inside the parentheses:
[tex]∫45t^3 dt = (45/4)t^4 + C[/tex] (the antiderivative of 45t^3 is (45/4)t∧4)
[tex]∫34t^2 dt = (34/3)t^3 + C[/tex] (the antiderivative of 34t^2 is (34/3)t∧3)
[tex]∫25t dt = (25/2)t^2 + C[/tex] (the antiderivative of 25t is (25/2)t∧2)
Now, we can substitute these results back into the original integral:
[tex]20(∫45t^3 dt - ∫34t^2 dt + ∫25t dt) = 20((45/4)t^4 + (34/3)t^3 + (25/2)t^2) + C= 20(15t^4-11t^3+25t^2) + C[/tex]
So, the evaluated integral for [tex]∫20(45t^3-34t^2+25t)dt[/tex] is [tex]20(15t^4-11t^3+25t^2) + C.[/tex]
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Which answers describe the shape below? Check all that apply.
0
A. Rectangle
B. Rhombus
C. Parallelogram
D. Trapezoid
E. Quadrilateral
F. Square
Answer:
Rectangle, parallelogram, trapezoid, quadrilateral
Step-by-step explanation:
because the properties of the rhomboid(that is what it is called) match with those different shapes listed above
Graph the function f(x)=√x+3.
What are the minimum and maximum values on the interval [-11, 5)?
Write your answers in the boxes.
Minimum:
Maximum:
The values of maximum and minimum values are 11i + 3 and √5+ 3 .
What is Function?Function can be defined in which it relates an input to the output. Functions are relationships in mathematics where each input has a particular output. Functions are often used in science and mathematics. In this section, we will learn the definitions of functions, various kinds of functions, and properties, along with examples to help us understand.
Given,
The function is f(x) = √x+3.
We have to find the maximum and minimum values on the interval [-11,5]
So,
f'(x) = 1/2√x
As when,
f'(x) = 0
has no solution than,
f(-11) = √-11 + 3
= 11i + 3
f(5) = √5+ 3 .
Therefore, The values of maximum and minimum values are 11i + 3 and √5+ 3 .
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Catarina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet above the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the cliff with the water at the back of her boat. Her eye level is 5½ feet above the ground. Approximately how far out from the cliff is Catarina’s boat? Round to the nearest foot.
If Catarina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet above the water in a lake. Approximately the distance from the cliff is Catarina’s boat is : 63.6ft.
What is distance?Distance can be defined as how far an object is from another object.
Now let find the distance:
AE/CD = BC/DE
Hence,
5.5/3.5 = 10/DE
Cross multiply
AE = 350/5.5
AE = 63.6ft
Therefore we can conclude that the distance is 63.6ft.
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The first and second terms of an exponential sequence (G.P.) are the first and third terms of a linear sequence (A.P.). The fourth term of the linear sequence is 10 and the sum of its first five terms is 60. Find the first four terms of the linear sequence.
Answer:
Step-by-step explanation:
A typical geometric sequence can be represented as
c
0
a
,
c
0
a
2
,
⋯
,
c
0
a
k
and a typical arithmetic sequence as
c
0
a
,
c
0
a
+
Δ
,
c
0
a
+
2
Δ
,
⋯
,
c
0
a
+
k
Δ
Calling
c
0
a
as the first element for the geometric sequence we have
⎧
⎪
⎨
⎪
⎩
c
0
a
2
=
c
0
a
+
2
Δ
→
First and second of GS are the first and third of a LS
c
0
a
+
3
Δ
=
10
→
The fourth term of the linear sequence is 10
5
c
0
a
+
10
Δ
=
60
→
The sum of its first five term is 60
Solving for
c
0
,
a
,
Δ
we obtain
c
0
=
64
3
,
a
=
3
4
,
Δ
=
−
2
and the first five elements for the arithmetic sequence are
{
16
,
14
,
12
,
10A typical geometric sequence can be represented as
c
0
a
,
c
0
a
2
,
⋯
,
c
0
a
k
and a typical arithmetic sequence as
c
0
a
,
c
0
a
+
Δ
,
c
0
a
+
2
Δ
,
⋯
,
c
0
a
+
k
Δ
Calling
c
0
a
as the first element for the geometric sequence we have
⎧
⎪
⎨
⎪
⎩
c
0
a
2
=
c
0
a
+
2
Δ
→
First and second of GS are the first and third of a LS
c
0
a
+
3
Δ
=
10
→
The fourth term of the linear sequence is 10
5
c
0
a
+
10
Δ
=
60
→
The sum of its first five term is 60
Solving for
c
0
,
a
,
Δ
we obtain
c
0
=
64
3
,
a
=
3
4
,
Δ
=
−
2
and the first five elements for the arithmetic sequence are
{
16
,
14
,
12
,
10
George is 48 inches tall and growing at a rate of 2 inches a year his sister is 42 inches tall and growing at a rate of 3 inches in a year if their growth rate continues how long will it be before Georgia sister is taller than he is
Answer: It will take 7 more years for George’s sister to be taller than him.
