Answer: The information in that "9" is the atomic number of the atom.
Step-by-step explanation:
For an atom of fluorine we have:
A = atomic weight = 18.99 amu.
Z = atomic number = 9.
(A - Z) = number of netrons = 10.
Then the "9" in the information of the atom of fluorine gives information about the atomic number.
Answer:
the atomic number
Step-by-step explanation:
Solve for x.
Enter the solutions from least to greatest.
3x^2 +4= 436
lesser x =
greater x =
Answer:
lesser x = -12
greater x = 12
Step-by-step explanation:
3x^2 +4= 436
3x^2 = 432
x^2 = 144
lesser x = -12
greater x = 12
108/40 in its simplest form
Answer:
27/10
Step-by-step explanation:
A calculator can tell you the fraction value is 2.7. As the ratio of integers, this is ...
108/40 = 27/10
Definitely has 152 dollars in the bank, she withdraws 20, then she deposit 80 writeaddition expression to represent this situation, then find the symbol and explain its meaning
━━━━━━━☆☆━━━━━━━
▹ Answer
152 - 20 + 80
▹ Step-by-Step Explanation
'Withdrawing' means taking out money which would be represented with a minus sign.
'Deposit' means you are adding money which would be represented with a plus sign.
Therefore, the answer is:
152 - 20 + 80
Hope this helps!
CloutAnswers ❁
━━━━━━━☆☆━━━━━━━
A coffee house blended 18 pounds of espresso flavored coffee beans with 17 pounds of vanilla flavored coffee beans. The 35 pound mixture cost $306.50. A second mixture included 19 pounds of espresso flavored coffee beans and 15 pounds of vanilla flavored coffee beans. The 34 pound mixture cost $298.50. Find the cost per pound of the espresso and vanilla flavored coffee beans.
Answer:
the price of the vanilla flavored coffee per pound is $8.50
the price of the espresso flavored coffee per pound is $9.00
Step-by-step explanation:
Let's give letters to the unknowns, so we can generate equations easily:
cost of espresso coffee beans per pound: "E"
cost of vanilla flavored coffee beans per pound : "V"
Now, the first statement:
18 pounds of E plus 17 pounds of V cost $306.50, can be written as:
18 E + 17 V = 306.5
The second statement:
19 pounds of E plus 15 pounds of V cost $298.50, can be written as:
19 E + 15 V = 298.5
Now, in order to solve this system of linear equations we can use substitution for example:
E = (306.5 -17 V)/18
and use this expression to substitute for E in the second equation:
19 (306.5 - 17 V)/ 18 + 15 V = 298.5
multiplying by 18 on both sides to eliminate denominators, we get:
19 (306.5 - 17 V) + 270 V = 5373
5823.5 - 323 V +270 V = 5373
5823.5 - 53 V = 5373
5823.5 - 5373 = 53 V
450.5 = 53 V
V = 8.5
Therefore the price of the vanilla flavored coffee per pound is $8.50
Now we use this found value in the substitution equation:
E = (306.5 -17 V)/18
E = (306.5 - 17 (8.5))/18
E = 9
Therefore the price of the espresso flavored coffee per pound is $9.00
What’s 12k-6z=36? (please explain)
Answer:
k=3 z=-6
Step-by-step explanation:
12k=36
k=36/12
k=3
-6z=36
z=36/-6
z=-6
12(3)-6(-6)=36
Explain why the sum or product of rational numbers is rational; why the sum of a rational number and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational. You can do this by providing examples of each
Answer:
Rational numbers are fractional numbers, whose numerator and denominator are integers and the denominator is ever zero.
Step-by-step explanation:
The sum of rational numbers gives a rational number;
[tex]\frac{1}{3}[/tex] + [tex]\frac{1}{3}[/tex] = [tex]\frac{2}{3}[/tex] , because the evaluation of the denominator always results to a non-zero integer.
The product of [tex]\frac{1}{3}[/tex] x [tex]\frac{1}{3}[/tex] = [tex]\frac{1}{9}[/tex], which multiply both numerator and denominator to give integer numbers.
