Answer:
the solution set of n2 - 14n = -45 is {5, 9}
Step-by-step explanation:
We have the following equation: n^2 - 14n = -45
Rearrange:
n^2 - 14n +45 = 0
Factorizing:
(n-9)(n-5) = 0
Therefore, the solution set of n2 - 14n = -45 is {5, 9}
What is the volume of a sphere with a diameter of 11ft? Round your answer to the nearest cubic foot.
[tex]d=11\text{ ft}\\V=\dfrac{4}{3}\pi r^3\\r=\dfrac{d}{2}=\dfrac{11}{2}=5.5\\\\V=\dfrac{4}{3}\pi \cdot (5.5)^3\approx697\text{ ft}^3[/tex]
The area of a circle is 16π m2. What is the radius of the circle?
Answer:
the radius of the circle= 4m
Step-by-step explanation:
Given:
Area of circle=16π m2
radius,r of circle=?
Formula of area of circle is given as
Area=πr^2
Putting values we get
16π=πr^2
r^2=16
r=4
the radius of the circle= 4m !
Given the area [tex]A=\pi r^2=16\pi[/tex] we can solve the formula for radius.
[tex]A=\pi r^2\Longrightarrow r=\sqrt{\dfrac{A}{\pi}}[/tex]
So,
[tex]r=\sqrt{\dfrac{16\cdot\not{\pi}}{\not{\pi}}}=\sqrt{16}=4[/tex]
The radius is 4m.
Hope this helps.
r3t40
In △ABC,a=34, b=18, and c=17. Find m∠A.
A. 14.1
B. 132.6
C. 152.5
D. 101.5
Answer:
C. m < A = 152.5 degrees.
Step-by-step explanation:
Use the Cosine Rule:
a^2 = b^2 + c^2 - 2.b.c.cos A
34^2 = 18^2 + 17^2 - 2.18.17 cos A
2.18.17 cos A = 18^2 + 17^2 - 34^2
cos A = (18^2 + 17^2 - 34^2) / (2.18.17)
cos A = -0.88725
m < A = 152.5 degrees.
Answer:
C.) 152.5
Step-by-step explanation:
I got it correct on founders edtell
Let f(x) = x^2-6 and g(x) =10x . Find (g ° f)(x)
Answer:
[tex](g \circ f)(x)=10x^2-60[/tex]
Answer:[tex]10x^2-60[/tex]
Step-by-step explanation:
[tex](g \circ f)(x)=g(f(x))[/tex]
Replace [tex]f(x)[/tex] with [tex]x^2-6[/tex].
This gives us:
[tex](g \circ f)(x)=g(f(x))[/tex]
[tex](g \circ f)(x)=g(x^2-6)[/tex]
This means to replace the old input variable with new input, [tex](x^2-6)[/tex].
Let's do that:
[tex](g \circ f)(x)=10(x^2-6)[/tex]
They probably want you to distribute:
[tex](g \circ f)(x)=10x^2-60[/tex]
Answer:
10x^2-60
Step-by-step explanation:
(G o F)(x) is the same as g(f(x)). We know that f(x)=x^2-6. So now you have to find g(x^2-6). To solve for that plug in x^2-6 in for x in the original equation for g(x). You get 10(x^2-6) or 10x^2-60
If a point is inside a circle, the distance from the center of the circle to that point ____.
A. is less than the radius
B. is perpendicular to that chord
C. passes through the center of the circle
D. bisects the radius
Answer:
A. is less than the radius.
Step-by-step explanation:
If a point is inside a circle, the distance from the center of the circle to that point is option (A) is less than the radius is the correct answer.
What is a circle?A circle is a collection of all points in a plane which are at a constant distance from a fixed point. A circle is a round-shaped figure that has no corners or edges.
For the given situation,
The distance from the center of the circle to any point on it's circumference is called radius.
So, when we consider the distance from the center of the circle, then the term should be related to radius.
Hence we can conclude that if a point is inside a circle, the distance from the center of the circle to that point is option (A) is less than the radius is the correct answer.
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In the diagram, find the measure of Angle y and Angle x.
PLEASE HELP ME!!