Step-by-step explanation:
Answer:
Step-by-step explanation:
7 years
7 = 3(5 + x) solve for x
Answer: x = -8/3
Step-by-step explanation:
7 = 3(5+x)
3(5+x)=7
apply the distributive property
15+3x=7
rearrange the variables to the left side of the equation
3x=7-15
3x=-8
divide on both sides
x=-8/3
Answer:
x = -8/3
Step-by-step explanation:
Now we have to,
→ Find the required value of x.
The equation is,
→ 7 = 3(5 + x)
Then the value of x will be,
→ 7 = 3(5 + x)
→ 7 = 3(5) + 3(x)
→ 7 = 15 + 3x
→ 7 - 15 = 3x
→ -8 = 3x
→ 3x = -8
→ [ x = -8/3 ]
Hence, the value of x is -8/3.
Find the two missing numbers in the following sequence: 1, 8, ?, 16, 17, 24, 25, 32, 33, ?
Answer:
7 and 40
Step-by-step explanation:
for you to get the next number you have to add 7 to that number...then add 1 to the number you have gotten then continue like that
eg: to get 8 you had to add 7 to 1 now to get the next number add 1 to 8 to get 9 then add 7 to 9 to get 16 then continue like that
The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014, the number of men receiving degrees (in thousands) is given by the equation y=3.6x+442, and for women, the equation is y=14.3x+318, where x is the number of years after 1970
PLEASE HELP IS AN EMERGENCY
The solution to the system of equations for this problem is given as follows:
(11.59, 479).
How to solve the system of equations?The system of equations for this problem is defined as follows:
y = 3.6x + 442.y = 14.3x + 318.The solution for x is obtained when the two equations are equal, hence:
14.3x + 318 = 3.6x + 442
10.7x = 124
x = 124/10.7
x = 11.59.
Applying the substitution in any of the equations, the solution for y is obtained as follows:
y = 3.2(11.59) + 442
y = 479.
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Seven less than the product of 9 and a number equals 8. Use the variable b for the unknown number
The equation formed by the sentence is 9b - 7 = 8.
What is equation ?
The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.
Let the variable be b
By the given sentence we get the equation
9b - 7 = 8
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Which compound inequality is represented by the graph?
Answer:
the third one
Step-by-step explanation:
an open circle means it does not include that number. therefore, it needs to be x is less than -2. a closed circle means it includes that number. therefore, x is greater than or equal to 4
What are the indices for the directions indicated by the two vectors in the following sketch?
The indices for the directions are indicated by the two vectors in the following sketch - [1 0 0] and [ 2 0 3].
Miller Indices of Direction Vector:
The Miller indices of direction are rationalized components of the direction vectors taken in all crystallographic directions. They are indicated by [[u v w ]]. They are always defined as the smallest integers, so if they occur in fractional or decimal form, we need to convert them into the smallest integral value by algebraic manipulation.
The indices of direction 1.
Let the length of direction 1 along the x-axis be X units. The length of the vectors along the y and z axes is 0.
[ X 0 0 ]
⇒ [tex]\left[\begin{array}{ccc}0&0&0\\X'&X'&X'\\\end{array}\right][/tex]
Divide each length by X to convert it into the smallest integer.