The sum and product of rational and irrational numbers are always irrational numbers, for instance,
[tex]\frac{1}{3}[/tex] x 7 = 2.3 , which is a number which decimal points that can only be represented by the product irrational number and rational number , where 7 is an irrational number.
[tex]\frac{1}{3}[/tex] + 7 = 7[tex]\frac{1}{3}[/tex] , which is a whole number and fractional number combined.
simplify 1/3 - 1/2 + 2/5
Answer:
[tex] \frac{1}{3} - \frac{1}{2} + \frac{2}{5} [/tex]
[tex] = \frac{10 - 15 + 2 \times 6}{30} [/tex]
[tex] = \frac{10 - 15 + 12}{30} [/tex]
[tex] = \frac{ - 5 + 12}{30} [/tex]
[tex] = \frac{7}{30} [/tex]
Answer:
[tex] \frac{1}{3} - \frac{1}{2} + \frac{2}{5} [/tex]
[tex] = \frac{10 - 15 + 2×6}{30} [/tex]
[tex] = \frac{10 - 15 + 12}{30} [/tex]
[tex] \frac{-5 +12}{30} [/tex]
= 7 /30
root 3 + root 2 /root 3- root 2 = a+b root 6
Answer:
5 + 2[tex]\sqrt{6}[/tex]
Step-by-step explanation:
Given
[tex]\frac{\sqrt{3}+\sqrt{2} }{\sqrt{3}-\sqrt{2} }[/tex]
Multiply numerator/ denominator by the conjugate of the denominator
The conjugate of [tex]\sqrt{3}[/tex] - [tex]\sqrt{2}[/tex] is [tex]\sqrt{3}[/tex] + [tex]\sqrt{2}[/tex]
= [tex]\frac{(\sqrt{3}+\sqrt{2})(\sqrt{3}+\sqrt{2}) }{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) }[/tex] ← expand numerator/ denominator using FOIL
= [tex]\frac{3+\sqrt{6} +\sqrt{6}+2 }{3-2}[/tex]
= [tex]\frac{5+2\sqrt{6} }{1}[/tex]
= 5 + 2[tex]\sqrt{6}[/tex] → with a = 5 and b = 2
what property is this 3[5(4)] + 3 = [3(5)]4 + 3
Answer:
Associative Property of Mulitplication
Step-by-step explanation:
The associative property of multiplication states that if all the operations are multiplication, then one can group and add them in whatever order they prefer.
Hope this helps! Tell me if I'm wrong!
A ball is dropped from a building at the same time a balloon rises from the ground. The heights, in feet, of the ball and balloon above the ground after x seconds are modeled by the functions below. Ball: f(x)=24−16x2Balloon: g(x)=4x After how many seconds are the ball and the balloon at the same height? Use a graphing calculator and round to the nearest hundredth. A. 1.36 B. 4.42 C. 1.11 D. 5.42
Answer:
C. 1.11
Step-by-step explanation:
Ball: f(x) = 24 − 16x^2
Balloon: g(x) = 4x
4x = 24 - 16x^2
16x^2 + 4x - 24 = 0
4x^2 + x - 6 = 0
[tex] x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} [/tex]
[tex] x = \dfrac{-1 \pm \sqrt{1^2 - 4(4)(-6)}}{2(4)} [/tex]
[tex] x = \dfrac{-1 \pm \sqrt{1 + 96}}{8} [/tex]
[tex] x = \dfrac{-1 \pm \sqrt{97}}{8} [/tex]
[tex] x = \dfrac{-1 + \sqrt{97}}{8} [/tex] or [tex] x = \dfrac{-1 - \sqrt{97}}{8} [/tex]
We discard the negative solution.
[tex] x = 1.11 [/tex]
Answer: C. 1.11
PLEASE HELP ME WITH THE QUESTION BELOW
Answer:
C: (-5, 1)
Step-by-step explanation:
If you put the equation in a graphing calculator and examine all the points, C is the only one not on the line.
help me and ill give you brainliest answer ;))) deal? (/ω＼*)……… (/ω•＼*)
Step-by-step explanation:
....