Answer:
Part 1) The measure of angle x is 24°
Part 2) The measure of angle y is 66°
Step-by-step explanation:
step 1
Find the measure of angle m∠IKL
we know that
m∠JKI+m∠IKL=180° ----> supplementary angles (form a linear pair)
we have
m∠JKI=48°
substitute
48°+m∠IKL=180°
m∠IKL=180°-48°=132°
step 2
Find the measure of angle x
we know that
The triangle IKL is an isosceles triangle
so
m∠KIL=m∠KLI=x
Remember that
The sum of the interior angles of a triangle must be equal to 180 degrees
m∠IKL+2x=180°
132°+2x=180°
2x=180°-132°
x=24°
step 3
In the right triangle JIL
Angles x and y are complementary
so
x+y=90°
24°+y=90°
y=90°-24°
y=66°
On a baseball field, the pitcher’s mound is 60.5 feet from home plate. During practice, a batter hits a ball 214 feet at an angle of 36° to the right of the pitcher’s mound. An outfielder catches the ball and throws it to the pitcher. Approximately how far does the outfielder throw the ball?
A. 129.7 ft
B. 168.8 ft
C. 127.7 ft
D. 212.4 ft
I believe the answer is B. I apologize if this is incorrect
Answer:
B) 168.8
Step-by-step explanation:
Amongst all the options given, B is the correct answer.
!!
Which angle is included between MP and PN?
Answer:
MPN
Step-by-step explanation:
A bicyclist rides the same number of miles every minute. The ratio table below shows the number of miles she rides during certain amounts of time.
Biking Times and Distances
Number of Minutes
Number of Miles
10
2.5
16
4
?
7.5
48
12
Which statement explains how to find the number of minutes it takes to bike 7.5 miles?
Find the ratio of minutes to miles, 4:1. Divide 7.5 by 4.
Find the ratio of minutes to miles, 4:1. Multiply 7.5 by 4.
Find the difference between 16 and 10. Add the difference to 16.
Find the difference between 16 and 10. Add the difference to 7.5.
Answer:
B) Find the ratio of minutes to miles, 4:1. Multiply 7.5 by 4
Step-by-step explanation:
A bicyclist rides the same number of miles every minute. The ratio table below shows the number of miles she rides during certain amounts of time.
Biking Times and Distances
Number of Minutes Number of Miles
10 2.5
16 4
? 7.5
48 12
Which statement explains how to find the number of minutes it takes to bike 7.5 miles?
B) Find the ratio of minutes to miles, 4:1. Multiply 7.5 by 4....
Answer:
Option B.
Step-by-step explanation:
A bicyclist rides the same number of miles in one minute.
Ratio table is given as
Number of minutes Distances (miles)
10 2.5
16 4
x 7.5
48 12
We have to find the number of minutes taken by a biker to bike 7.5 miles
So the ratio between minutes to the distance biked will be = [tex]\frac{10}{2.5}=4:1[/tex]
Now this ratio will be similar to the ratio = [tex]\frac{x}{7.5}=x:7.5[/tex]
Now [tex]\frac{4}{1}=\frac{x}{7.5}[/tex]
x = 7.5×4 = 30 minutes
Therefore, Option B will be the answer.
How many moles of mgci2 are there in 318 g of the compound?
Answer:
there are 3.3399 moles in 318 grams! ✔️
Step-by-step explanation:
So we know that 100 grams MgCl2 to mol = 1.0503 mol.
Therefore, to find the number of moles in 318g of the compound, we use the rule of three:
if 1.0503 mol -------------> 100 grams
X <------------- 318 grams
The solution is: X = (318*1.0503)/100 = 3.3399 mol.
Summarizing, there are 3.3399 moles in 318 grams! ✔️
In the circle graph what is the measure of the central angle for corn?
Round your answer to the nearest whole number.
Answer:
54°
Step-by-step explanation:
Corn made up 15 % of the vegetables, so
Corn makes up 15 % of the degrees in the circle.
There are 360 degrees in a circle, so
15/100 × 360° = 54 °
The measure of the central angle for corn is 54°.
Question 3 of 33
1 Point
Which type of tessellation uses one type of regular polygon to cover a surface
completely?
O
A. Rotating tessellation
O
B. Regular tessellation
O
C. Semi-regular tessellation
O
D. Symmetrical tessellation
The type of tessellation used to cover a surface completely using one type of regular polygon is regular tessellation.