⇒ [1 0 0]
And, this is the Miller indices of the direction vector 1.
So, the Miller indices of the direction vector 1 are [ 1 0 0].
Direction 2
The indices of direction 2.
Let the length of direction 2 along the x-axis be X units. The length of the vector along the y and z axes are 0.
[ 0.2 0 0.3 ]
⇒[ 2 0 3 ]
Multiply each fractional length by 10 to convert it into the smallest integer.
So, the Miller indices of the direction vector 2 are [ 2 0 3].
Therefore, The indices for the directions are indicated by the two vectors in the following sketch - [1 0 0] and [ 2 0 3].
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Please help me on this asap
The new points of the trapezoid are P'(4, -8), Q'(8, -8), R'(8, 4) and S'(-8, 4)
What is an equation?An equation is an expression relating two or more numbers and variables.
Dilation is the increase or decrease in size of a figure by a scale factor. The rule for dilation is:
(x, y) ⇒ (kx, ky)
The vertices of the trapezoid is P(1, -2), Q(2, -2), R(2, 1) and S(-2, 1)
If the trapezoid is dilated by a scale factor of 4 about the origin, the new points are:
P'(4, -8), Q'(8, -8), R'(8, 4) and S'(-8, 4)
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The undergraduate
grade point
averages (UGPA) of students taking
an admissions test in a recent year
can be approximated by a normal
distribution, as shown in the figure.
(a) What is the minimum UGPA that
would still place a student in the top
10% of UGPAS?
(b) Between what two values does the
middle 50% of the UGPAS lie?
Mean=3.24
Std.=0.17
a) The Corresponding minimum UGPA = 3.70
b) The range of middle 50% values will be: (3.28, 3.52).
What is the z-score?
A z-score (also known as a standard score) is a measure of how many standard deviations an individual data point is from the mean of a distribution. A z-score is calculated by subtracting the population mean from an individual data point, and then dividing that number by the population standard deviation.
a) z score corresponding to top 5% area = 1.645
Hence,
Corresponding minimum UGPA = 3.40 + 1.645*0.18 = 3.70
b) z score corresponding to middle 50% area = -0.67, 0.67
Hence,
The range of middle 50% values will be:
(3.40 - 0.67*0.18, 3.40 + 0.67*0.18)
(3.28, 3.52)
Hence,
a) The Corresponding minimum UGPA = 3.70
b) The range of middle 50% values will be: (3.28, 3.52).
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please help me with math i’ll give you brainlist
Answer:
it's a function
Step-by-step explanation:
to test if it's a function or not you do the vertical line test if it intersects the function at more than one point then it's not a function,, but here in the given function when you do the vertical test it'll only intersect it at one point so it's a function
Which other expression has the same value as(-14)-8 ? Explain your
reasoning.
a. (-14)+8
b. (-14)- (-8)
c. (-14)+(-8)
d. (-14)+ (-8)
Pls help needed tyyy who lives farther? Easy
Sawyer lives 12.7 km farther from the soccer field.
How to find who lives farther from the soccer fields?Jabar lives 5 miles from the soccer fields, Sawyer lives 12.7 km from the same soccer fields .
The person that lives farther form the soccer field can be calculated as follows;
Let's convert Jabar distance from the soccer field to km.
1 miles = 1.60934
5 miles = ?
cross multiply
distance of Jabar from the soccer field = 5 × 1.60934
distance of Jabar from the soccer field = 8.0467 km
Therefore, jabar lives 8.0467 km from the soccer field.
Hence, Sawyer lives 12.7 km farther from the soccer field.
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2b.) Determine the average rate of change between the
sixth and fourteenth hours, explain what it means in
context of the problem.