.........
.
An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What should be the length of the side x of the square cut out of each corner to get a box with the maximum volume
Answer:
x = 7.45 inch for the maximum volume.
The area of the square = x² = 7.45² = 55.5025 inch²
Step-by-step explanation:
From the given information:
An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side.
The objective is to determine the length of the side x of the square cut out of each corner to get a box with the maximum volume
The volume of the box = l×b×h
The volume of the box = [tex](42 - 2x) \times (48-2x) \times (x)[/tex]
The volume of the box = [tex](2016 - 84x - 96x +4x^2)x[/tex]
The volume of the box = [tex](2016 -180x+4x^2)x[/tex]
The volume of the box = [tex](2016x -180x^2+4x^3)[/tex]
The volume of the box = [tex]4x^3 - 180x^2 +2016x[/tex]
For the maximum volume V' = 0
V' = [tex]12x^2 - 360x + 2016[/tex]
Using the quadratic formula; we have:
[tex]= \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}[/tex]
where;
a = 12 , b = -360 c = 2016
[tex]= \dfrac{-(-360) \pm \sqrt{(-360)^2 -4(12)(2016)}}{2(12)}[/tex]
[tex]= \dfrac{360 \pm \sqrt{129600 -96768}}{24}[/tex]
[tex]= \dfrac{360 \pm \sqrt{32832}}{24}[/tex]
[tex]= \dfrac{360 \pm 181.196}{24}[/tex]
[tex]= \dfrac{360 + 181.196}{24} \ \ \ OR \ \ \ \dfrac{360 - 181.196}{24}[/tex]
[tex]= \dfrac{541.196}{24} \ \ \ OR \ \ \ \dfrac{178.804}{24}[/tex]
[tex]= 22.55 \ \ \ OR \ \ \ 7.45[/tex]
For the maximum value , we check the points in the second derivative term
V'' = 24x - 360
V'' ( 22.55) = 24(22.55) - 360
V'' ( 22.55) = 541.2 - 360
V'' ( 22.55) = 181.2 (minimum)
V'' ( 7.45) = 24(7.45) - 360
V'' ( 7.45) = 178.8 - 360
V'' ( 7.45) = -181.2 < 0 (maximum)
Therefore, x = 7.45 inch for the maximum volume.
The area of the square = x² = 7.45² = 55.5025 inch²
The maximum volume of a box is the highest volume the box can take.
The side length that ensures maximum volume is 22.55 inches or 7.45 inches.
The dimension of the plastic is:
[tex]\mathbf{Length = 42}[/tex]
[tex]\mathbf{Width = 48}[/tex]
Assume the side length cut-out is x
So, the dimension of the box is:
[tex]\mathbf{Length = 42 - 2x}[/tex]
[tex]\mathbf{Width = 48 - 2x}[/tex]
[tex]\mathbf{Height = x}[/tex]
So, the volume of the box is:
[tex]\mathbf{Volume = Length \times Width \times Height}[/tex]
This gives;
[tex]\mathbf{Volume = (42 - 2x) \times (48 - 2x) \times x}[/tex]
Expand
[tex]\mathbf{Volume = (42 - 2x) \times (48x - 2x^2)}[/tex]
Expand
[tex]\mathbf{Volume = 2016x - 96x^2 - 84x^2 + 4x^3}[/tex]
Differentiate
[tex]\mathbf{V' = 2016 - 192x - 168x + 12x^2}[/tex]
[tex]\mathbf{V' = 2016 -360x + 12x^2}[/tex]
Rewrite as:
[tex]\mathbf{V' = 12x^2 -360x + 2016}[/tex]
Set to 0
[tex]\mathbf{12x^2 -360x + 2016 = 0}[/tex]
Divide through by 12
[tex]\mathbf{x^2 -30x + 168 = 0}[/tex]
Using a calculator, the values of x are:
[tex]\mathbf{x = 22.55\ or\ x = 7.45}[/tex] ------ approximated to 2 decimal places
Hence, the side length that ensures maximum volume is 22.55 inches or 7.45 inches.