What is Tessellation?Tessellation is defined as the process of covering of a surface or a plane using the geometric shapes.
One geometric shape will not overlap with the other and there will not be any gaps in between.
Regular tessellation is the tiling which uses one type of regular polygon to form a pattern.
In semi-regular tessellation, two or more regular polygons are used such that each vertex is the same.
Here, it is only used one type of regular polygon. No other polygons are used.
So it can't be semi-regular tessellation.+
So it is regular tessellation.
Hence the tessellation is regular tessellation.
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A pen and a pencil together cost $5.10. The pen cost $5 more than the pencil. How much does each cost?
Answer:
The pen costs $5.05, and the pencil costs $0.05
Step-by-step explanation:
First, we define 2 variables for the two prices.
Let x = price of the pen.
Let y = price of the pencil.
Now we use the given information to write two equations.
"A pen and a pencil together cost $5.10. "
x + y = 5.1
"The pen cost $5 more than the pencil. "
y = x + 5
The system of equations is:
x + y = 5.1
y = x + 5
Since the second equation is already solved for y, we will use the substitution method to solve the system of equations.
Substitute y of the first equation by x + 5.
x + x + 5 = 5.1
2x + 5 = 5.1
2x = 0.1
x = 0.05 (price of the pencil)
Now substitute 0.05 for x in the second equation.
y = 0.05 + 5
y = 5.05 (price of the pen)
The pen costs $5.05, and the pencil costs $0.05
Final answer:
The pencil costs $0.05 and the pen costs $5.05, with the pen being $5 more expensive than the pencil and the total cost for both being $5.10.
Explanation:
To solve this problem, we need to create equations based on the information given. Let's assign x as the cost of the pencil and x + $5.00 as the cost of the pen, since the pen costs $5 more than the pencil.
According to the problem, the total cost of both items is $5.10, so we can write the equation as follows:
x + (x + $5.00) = $5.10Combining the like terms, we get:
2x + $5.00 = $5.10Subtracting $5.00 from both sides, we find:
2x = $0.10Dividing both sides by 2 to solve for x:
x = $0.05Thus, the pencil costs $0.05, and the pen, which costs $5 more, is $5.05.
The perimeter of a triangle is 30 in. The longest side is 2 less than 3 times the shortest side and the other side is 2 more than twice the shortest side. Find the length of each side
Answer:
Shortest side=5, Largest side=13, Other side=12
Step-by-step explanation:
Let's define the shortest side of the triangle as x, since the other two sides are defined in terms of this one.
The longest side is 2 less than 3 times the shortest side, then it is 3 * x - 2.
The other side is 2 more than twice the shortest side, then it is 2 * x + 2.
By the problem directions, we know that the perimeter or sum of sides is 30.
x + 3x - 2 + 2*x + 2 = 30
Adding all the terms "with x" together and all terms "without x" together, we get...
6*x = 30
Solving for x (dividing both sides by 6), we get...
x = 30/6 = 5
Finally,
Shortest side: 5
Largest side: 3 * 5 - 2 = 13
Other side: 2 * 5 + 2 = 12
The length of each side of the triangle are 5 in, 12 in and 13 in respectively
Given:
Perimeter of the triangle = 30 in
let
shortest side = x
other side = 2x + 2
Longest side = 3x - 2
Perimeter of the triangle = shortest side + other side + Longest side
30 = x + (2x + 2) + (3x - 2)
30 = x + 2x + 2 + 3x - 2
30 = 6x
divide both sides by 6
x = 30 / 6
x = 5
Therefore,
shortest side = x
= 5 in
other side = 2x + 2
= 2(5) + 2
= 10 + 2
= 12 in
Longest side = 3x - 2
= 3(5) - 2
= 15 - 2
= 13 in
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What is the explicit formula for this geometric sequence?
27, 9, 3, 1, ...
Answer:
D.) 27 * ( 1/3) ^ (x-1)
Step-by-step explanation:
This is because the first value, 27, indicates the rest state, or the y intercept. The second value, 1/3, is derived from the pattern of the values decreasing by the previous value being divided by 3, or technically multiplied by (1/3). The (x-1) is derived from the original formula for the geometric series.
f(x) = x * r ^ (n-1)
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For any questions or more information, please comment below and I'll respond as soon as possible.
