Answer:
what is the Question
Step-by-step explanation:
math is ezy :3
Yogi's yoga studio charges members $79 for Enrollment and $45 per month Write an equation to represent the relationship between x, the number of months and y, the total cost of membership
Data:
Enrollment: $79
Charge per Month: $45/month
x: number of months
y: Total cost
You can follow the next general expression:
[tex]y=kx+b[/tex]Where k is the constant of change, in this case the charge per month, and b is the charge at time 0, in this case the charge per enrollment.
Then, You get the next expression that represents the relationship:[tex]y=45x+79[/tex]Calculate the value of the expression:1+1x100+2
In order to calculate the value of this expression, first we need to calculate the multiplication between 1 and 100. Then, w
graph the system of quadratic Inequalities. (please show how you find the points to graph)
Points you need to find to graph quadratic inequalities:
Vertex of each parabola:
1-Write each ineqaulity as an equation:
[tex]\begin{gathered} y=x^2-4x+8 \\ y=-x^2+4x+2 \end{gathered}[/tex]Vertex:
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ x-coordinate\text{ of the vertex:} \\ x=-\frac{b}{2a} \\ \\ y-coordinate\text{ of the vertex:} \\ f(-\frac{b}{2a}) \end{gathered}[/tex]First equation: the leding coefficient is 1 then the parabola opens up.
Vertex of first equation:
[tex]\begin{gathered} x=-\frac{-4}{2(1)}=\frac{4}{2}=2 \\ \\ y=2^2-4(2)+8 \\ y=4-8+8 \\ y=4 \\ \\ \text{Vertex: (2,4)} \end{gathered}[/tex]Second equation: the leading coefficient is -1 then the parabola opens down.
Vertex of the second equation:
[tex]\begin{gathered} x=-\frac{4}{2(-1)}=\frac{-4}{-2}=2 \\ \\ \\ y=-(2)^2+4(2)+2 \\ y=-4+8+2 \\ y=6 \\ \\ \text{Vertex: (2,6)} \end{gathered}[/tex]Points of interception:
Equal the equations and solve x:
[tex]\begin{gathered} x^2-4x+8=-x^2+4x+2 \\ \\ x^2+x^2-4x-4x+8-2=0 \\ 2x^2-8x+6=0 \\ \\ \text{Quadratic formula:} \\ ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ \\ x=\frac{-(-8)\pm\sqrt[]{(-8)^2-4(2)(6)}}{2(2)} \\ \\ x=\frac{8\pm\sqrt[]{64-48}}{4} \\ \\ x=\frac{8\pm\sqrt[]{16}}{4} \\ \\ x=\frac{8\pm4}{4} \\ \\ x_1=\frac{8+4}{4}=\frac{12}{4}=3 \\ \\ x_2=\frac{8-4}{4}=\frac{4}{4}=1 \end{gathered}[/tex]The parabolas intersect in x=1 and x=3 (use one of the equations to find the y-value of the intersection):
[tex]\begin{gathered} y=1^2-4(1)+8 \\ y=1-4+8 \\ y=5 \\ \\ \text{point: (1,5)} \\ \\ y=3^2-4(3)+8 \\ y=9-12+8 \\ y=5 \\ \\ \text{point: (3,5)} \end{gathered}[/tex]Then, you have the next points:
Vertex: (2,4) opens up; (2,6) opens down
Intersection points: (1,5) and (3,5)
First parabola has the inequality sing > : the border line is a dotted line and the shadow area is under the parabola.
Second parabola has the inequality sing ≤ : the border line is a full line and the shadow area is over the parabola
Graph:
11) Tell whether (3, 20) is a solution of y = 4x +8 A) Yes B) No
y = 4x + 8
To tell if (3, 20) is a solution, we will substitute the value of x = 3 and y= 20
into the equation to see if the left-hand side equals the right-hand side of the equation
So, upon substituting
20 = 4 x 3 + 8
20 = 12 + 8
20 = 20
Since the left-hand side of the equa
AcellusConvert this decimal into its fractionalform, simplified completely.0.450
we have the following:
[tex]0.450=\frac{45}{100}=\frac{9}{20}[/tex]therefore, the answer is 9/20
I need help with math
We have the following problem:
Using the kinematic equation, we know that for movements with constant acceleration the position is
[tex]h(t)=h_0+v_0t-\frac{gt^2}{2}[/tex]I already wrote the variables convenient for our problem, where
t = time (s)
h₀= initial height (m)
v₀ = initial velocity (m/s)
g = gravity acceleration (m/s²)
h = height after t seconds
We also know that
t = independent variable
h₀= 10m
v₀ = 56 m/s
g ≅ 10 m/s²
h = dependent variable
Therefore our quadratic will be
[tex]\begin{gathered} h(t)=10+56t-\frac{10t^2}{2} \\ \\ h(t)=10+56t-5t^2 \end{gathered}[/tex]Now we can answer the question in fact.