Read more about maximizing volumes at:
https://brainly.com/question/11599887
Given: AE ≅ CE ; DE ≅ BE Prove: ABCD is a parallelogram. Parallelogram A B C D is shown. Diagonals are drawn from point A to point C and from point D to point B and intersect at point E. Line segments A E and E C are congruent. Line segments D E and E B are congruent. We have that AB || DC. By a similar argument used to prove that △AEB ≅ △CED, we can show that △ ≅ △CEB by. So, ∠CAD ≅ ∠ by CPCTC. Therefore, AD || BC by the converse of the theorem. Since both pair of opposite sides are parallel, quadrilateral ABCD is a parallelogram.
Answer:
1.AED
2.SAS
3.ACB
4.Aternate interior angles
Step-by-step explanation:
Answer:
edg just did it
Step-by-step explanation:
5x + 2y + 8 + 7x −6y + 8 simplify
Answer:
12x+16-4y
Step-by-step explanation:
Answer:
lets start with x = 5x+7x=12x
now y = 2y-6y= -4y
now reg numbers = 8+8=16
now add all of them not litteraly add by add as in make them into a expression
so
12x-4y+16
Which square root represents 6.92?
Answer:
Step-by-step explanation:
Answer:
2.6305
Step-by-step explanation:
please help me on this
Answer:
A: (x-3, y+4)
Step-by-step explanation:
A coordinate plane is basically just a number line with two axes.
If we move a numbet right 3, to return it back we’d need to bring it left 3.
If we move a number down 4, we need to bring it up 4 to bring it back to it’s original position.
So to get it back, we need to add the opposite.
(x + 3), 3’s opposite is -3
(y - 4), -4’s opposite is 4.
Hope this helped!
What’s 67x59 answer quick please
Answer:
3953
Step-by-step explanation:
67x59 = 3953
graph 7/4 on the number line
Answer:
Step-by-step explanation:
First you have to change your fraction:
Change 7/4 into a mixed number
7/4 = 1 3/4
Then you 'guestamate' about 3/4 of the distance between 1 and 2 and mark your point. Like this:
Hope this helps
The number is expressed as the equation A = 7/4 or A = 1.75
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
A = 7/4 be equation (1)
On simplifying the equation , we get
A = 1.75
Now , the number 1.75 lies on the positive side of the number line and is between 1 and 2
So , the number 1.75 lies between 1 < 1.75 < 2
Hence , the number is A = 1.75
To learn more about equations click :
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I'm taking a homeschooling course and my parents put me in a couple grades before my level, so its a little difficult for me sometimes. So if someone could explain this it would be super helpful. ^_^ Select the correct scientific notation form of this numeral using only 2 significant figures. 8,421,032.9266
Answer: 8.4 * 10^6
Step-by-step explanation:
Hope this helps^^
two APS have the same common difference the first term of one of these is -1 and that of the other is -8. then the difference between their 4th term is
Answer:
The difference between the 4th term of both series is 7
Step-by-step explanation:
The formula for the nth term of an APS is a + (n - 1)×d
The first term of one of the series = -1
The first term of the other of the series = -8
The fourth term of both series is thus;
4th term first series = -1 + (4 - 1)× d = -1 + 3·d
4th term second series = -8 + (4 - 1)× d = -8 + 3·d
The difference between the 4th term of both series = -1 + 3·d - (-8 + 3·d)
-1 + 3·d - (-8 + 3·d) = -1 + 3·d + 8 - 3·d = 8 - 1 = 7
The difference between the 4th term of both series = 7.
PLEASE HELP The domain is {x∈R| x≠1}, and the range is {y∈R| y≠0}. The domain is all real numbers, and the range is {y∈R| y≠0}. The domain is {y∈R| y≠1}, and the range is {x∈R| x≠0}. The domain is {x∈R| x≠1}, and the range is all real numbers.