The explicit formula for the geometric sequence is:[tex]a_n = 27 \cdot \left(\frac{1}{3}\right)^{(n-1)}[/tex]
To find the explicit formula for the geometric sequence 27, 9, 3, 1, ..., we first need to identify key components of a geometric sequence.
First Term (a): The first term of this sequence, denoted as [tex]a[/tex], is 27.
Common Ratio (r): The common ratio [tex]r[/tex] is found by dividing the second term by the first term:
[tex]r = \frac{9}{27} = \frac{1}{3}[/tex]
We can check this by calculating further:
[tex]\frac{3}{9} = \frac{1}{3} \quad \text{and} \quad \frac{1}{3} = \frac{1}{3}[/tex]
Thus, the common ratio is consistent at [tex]r = \frac{1}{3}[/tex].
Explicit Formula: The explicit formula for a geometric sequence is given by:
[tex]a_n = a \cdot r^{(n-1)}[/tex]
where [tex]a_n[/tex] is the [tex]n[/tex]-th term, [tex]a[/tex] is the first term, and [tex]r[/tex] is the common ratio.
Substituting in our values:
[tex]a_n = 27 \cdot \left(\frac{1}{3}\right)^{(n-1)}[/tex]
Let f(x) = x^2 +1 and g(x)= x-2 Evaluate (f x g) (0). I need help i know the answer is 5 i just don't know how to get that answer !
Answer:
(f*g)(0)=-2
f(g(0))=5 I think you meant this one based on your answer. Please read lower half of explanation for this answer. Please ask me any question on this that you have.
Step-by-step explanation:
(f*g)(0) means f(0)*g(0).
f(0) means to replace x with 0 in x^2+1 (since g(x)=x^2+1).
g(0) means to replace x with 0 in x-1 (since g(x)=x-2).
f(0)=0^2+1=0+1=1
g(0)=0-2=-2
So (f*g)(0)=f(0)g(0)=(1)(-2)=-2.
But I guess you didn't mean this because you said the answer is 5...
Oh maybe you mean
[tex](f \circ g)(0)[/tex]?
[tex](f \circ g)(0)[/tex] means [tex]f(g(0))[/tex].
So f(g(0))...we start with the inside first... that is g(0)?
g(0)=-2 (we found this above)
f(g(0))
f(-2) ->I replaced g(0) with -2
Now f(-2) means to replace x with (-2) in x^2+1 (since f(x)=x^2+1)
So let's do that:
(-2)^2+1
4+1
5
In triangle ABC, c = 9 m/_B = 65° and a = 105. Find b.
Answer:
b = 101.52
Step-by-step explanation:
b^2 = a^2 + c^2 - 2ac CosB
b^2 = 105^2 + 9^2 - 2(105)(9) Cos 65°
b^2 = 11 106 - 798.75
b^2 = 10 307.25
b = 101.52
Answer:101.52
Step-by-step explanation:
Given
a=105
c=9
[tex]B=65^{\circ}[/tex]
using cosine rule
[tex]2acCosB=a^2+c^2-b^2 [/tex]
[tex]2(105)(9)Cos65=105^2+9^2-b^2[/tex]
[tex] b^2=11025+81-798.7485[/tex]
b=101.524
Simplify −2xy + 3x − 2xy + 3x
A. 2xy
B. 4xy +6x
C. 4xy − 6x
D. −4xy + 6x
-2xy+3x-2xy+3x ( original equation)
-2xy-2xy+3x+3x ( just rearranging)
-4xy+6x
Answer :-4xy+6x- D.
Answer:
The correct answer is option D. -4xy + 6x
Step-by-step explanation:
It is given an expression, −2xy + 3x − 2xy + 3x
To simplify the given expression
Let the expression be −2xy + 3x − 2xy + 3x
−2xy + 3x − 2xy + 3x = -2xy - 2xy + 3x + 3x [write similar terms together]
= -4xy + 6x
Therefore the correct answer is option D.
-4xy + 6x
A. Figure A
B. Figure B
Answer:
A.) Figure A
Step-by-step explanation:
because the first part "If..." always goes on the inside of the 2 circles
The perimeter of a rectangle is 16 inches. The equation that represents the perimeter of the rectangle is , where l represents the length of the rectangle and w represents the width of the rectangle. Which value is possible for the length of the rectangle?