a)
The rocket will hit the ground when the height is equal to zero, hence, h(t) = 0. In fact, we are looking for the zeros of the quadratic
[tex]\begin{gathered} 10+56t-5t^2=0 \\ \\ t=\frac{-56\pm\sqrt{56^2-4\cdot(-5)\cdot(10)}}{2\cdot(-5)} \\ \\ t=\frac{-56\pm\sqrt{3136+200}}{-10} \\ \\ t=\frac{56\pm\sqrt{3336}}{10} \\ \\ t=11.38\text{ s or }-0.18\text{ s} \end{gathered}[/tex]See that we have one negative zero, but we will ignore that because it's physically impossible, therefore the rocker will reach the ground after 11.38 seconds
b)
To find the maximum height we must find the max value of the quadratic, we know that the vertex of a quadratic is its max/min, also, we can write it as
[tex]\begin{gathered} y_V=-\frac{\Delta}{4a} \\ \\ x_V=-\frac{b}{2a} \end{gathered}[/tex]Where
[tex](x_V,y_V)[/tex]Is the vertex. Here, we want to find the y coordinate, because y here is the height, therefore
[tex]\begin{gathered} \max h=-\frac{\Delta}{4a} \\ \\ \operatorname{\max}h=-\frac{3336}{4\cdot(-5)} \\ \\ \operatorname{\max}h=\frac{3336}{20} \\ \\ \operatorname{\max}h=166.8\text{ m} \end{gathered}[/tex]The max height of the rocket is 166.8m
c)
Now we have a similar problem, it's also the vertex but now the coordinate x, that here, represents the time, then
[tex]\begin{gathered} t_{\max}=\frac{-b}{2a} \\ \\ t_{\operatorname{\max}}=\frac{-56}{2\cdot(-5)} \\ \\ t_{\operatorname{\max}}=\frac{56}{10} \\ \\ t_{\operatorname{\max}}=5.6\text{ s} \end{gathered}[/tex]The rocket will reach the maximum height after 5.6 seconds.
fill in the blanks.6x20=6x2x_____5x100=______x10x10
ANSWER
1. 10
2. 5
EXPLANATION
We want to expand the expressions given:
1. 6 x 20 = 6 x 2 x ____
To do this, we have to divide 20 by 2, because 20 was expanded to give 2 x __.
20 divided by 2 is 10, so the answer is:
6 x 20 = 6 x 2 x 10
2. 5 x 100 = ___ x 10 x 10
This is straightforward, since 10 x 10 is 100.
The answer is:
5 x 100 = 5 x 10 x 10
Cecil wrote the fraction 6/4. Susie wants to write anequivalent fraction. Whichof the following could be herfraction? A. 2/3 B. 6/9 C. 8/12 D. All of the above.
ANSWER
8/12. Option C
EXPLANATION
In a simple term: Equivalent fraction can be determined by simply multiplying the numerator and the denominator by the SAME NUMBER.
That is,
if you have 2/3 the equivalent fraction will be 4/6 (when multiplied by 2) or 6/9 (when multiplied by 3) or 8/12 (when multiplied by 4) etc.
So, from the question above:
The equivalent fraction of 4/6 (when multiplied by 2) is 8/12
How can we tell when every point on the graph is a solution to the problem?
One way to verify that if a point exist on both lines is to substitute the x- and y-values of the ordered pair into the equation of each line. If the substitution results in a true statement, then you have the correct solution!