Answer:
The domain is x cannot equal 1 and range cannot equal 0. The first option you have is correct
Step-by-step explanation:
Un grupo de amigos quiere realizar una actividad de recaudación de fondos, para ello han decidido vender pizzas. La primera pizzería vende pizzas de 40 cm de diámetro a S/ 40 y otra vende pizzas de 25 cm de diámetro a S/ 25. ¿Qué pizza será recomendable que los amigos adquieran para lograr mejores ganancias?
Answer:
Los Amigos deberían comprar pizzas vendidas por la primera pizzería que cuestan S / 0.032 por cm² para obtener más ganancias.
Step-by-step explanation:
Los parámetros dados son;
El diámetro de las pizzas vendidas por la primera pizzería = 40 cm
El precio de las pizzas vendidas por la primera pizzería = S / 40
El diámetro de las pizzas vendidas por la segunda pizzería = 25 cm
El precio de las pizzas vendidas por la segunda pizzería = S / 25
El área de las pizzas vendidas por la pizzería = π × D² / 4
El área de las pizzas vendidas por la primera pizzería = π × 40² / 4≈1256.64 cm²
El costo por unidad de superficie de las pizzas vendidas por la primera pizzería = S / 40 / (1256.64 cm²) ≈ S /0.032/cm²
El área de las pizzas vendidas por la segunda pizzería = π × 25² / 4≈490.87 cm²
El costo por unidad de superficie de las pizzas vendidas por la primera pizzería = S / 25 / (490.87 cm²) ≈ S /0.051/cm²
Las pizzas vendidas por la primera pizzería que cuestan S /0.032/cm² s más baratas y producirán mejores ganancias que las pizzas vendidas por la segunda pizzería que cuestan S /0.051/cm²
Por lo tanto, los amigos deben comprar pizzas vendidas por la primera pizzería que cuestan S /0.032 / cm² para obtener más ganancias.
Grant is painting a rectangular board that has a width of į foot. He has enough
paint to cover 3 square feet. If he is able to cover the whole board by using all of
his paint, what is the length of the board in feet?
Answer:
how you convert feet to square feet
length X width = square feet
if the board is 3ft long and 1 foot wide
3ft X 1ft = 3square feet
so the length is 3 feet
Answer:
Step-by-step explanation:
the answer is 9
Simplify the equation step by step listing the property or operations used -5/6*(-5/8)*6/5
Answer:
5/8
Decimal Form: 0.625
Step-by-step explanation:
Cancel 6.
−5×-5/8*1/5
cancel 5
- -5/8
Move the negative sign to the left.
- (-5/8)
remove parentheses
answer is 5/8
hope i helped
-lvr
PLZ SOMEONE ANSWER I GIVE 5/5 RATING AND WILL NAME BRAINLIEST!!!!!
Answer:
100. 120
101. 28
102. 1
103. 24
Step-by-step explanation:
Number of Permutation (P) can be obtained as follow:
nPr = n! / (n– r)!
Number of Combination (C) can be obtained as follow:
nCr = n! / (n– r)! r!
With the above ideas in mind, let us determine the answers to the questions given above.
100. Evaluation of ₅P₄
nPr = n! / (n– r)!
₅P₄ = 5! /(5 – 4)!
₅P₄ = 5! / 1!
₅P₄ = 5 × 4 × 3 × 2 × 1 / 1
₅P₄ = 120
101. Evaluation of ₈C₂
nCr = n! / (n– r)! r!
₈C₂ = 8! / (8 – 2)!2!
₈C₂ = 8! / 6!2!
₈C₂ = 8 × 7 × 6!/ 6! × 2 × 1
₈C₂ = 8 × 7 / 2 × 1
₈C₂ = 56/2
₈C₂ = 28
102. Evaluation of ₇C₀
nCr = n! / (n– r)! r!
₇C₀ = 7! / (7 – 0)!0!
₇C₀ = 7! / 7!0!
Note: 0! = 1
₇C₀ = 7! / 7!