7 in.
8 in.
9 in.
10 in.
Answer:
A. 7 in.
Step-by-step explanation:
We have been given that the perimeter of a rectangle is 16 inches. The equation that represents the perimeter of the rectangle is , where l represents the length of the rectangle and w represents the width of the rectangle.
We know that perimeter of rectangle is 2 times the sum of width and length of rectangle.
[tex]\text{Perimeter}=2(l+w)[/tex]
[tex]\text{16 in}=2(l+w)[/tex]
[tex]\frac{\text{16 in}}{2}=\frac{2(l+w)}{2}[/tex]
[tex]\text{8 in}=l+w[/tex]
To be a rectangle length cannot be 8 as length and width of the rectangle is 8 inches.
Therefore, 7 inches the possible value for the length of the rectangle.
Answer:
A
Step-by-step explanation:
Lorelei evaluates the expression to determine how many different groups of ten she can make out of twelve items. Her solution: Subtract within parentheses and simplify: Expand: Divide out common factors: Because 6 divided by 2∙1 is 3, there are 3 ways to choose the groups. Which statements describe Lorelei’s solution? Check all that apply. Her work is correct. Her answer is correct. In step 1, the subtraction cannot be completed before the factorial of each number is calculated. In step 1, 12! divided by 10! is not equivalent to 6! divided by 5!. In step 3, the dividing out of common factors was performed incorrectly. There are sixty-six ways to choose ten items from twelve.
Answer:
D. In step 1, 12! divided by 10! is not equivalent to 6! divided by 5!.
F. There are sixty-six ways to choose ten items from twelve.
Step-by-step explanation:
The expressions that determine Lorelei's solution include in step 1, 12! divided by 10! is not equivalent to 6! divided by 5! as well option F.
How to illustrate the expression?From the information given, Lorelei evaluates the expression to determine how many different groups of ten she can make out of twelve items.
In this case, the expressions that determine Lorelei's solution include in step 1, 12! divided by 10! is not equivalent to 6! divided by 5! and that there are sixty-six ways to choose ten items from twelve.
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A line passes through the points (-5,2) and (10,-1) which is the equation the line
Step-by-step explanation:
To find the equation of the line, start by finding the slope. You can do this by using the slope formula below.
m(slope) = (y2 - y1)/(x2 - x1)
m = (2 - -1)/(-5 - 10)
m = 3/-15
m = -1/5
Now that we have the slope, we can use it along with either point in point-slope form to get the equation.
y - y1 = m(x - x1)
y - 2 = -1/5(x + 5)
y - 2 = -1/5x - 1
y = -1/5x + 1
Answer: In slope intercept formula it is y=-1/5x+3
Step-by-step explanation:
Use the elimination method to solve the system of equations. Choose the
correct ordered pair.
-3y = x-5
x+ 5y = 7
O A. (-1,2)
O B. (5,0)
O C. (2.1)
O D. (-4,3)
Answer:
C. (2, 1)
Step-by-step explanation:
-3y = x-5
x+ 5y = 7
Subtract x from both sides in the first equation. Write the second equation below it.
-x - 3y = -5
x + 5y = 7
Add the two equations above.
2y = 2
Divide both sides by 2.
y = 1
Substitute y with 1 in the second original equation and solve for x.
x + 5(1) = 7
x + 5 = 7
Subtract 5 from both sides.
x = 2
Answer: C. (2, 1)
Answer:
○ C. (2, 1)
Step-by-step explanation:
{-3y = x - 5 [Move -x to the left of the equivalence symbol]
{x + 5y = 7
{-x - 3y = -5
{x + 5y = 7
____________
2y = 2
__ _
2 2
y = 1 [Plug this back into both equations to get the x-coordinate of 2]; 2 = x
I am joyous to assist you anytime.
Help me please and thank you !
:
starting value
-- :
the y-intercept is the starting value of the coin after no time has passed. :)
Solve 18x + 6 > 12x + 18.
-
The solution of the inequation 18x + 6 [tex]>[/tex] 12x + 18 is x [tex]>[/tex] 2.