Which number has a repeating decimal form? A [tex] \sqrt{15} [/tex]B 11/ 25 C. 3/20 D. 2/6
Answer
Explanation
To know which is correct, we simply write the given numbers in decimal form
√15 = 3.8729
(11/25) = 0.44
(3/20) = 0.15
(2/6) = 0.333333333
We can easily see that
Re-arrange this vertex equation y = 2 (x + 1)2 - 6 in standard form?
When we have a quadratic equation, we can have it in vertex and standard form.
The vertex form comes in the form:
[tex]y=a\mleft(x-h\mright)^2+k[/tex]The standard form comes in the form:
[tex]y=ax^2+bx+c[/tex]Converting to/from either simply requires some manipulations via expansion of the bracket as will be seen.
[tex]\begin{gathered} y=2(x+1)^2-6 \\ y=2(x^2+2x+1)-6 \\ y=2x^2+4x+2-6 \\ y=2x^2+4x-4 \end{gathered}[/tex]Hence, we have our standard form.
Write each expression without the absolute value symbol.(x+7)
We must write the following expression without the absolute value symbol:
[tex]|x+7\left|\right?.[/tex]We have two cases:
1) If (x + 7) ≥ 0 or x ≥ -7, the expression is (x + 7).
2) If (x + 7) < 0 or x < -7, the expression is -(x + 7).
Combining these results, we have:
[tex]|x+7\left|=\right?\begin{cases}x+7\text{ if }x\ge-7 \\ -x-7\text{ if }x<-7\end{cases}[/tex]AnswerThe equivalent expression to |x + 7| without an absolute value symbol is:
[tex]|x+7\left|=\right?\begin{cases}x+7\text{ if }x\ge-7 \\ -x-7\text{ if }x<-7\end{cases}[/tex]Which graph shows point pas (-5,6)and point q as (3,-4)?
Answer
Option A is correct.
From the explanation, we can easily see that the first graph shows point P as (-5, 6) and Point Q as (3, -4).
Explanation
The key to marking points on the graph is to know that the coordinates are named as (x, y)
And to mark a point (-5, 6), it means x = -5 and y = 6
So, we move 5 units to the left from the origin along the negative x-axis and 6 units upwards along the y-axis.
And for (3, -4), x = 3, y = -4
We move 3 units to the right from the origin along the positive x-axis and 4 units downwards along the y-axis.
Hope this Helps!!!
The Harrisburg Recreation Center recently changed its hours to open 1 hour later and close 3 hours later than it had previously. Residents of Harrisburg age 16 or older were given a survey, and 560 residents replied. The survey asked each resident his or her student status (high school, college, or nonstudent) and what he or she thought about the change in hours (approve, disapprove, or no opinion). The results are summarized in the table below. Student status Approve Disapprove | No opinion 30 High school College Nonstudent 4 10 353 6 85 Total 129 367 38. What fraction of these nonstudent residents replied that they disapproved of the change in hours? F. } HAWI- G. J. 353 367 K. 353 485
Explanation
to get the fraction of Nonstudents that disaproved
[tex]\text{fraction}=\frac{total\text{ nonstudents that disaproved}}{\text{total nonstudents}}[/tex]then
let
total nonstudents that disaproved=353
total nostudents=85+353+47=485
now, replace
[tex]\text{fraction}=\frac{353}{485}\rightarrow k[/tex]so,the answer is k
what is the surface area for a m rectangular prism. with the measurements as: height = 9 length = 3width = 7
Answer:
Surface area = 222 square cm
Explanation:
Given the following data
Length = 3 cm
Height = 9 cm
Width = 7 cm
Surface area = 2(wl + hl + hw)
Surface area = 2(7 * 3 + 9 * 3 + 9 * 7)
Surface area = 2( 21 + 27 + 63)
Surface area = 2( 111)
Surface area = 222 square cm
Therefore, the surface area is 222 square cm
-a+9bA=4B= - 4 I forgot how this thing works? Please someone help!