₇C₀ = 1
103. Evaluation 4!
4! = 4 × 3 × 2 × 1
4! = 24
A park is a rectangle a fence runs along all four sides the area of the park is 5,400 yards squared the length is 1.5 times longer then the width what is the perimeter of the park
Answer:
Perimeter of the park = 300 yards
Step-by-step explanation:
Area of a rectangle= length * width
Area of the rectangular park = 5,400 yards squared
Let
Width= x
Length= 1.5 * x = 1.5 x
Area of a rectangle= length * width
5,400= 1.5x * x
5400= 1.5x^2
x^2= 5400 / 1.5
x^2 = 3600
Take the Square root of both sides
√x^2 = √3600
x=60
Therefore,
Width = x = 60 yards
Length = 1.5x
=1.5(60)
=90 yards
Perimeter of the rectangular park = 2(length + width)
=2(90+60)
=2(150)
= 300 yards
if a motercycle is moving at a constant speed down the highway of 40 km/hr, how long would it the motorcycle to travel 10 km
Answer:
15 minutes
Step-by-step explanation:
First, the motorcycle goes at a speed of 40 km/hr.
The question asks for how long it would take to travel 10 km.
Well, there are 60 minutes in an hour, since we will be translating into minutes.
Also, 10 km is 1/4 of 40 km, so it would make sense that the time length would be 1/4 of an hour as well.
1/4 of 60 minutes is 15 minutes. So it takes 15 minutes for the motorcycle to travel 10 km.
Now, if all this wordy stuff is too much to comprehend, you can also solve using proportional relationships.
[tex]\frac{40km}{60min}=\frac{10km}{xmin}[/tex]
Now cross multiply:
[tex]40km*xmin=10km*60min\\40x=600[/tex]
Divide both sides by 40:
[tex]\frac{40x}{40}=\frac{600}{40}\\x=15[/tex]
Again, this shows that it wouls take 15 minutes for the motorcycle to travel 10 km.
х+у= 9
2x – Зу = 8
someone plz solve
Step-by-step explanation:
x+y=9
x=9-y ----------(i)
Now in second eqn put the value of x
2x -3y=8
2(9-y) - 3y=8
18 - 2y - 3y=8
18-8=5y
10=5y
y=2
Then put the value of y in eqn (i)
x=9-y
x=9-2
x=7
I hope this will be helpful for you.
Answer:
[tex] \huge {\boxed{ \bold{ \boxed{(7 \:, 2)}}}}[/tex]Step-by-step explanation:
Given,
x + y = 9
2x - 3y = 8
Using substitution method:
In this method , a variable is expressed in terms of another variable from one equation and it is substituted in the remaining equation.
[tex] \mathsf{x + y = 9}[/tex]
Move y to right hand side and change it's sign
⇒[tex] \mathsf{x = 9 - y}[/tex]⇔equation ( i )
Now, put the value of X from equation ( i )
⇒[tex] \mathsf{ 2 (9 - y) - 3y = 8}[/tex]
Distribute 2 through the parentheses
⇒[tex] \mathsf{18 - 2y - 3y = 8}[/tex]
Collect like terms
⇒[tex] \mathsf{18 - 5y = 8}[/tex]
Move constant to right hand side and change it's sign
⇒[tex] \mathsf{ - 5y = 8 - 18}[/tex]
Calculate
⇒[tex] \mathsf{ - 5y = - 10}[/tex]
Divide both sides of the equation by -5
⇒[tex] \mathsf{ \frac{ - 5y}{ - 5} = \frac{ - 10}{ - 5} }[/tex]
Calculate
⇒[tex] \mathsf{y = 2}[/tex]
Now, Put the value of y in the equation ( i )
⇒[tex]x = 9 - 2[/tex]
Calculate the difference
⇒[tex] \mathsf{x = 7}[/tex]
Hence, x = 7 and y = 2
The possible solution of the system is the ordered pair ( x , y )
[tex] \mathsf{(x \: , y \: ) = (7 , \: 2)}[/tex]
---------------------------------------------------------
Check if the given ordered pair is the solution of the system of equation
⇒[tex] \mathsf{7 + 2 = 9}[/tex]
⇒[tex] \sf{2 \times 7 - 3 \times 2 = 8}[/tex]
Simplify the equalities
⇒[tex] \sf{9 = 9}[/tex]
⇒[tex] \mathsf{8 = 8}[/tex]
Since all of the equalities are true , the ordered pair is the solution of the system.