The average rate of change from the sixth to the fourteenth hours indicates that, on average, 15 fewer pairs of shoes were sold every hour.
what is slope ?A line's slope determines how steep it is. A mathematical equation for the gradient is called "gradient overflow" (the change in y divided by the change in x). The slope is the ratio of the vertical change (rise) between two points to the horizontal change (run) between those same two spots. When a straight line's equation is written as y = mx + b, the slope-intercept form of an equation is employed to express it. The y-intercept is located at a location where the line's slope is m, b is b, and (0, b). For instance, the slope of the equation y = 3x - 7 and the y-intercept (0, 7).
given
slope = y2- y1 / x2 - x1
= -120/8 = -15
( 6 , 120 )
( 14 , 0 )
The average rate of change from the sixth to the fourteenth hours indicates that, on average, 15 fewer pairs of shoes were sold every hour.
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Marcella has a red, square tablecloth with a diagonal of ^6 feet. She also has a white, square tablecloth with a diagonal of ^12 feet. Jordan says that the length of the diagonal of the white tablecloth is twice the length of the diagonal of the red tablecloth. Is he correct.
Answer:
Out of all possible choices you've shown, I'd say the best answer would be C
Step-by-step explanation:
Which of the following are irrational numbers?
Choose the two correct answers.
Responses
100−−−√ 100 ,
28−−√ 28 ,
11−−√ 11 ,
2.78¯¯¯¯
The value of an irrational numbers are,
⇒ √28
⇒ √11
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The numbers are,
⇒ √100
⇒ √28
⇒ √11
⇒ 2.78⁻⁻⁻
Now, We know that;
An irrational number is a number that cannot be expressed as a fraction for any integers.
Hence, We get;
The correct value of an irrational numbers are,
⇒ √28
⇒ √11
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12
R
T
37
35
Enter the ratio equivalent to tan(R).
>S
Answer:
[tex]\frac{35}{12}[/tex]
Step-by-step Explanation:
The tangent of an acute angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
Selkirk Company obtained a $15,000 note receivable from a customer on January 1, 2021. The note, along with interest at 9%, is due on July 1, 2021. On February 28, 2021, Selkirk discounted the note at Unionville Bank. The bank’s discount rate is 12%.
Required:
Prepare the journal entries required on February 28, 2021, to accrue interest and to record the discounting for Selkirk. Assume that the discounting is accounted for as a sale. (Do not round intermediate calculations. If no entry is required for a transaction/event, select "No journal entry required" in the first account field.)
The journal entries required on February 28, 2021, for Selkirk Company to accrue interest and to record the discounting are as follows:
Journal Entries:Debit Interest Receivable $225
Credit Interest Revenue $225
Debit Cash $15,048
Debit Interest Expense $177
Credit Note Receivable $15,000
Credit Interest Receivable $225
What are the steps in recording a discounted note receivable?When a note receivable is discounted, the following steps are followed:
Compute the maturity valueCompute the discount (discount rate times maturity value)Compute the proceeds (maturity value less discount)Compute the net interest income or expense (proceeds less carrying value)Prepare the journal entries.Transaction Analysis:February 28, 2021
Interest Receivable $225 Interest Revenue $225
Cash $15,048 Interest Expense $177 Note Receivable $15,000 Interest Receivable $225
Note receivable = $15,000
Interest = 9%
Maturity period = 6 months (Jan 1, 2021, to July 1, 2021) or 180 days
Discount date = February 28, 2021
Maturity value = $15,675 ($15,000 + $15,000 x 9% x 6/12)
Discount = $627 ($15,675 x 12% x 120/360)
Proceeds = $15,048 ($15,675 - $627)
Accrued interest = $225 ($15,000 x 9% x 60/360)
Note's carrying value on February 28, 2021, = $15,225 ($15,000 + $225)
Interest expense = $177 ($15,048 - $15,225)
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What shape results from a plane passing through a cone that is perpendicular to the base?
Isosceles Triangle
Circle
Parabola
Ellipse
The shape that results from a plane passing through a cone that is perpendicular to the base is a B. Circle
How is the Circle Formed?A circle intersects a cone when a plane passes through it perpendicular to the base. This is due to the plane cutting through the cone's circular cross-section at a 90-degree angle to the base of the cone.
As a result, if the plane were to be slanted or angled, the intersection would have the shape of an ellipse, which is a circle that has been flattened.
This is due to the fact that an ellipse is the shape created when a plane crosses a cone at an angle other than perpendicular to the base.