-a+9b
a=4
b=-4
Replace a by 4 and b by -4 in the expression, then solve it
-(4)+9(-4)
-4 -36
-40
21.Find the indicated angle measure to the nearest degree.1613?А.510B. 39С C36549
using trigonometric ratio,
let
the unknown angle = x degrees
[tex]\tan x=\frac{opposite}{\text{adjacent}}[/tex][tex]\begin{gathered} \tan x=\frac{13}{16}^{} \\ x=\tan ^{-1}0.8125 \\ x=39.0938588862 \\ x\approx39^{\circ} \end{gathered}[/tex]1. Corinne has a cell phone plan that includes 200 minutes for phone calls and unlimited texting. An additional fee is charged for using more than 200 minutes for phone calls. The figure below is the graph of C = f(m), where C is the monthly cost after m minutes used. Part A What is the minimum monthly cost for Corinne's cell phone plan? Show or explain your work. Part B What is the value of f(150). Explain its meaning in terms of the cell phone plan. Part C For what mis f(m) = 55? Explain its meaning in terms of the cell phone plan. Part D What is the cost per minute after Corinne uses her monthly allowance of 200 minutes? Show or explain your work.
Part A) Minimum cost = $30
Part B) Value of f(150) = $30
Part C) m = 275 minutes
Part D) Cost per minute after 200 minutes = $0.2
Explanations:From the graph shown:
Monthly rate for 200 minutes for phone calls = $30
An additional fee is charged for more than 200 minutes for phone calls
Part A) The minimum monthly cost of Corinne's cell phone plan.
Note that the minimum monthly cost of Corinne's cell phone plan will be when he does not use more than 200 minutes for phone calls.
Therefore, the minimum monthly cost, C = f(200) = $30
Part B)
The value of f(150)
f(150) means the cost of Corinne's cell phone plan when 150 minutes is spent for phone calls, i.e. m = 150
Since there is a flat rate of $30 for 0 to 200 minutes, f(150) = $30
Part C)
For what m is f(m) = 55
This means that we should find the number of minutes spent when the cost of the plan is $55
From the graph, $55 is charged at 275 minutes
Therefore, when f(m) = 55, m = 275 minutes
Part D)
Cost per minutes after the monthly allowance of 200 minutes
After the monthly allowance of 200 minutes, we would notice that, for every 50 minutes, there is a $10 charge. That means that for every 1 minute, there will be a charge of 10/50 = $0.2
Cost per minute = $0.2
The sales tax on a table is $15.96find the purchase price The total price
Answer:
[tex]\begin{gathered} a)\text{ Purchase Price = \$190} \\ b)\text{ Total Price = \$205.96} \end{gathered}[/tex]Explanation:
Here, we want to get the purchase price and the total price
a) The purchase price before tax
In the question, we have it that the tax is 8.4% of the purchase price
Let the purchase price be $P
8.4% of this is $15.96
Mathematically:
[tex]\begin{gathered} \frac{8.4}{100}\times\text{ P = 15.96} \\ \\ 8.4P\text{ = 100}\times15.96 \\ P\text{ = }\frac{100\times15.96}{8.4} \\ P\text{ = \$190} \end{gathered}[/tex]b) The total price is the sum of the tax and the purchase price
Mathematically, we have this as:
[tex]\text{ 190 + 15.96 = \$205.96}[/tex]Find the volume of a cone with a slant height of 15 inches and a radius of 9 inches. Leave your answers in terms of π
Given:
height(h)=15 inches
radius(r)=9 inches
Volume of cone:
[tex]V=\pi\times r^2\times\frac{h}{3}[/tex][tex]V=\pi\times9^2\times\frac{15}{3}=\pi\times81\times5[/tex][tex]V=\pi\times405[/tex][tex]V=405\pi\text{ cubic inches}[/tex]Consider the following graph. List the ordered pairs corresponding to the points in the graph?
Answer:
A(-6, 1)
B(-6, -7)
C(7, -9)
D(-8, -8)
Explanation:
From the graph, we can see that at point A, x = -6 and y = 1. Therefore, the ordered pair can be written as A(-6, 1)
At point B, x = -6 and y = -7. The ordered pair can be written as B(-6, -7)
At point C, x = 7 and y = -9. Its ordered pair will be C(7, -9)
At point D, x = -8 and y = -8. Its ordered pair will be D(-8, -8)
Given:• VU is tangent to Circle Q• UV = 20• UT = 8U2000TSFind the length of the radius of the circle.6O 1202142
To solve this question and find the radius of the circle, we will use the s
The runaway success of the switch prompted the company to raise it's sales and earnings by forecast for the second time since November. It now expects a 24% jump in profit from what it projects just three months ago, with 560 billion yen (5.6 billion) estimated for the year ending in March. What is the total now?