[tex] \mathsf{ \underline{ \bold{(x \:, y \: ) = (7 \:, 2)}}}[/tex]
Hope I helped!
Best regards!!
*Silly and/or spam answers will not be tolerated*
Evaluate the limit: [tex]\lim_{x \to -6} \frac{\sqrt{10 - x}-4 }{x + 6}[/tex]
Please explain how to rationalize and solve for the limit.
Answer:
-1/8
Step-by-step explanation:
lim x approaches -6 (sqrt( 10-x) -4) / (x+6)
Rationalize
(sqrt( 10-x) -4) (sqrt( 10-x) +4)
------------------- * -------------------
(x+6) (sqrt( 10-x) +4)
We know ( a-b) (a+b) = a^2 -b^2
a= ( sqrt(10-x) b = 4
(10-x) -16
-------------------
(x+6) (sqrt( 10-x) +4)
-6-x
-------------------
(x+6) (sqrt( 10-x) +4)
Factor out -1 from the numerator
-1( x+6)
-------------------
(x+6) (sqrt( 10-x) +4)
Cancel x+6 from the numerator and denominator
-1
-------------------
(sqrt( 10-x) +4)
Now take the limit
lim x approaches -6 -1/ (sqrt( 10-x) +4)
-1/ (sqrt( 10- -6) +4)
-1/ (sqrt(16) +4)
-1 /( 4+4)
-1/8
Answer:
[tex]\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} =-\frac{1}{8}[/tex]
Step-by-step explanation:
We want to evaluate the limit:
[tex]\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} \\[/tex]
When attempting to evaluate a limit, we should always try direct substitution. This yields:
[tex]\displaystyle \begin{aligned} &\Rightarrow \frac{\sqrt{10-(-6)}-4}{(-6)+6} \\ \\ &=\frac{\sqrt{16}-4}{-6+6}\\ \\ &=\frac{4-4}{-6+6}\\ \\ &=\underbrace{\frac{0}{0}}_{\text{Indeterminate}} \end{aligned}[/tex]
Since the result is an indeterminate form, we can try simplifying the limit.
Let's cancel the square root in the numerator. We can use the difference of two squares. Recall that:
[tex](a-b)(a+b)=a^2-b^2[/tex]
The expression in the numerator is:
[tex]\sqrt{10-x}-4[/tex]
Therefore, to cancel it out, we will multiply it by:
[tex]\sqrt{10-x}+4[/tex]
Multiply. This yields:
[tex]=\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6}\cdot\frac{\sqrt{10-x}+4}{\sqrt{10-x}+4} \\[/tex]
Simplify:
[tex]\displaystyle \begin{aligned} &= \lim_{x \to -6}\frac{(\sqrt{10-x})^2-(4)^2}{(x+6)(\sqrt{10-x}+4)}\\ \\ &=\lim_{x\to-6}\frac{(10-x)-(16)}{(x+6)(\sqrt{10-x}+4)}\\ \\ &=\lim_{x \to \ -6}\frac{-x-6}{x+6(\sqrt{10-x}+4)}\end{aligned}[/tex]
Factor:
[tex]\displaystyle \lim_{x \to \ -6}\frac{-(x+6)}{(x+6)(\sqrt{10-x}+4)}[/tex]
Cancel:
[tex]\displaystyle \lim_{x \to \ -6}-\frac{1}{\sqrt{10-x}+4}[/tex]
Now, we can attempt direct substitution again. Thus:
[tex]\displaystyle \begin{aligned} &\Rightarrow -\frac{1}{(\sqrt{10-(-6)}+4)}\\ \\ &=-\frac{1}{\sqrt{16+4}}\\ \\ &=-\frac{1}{(4+4)} \\ \\ &=-\frac{1}{8}\end{aligned}[/tex]
Therefore:
[tex]\displaystyle \lim_{x \to -6}\frac{\sqrt{10-x}-4}{x+6} =-\frac{1}{8}[/tex]