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7. Choose all the expressions that are equal
to 1.25 10.
0 12.5-
12.5 ÷ 10²
0.125 100
O 1,250 104
0 12.5 ÷ 1
125 - 1,000
Upon solving the decimals, The expression which is equivalent to 0.125 are -12.5 ÷ 10²⇒ 0.125 and 1,250 ÷ 10⁴ ⇒ 0.125.
A decimal is a number that is divided into two parts: a whole and a fraction. Between integers, decimal numbers are used to express the numerical value of complete and partially whole quantities.
Decimal fractions and recurring (repeating or non-terminating, like) decimals are the three different sorts of decimals (a.k.a. converting a decimal into a fraction whose denominator is a power of ten).
1.25 ÷ 10 ⇒ 0.125
Upon solving the decimals,
12.5 ÷ 10² ⇒ 0.125
0.125 ÷ 100 ⇒ 0.00125
1,250 ÷ 10⁴ ⇒ 0.125
12.5 ÷ 1 ⇒ 12.5
125 ÷ 1,000 ⇒ 0.0125
The expression which is equivalent to 0.125 are -
12.5 ÷ 10² ⇒ 0.125
1,250 ÷ 10⁴ ⇒ 0.125
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The dimensions of a rectangle are shown. Write the area of the rectangle as a sum of cubes. Question content area bottom Part 1 The area of the rectangle as a sum of cubes is enter your response here. x^2-3x+9 x+3
The area of the rectangle represented as the sum of cubes is x³ + 3³
What is an equation?An equation is an expression showing the relationship between numbers and variables.
The sum of cubes of a polynomial is in the form a³ + b³ while A the difference of cubes of a polynomial is a³ - b³
The area of rectangle = length * width
From the diagram, length = x² - 3x + 9; width = x + 3, hence:
The area of rectangle = length * width = (x² - 3x + 9)(x + 3)
Area = x³ - 3x² + 9x + 3x² - 9x + 27 = x³ + 27
Area = x³ + 3³
The area of the rectangle is x³ + 3³
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What are the measurement’s of questions 1,2 and 3
Angles in parallel lines are formed when two parallel lines intersect with another line known as a transversal. So here ∠y is 40° and ∠z is 140°.
What are the angles over parallel lines?Parallel lines form a pair of angles when cut by a transversal. As a result, corresponding angles are equal, as are alternate interior angles, alternate exterior angles, vertical angles, and the sum of interior angles on the same side of the transversal.We know that sum of angles over a straight line is 180°
So here ∠x + 140° = 180°
So ∠x = 180 - 140
Therefore ∠x = 40°
So in this figure ∠x = ∠y , Since they are corresponding angles are equal.
Therefore ∠y = 40°
So ∠z will be 140° . Since ∠y = 40° its opposite angle will also be 40°.
So ∠z + 40° = 180°
Therefore ∠z = 140°
So the angles ∠y = 40° and ∠z = 140°.
To learn more about angles in parallel lines refer to :
https://brainly.com/question/28340326
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HELP ME PLEASE!!
i need to find the j and k values and if you could explain that would be amazing!! Please and thank youu!! :))
Answer:
[tex]j = \dfrac{9}{2} = 4,5 \\\\\\k = 2[/tex]
Step-by-step explanation:
I am working on the assumption that the figure shown is a parallelogram, otherwise my answer will not be valid
The diagonals of a parallelogram bisect each other
In the above figure, the top left-bottom right diagonal is bisected by the top right - bottom-left diagonal
This means
[tex]3j = 5j - 9[/tex]
Subtract 5j from both sides:
[tex]3j - 5j = -9[/tex]
[tex]-2j = -9[/tex]
[tex]j = \dfrac{-9}{-2} = \dfrac{9}{2} = 4,5 \\[/tex]
I am not sure if you want it in fraction or decimal
Similarly
[tex]k + 10 = 6k\\\\[/tex]
Subtract k from both sides:
[tex]10 = 6k-k\\\\\\10 = 5k\\\\\\\textrm{Switching sides,}\\5k = 10\\\\k = \dfrac{10}{5} = 2\\\\[/tex]