If the new forecast is 24% more than the 5.6 billion estimated, we can calculate this as:
[tex]\begin{gathered} Y_{\text{new}}=Y_{\text{old}}+0.24\cdot Y_{\text{old}}=(1+0.24)Y_{\text{old}}=1.24\cdot Y_{\text{old}} \\ Y_{\text{new}}=1.24\cdot5.6=6.944\approx6.9 \end{gathered}[/tex]Answer: The total now is approximately 6.9 billion.
can you break it down and help me out please?
Given
[tex]x^2+x-2\ge0[/tex]To find the solution.
Now,
It is given that,
[tex]x^2+x-2\ge0[/tex]Using factorization method,
[tex]\begin{gathered} x^2+x-2=0 \\ x^2-x+2x-2=0 \\ x(x-1)+2(x-1)=0 \\ (x+2)(x-1)=0 \end{gathered}[/tex]That implies,
[tex]\begin{gathered} x+2\ge0,x-1\ge0 \\ x\ge-2,x\ge1 \end{gathered}[/tex]Hence, the solution set is,
[tex]undefined[/tex]The circumference of a big circle is 36 pi. The area of a smaller circle located inside the bigger circle is 16 pi. If you randomly pick a point inside the big circle, what is the probability the point lands in the smaller one?
Given:
a.) The circumference of a big circle is 36 pi.
b.) The area of a smaller circle located inside the bigger circle is 16 pi.
The probability that the point lands in the smaller one is,
[tex]\text{ Probability = }\frac{Area_{Small\text{ Circle}}}{Area_{Big\text{ Circle}}}[/tex]However, only the circumference of the big circle is given. To be able to get the probability, we must first determine the area of the circle.
a.) Area of the big circle.
[tex]\begin{gathered} \text{ Circumference = }2\pi r \\ 36\pi\text{ = 2}\pi r \\ \frac{36\pi}{2\pi}\text{ = r} \\ 18\text{ = r} \end{gathered}[/tex][tex]\begin{gathered} \text{ Area = }\pi r^2 \\ \end{gathered}[/tex][tex]\begin{gathered} \text{ = }\pi(18)^2 \\ \text{ = }\pi(324) \\ \text{ Area = 324}\pi \end{gathered}[/tex]b.) Let's now determine the probability.
[tex]\text{ Probability = }\frac{Area_{Small\text{ Circle}}}{Area_{Big\text{ Circle}}}[/tex][tex]\text{ = }\frac{16\pi}{324\pi}[/tex][tex]\text{ = }\frac{16}{324}\text{ = }\frac{\frac{16}{4}}{\frac{324}{4}}\text{ = }\frac{4}{81}[/tex][tex]\text{ Probability = }\frac{4}{81}[/tex]Therefore, the probability that the point lands in the smaller one is 4/81.
Which of the following gives the correct range for the graph?
A coordinate plane with a segment going from the point negative 4 comma negative 2 to 0 comma negative 1 and another segment going from the point 0 comma negative 1 to 3 comma 5.
−2 ≤ x ≤ 5
−2 ≤ y ≤ 5
−4 ≤ x ≤ 3
−4 ≤ y ≤ 3
Answer:
The correct range is -2 < y < 5.
What are the values of w and x in the triangle below? Round the answers to the nearest tenth.thank you ! :)
Answer:
w = 14.4
x = 11.2
Explanation:
We would consider the smaller and larger right angle triangles.
For the smaller right triangle, taking 48 as the reference angle,
opposite side = 16
adjacent side = w
We would find w by applying the tangent trigonometric ratio which is expressed as
tanθ = opposite side/adjacent side
Thus,
tan48 = 16/w
By cross multiplying,
wtan48 = 16
w = 16/tan48
w = 14.4
For the larger right triangle, taking 32 as the reference angle,
opposite side = 16
adjacent side = w + x = 14.4 + x
We would find w by applying the tangent trigonometric ratio which is expressed as
tanθ = opposite side/adjacent side
Thus,
tan32 = 16/(14.4 + x)
By cross multiplying,
(14.4 + x)tan32 = 16
(14.4 + x) = 16/tan32
14.4 + x = 25.6
x = 25.6 - 14.4
x = 11.2
-3|2x+1|<4 how is this solved
Answer:
True for all x
Step-by-step explanation:
Multiply by -1. This reverses the inequality.[tex]-3\left|2x+1\right| < 4\\\left(-3\left|2x+1\right|\right)\left(-1\right) > 4\left(-1\right)[/tex]
Remember: Negative * Positive = NegativeRemember: Negative * Negative = Positive[tex]3\left|2x+1\right| > -4:\:Solve\\4\times -1 = -4\\\left(-3|2x+1|\right)\left(-1\right)\\\left(-3\times -1\right)\\\left(-3|2x+1|)[/tex]
Divide by 3[tex]=\frac{3\left|2x+1\right|}{3} > \frac{-4}{3}[/tex]
___________
[tex]\frac{3\left|2x+1\right|}{3}\\\frac{3}{3}\\= \frac{1}{1}[/tex]
___________
[tex]= \frac{1}{1} > \frac{-4}{3}[/tex]
___________
[tex]-\frac{4}{3}\\= -\frac{4}{3}[/tex]
Because absolute values are greater than or equal to 0, the problem will be classified as "True" for any value of x no matter what.Hope this helps.
Question 4Task 2: Nee how (hello)Business is projected to be booming after the latest release of The Fast and the Furious3.14159265359... Carver's Auto Custom must determine how many cans of paint and rims tostock at their Shanghai location.The Carver Family did choose Warehouse Space A. The warehouse includes 8000 sq. ft. ofshowroom and workshop space. One half of this warehouse space will be used to stock paintcans and rims. The warehouse has a height of 20 ft.Tell how many of cans you will stock. You must have exactly 4 cans ofpaints for every rim you stock.
The area of the warehouse is
[tex]A=8000ft^2[/tex]Half of this area stock paint, cans and rims:
[tex]\begin{gathered} A_{\text{stock}}=4000ft^2 \\ \text{then, the volume of the room is} \\ V_{\text{stock}}=4000\times20 \\ V_{\text{stock}}=80000ft^3 \end{gathered}[/tex]thats because the heigth of the stock room is equal to 20 ft.
On the other hand, we know that there are 2 cans in a box which volume
[tex]\begin{gathered} V_{\text{box}}=15\times7\times6inches^3 \\ \text{then for one can, the volume is} \\ V_{\text{can}}=\frac{V_{box}}{2}=\frac{15\times7\times6}{2}=15\times7\times3inches^3 \\ V_{\text{can}}=315in^3 \end{gathered}[/tex]and a rim is inside a box with measures
[tex]\begin{gathered} V_{\text{rim box}}=36\times36\times15inches^3 \\ V_{\text{rim box}}=19440in^3 \end{gathered}[/tex]Then, we need to find the ratio V_total to V_stock in order to find the number of rims in the room.
Then, V_total is the sum of 4 times the volume of one can plus the volume of 1 rim, that is,
[tex]V_{\text{total}}=4\cdot V_{\text{can}}+V_{\text{rim}}[/tex]because we need 4 cans and 1 rim in our room. This total volume is given by
[tex]V_{\text{total}}=4\cdot315+19440inches^3[/tex]which gives
[tex]V_{\text{total}}=20700inches^3[/tex]The last step is convert the V_total from cubic inches to cubic feets. We can do that by means of
[tex]V_{\text{total}}=20700inches^3(\frac{1ft^3}{12^3inches^3})[/tex]because 1 feet is equal to 12 inches. It yields,
[tex]\begin{gathered} V_{\text{total}}=20700(\frac{1}{144}) \\ V_{\text{total}}=143.75ft^3 \end{gathered}[/tex]Finally, we can find the ratio mentioned above:
[tex]\text{ratio}=\frac{V_{stock}}{V_{total}}=\frac{80000}{143.75}=556.52[/tex]By rounding down to the nearest interger, the ratio is 556. This means that we can stock 556 rims in the warehouse.
Cuánto es 71/4 menos un entero 3/4
7 1/4 - 1 3/4
[tex]7\frac{1}{4}-1\frac{3}{4}=\frac{28+1}{4}-\frac{4+3}{4}=\frac{29}{4}-\frac{7}{4}=\frac{22}{4}=5\frac{2}{4}=5\frac{1}{2}[/tex]Respuesta:
5 1/2
[tex]5\frac{1}{2}[/tex]