An inequation is a mathematical statement that represents an inequality relationship between two expressions. It compares the relative values of the two expressions, indicating whether one is greater than, less than, or not equal to the other.
Solving an inequation involves finding the values of the variables that satisfy the inequality. This can often be done by using algebraic techniques, such as simplifying and rearranging the expressions, isolating the variable, and applying appropriate rules and operations.
To solve the above inequation, let's assume that it is an equation.
Then,
18x + 6 = 12x + 18
18x - 12x = 18 - 6
6x = 12
x = [tex]\frac{12}{6}[/tex]
x = 2.
Thus the inequation in simplifying is x [tex]>[/tex] 2.
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Final answer:
The result to the inequality 18x 6> 12x 18 is x> 2. This is set up by abating 12x from both sides, also abating 6, and also dividing by 6 to insulatex.
Explanation:
To break the inequality 18x 6> 12x 18, we need to insulate the variable x on one side.
Abate 12x from both sides 18x- 12x 6> 18.
Simplify the equation 6x 6> 18.
Abate 6 from both sides 6x> 12.
Divide both sides by 6 to break for x x> 2.
The result to the inequality is x lesser than 2. This gives a range of values for x that will satisfy the original inequality. thus, any number lesser than 2 for x makes the inequality true.
I can’t get the answer that it’s either there, welpppp!
Answer: D. Y=3/2x
Step-by-step explanation: When going from left to right, the line is going up. This means that the slope is positive. We can eliminate answers C and E because they are negative numbers. The slope is rise over run. From the bottom point, the line goes up 3 and to the right 2. Therefore, the answer is D. Y=3/2x.
According to an informal poll in Glenview 1/3
of the men and 2/3 of the women said they
would vote for John Smith. On election day,
one and a half times as many men as women
voted. What fraction of the total vote,
according to the poll, should be cast for John
Smith?
Answer:
Percent form: 41.65%
Fraction form: [tex]\frac{833}{2000}[/tex]
Step-by-step explanation:
33% of men support John Smith.
67% of women support John Smith.
Lets say, for instance, that there are 20 polled individuals -- 10 women and 10 men.
However, since 1.5 times as many men voted as women, we have to apply our poll to 15 men and 5 women.
33% of the 15 men makes 5 men who voted for Smith and 10 who did not.
67% of the 5 women makes 3.33 women who voted for Smith and 1.67 who did not.
Add these numbers together.
8.33 total voters were cast for John Smith, while 11.67 were not.
Now, divide 8.33 by 20 and multiply by 100 to obtain a percentage.
[tex]\frac{8.33}{20} \\.4165\\41.65[/tex]
41.65% of voters voted for John Smith according to the results of the poll.
According to an informal poll, with 1/3 of men and 2/3 of women supporting John Smith, and 1.5 times more men than women voting, 2/3 of the total vote should be cast for John Smith.
We need to calculate the fraction of the total vote that would be cast for John Smith according to the informal poll. Let's denote the number of women voters as w and the number of men as m. According to the poll, 1/3 of men and 2/3 of women will vote for John Smith. On election day, 1.5 times as many men as women voted, which means m = 1.5w.
The total votes for John Smith would be (1/3)m + (2/3)w. Substituting m with 1.5w, we get:
Total votes for John Smith = (1/3)(1.5w) + (2/3)w
This simplifies to:
Total votes for John Smith = 0.5w + (2/3)w = (1.5/3)w + (2/3)w = (1.5/3 + 2/3)w = (3/3 + 2/3)w = (5/3)w
The total votes cast would be m + w, substituting again we get 1.5w + w = 2.5w.
To find the fraction of total votes that would be for John Smith, we divide the total votes for John Smith by the total votes cast:
Fraction for John Smith = (5/3)w / (2.5)w
Since w is in both the numerator and denominator, it cancels out, and we're left with:
Fraction for John Smith = (5/3) / (2.5)
Converting 2.5 to a fraction gives us 5/2, so:
Fraction for John Smith = (5/3) / (5/2)
By dividing fractions, we invert and multiply:
Fraction for John Smith = (5/3) * (2/5) = 2/3
Therefore, according to the poll, 2/3 of the total vote should be cast for John Smith.
PLEASE BE QUICK AND ACCURATE
After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. You notice the rainbow is the shape of a parabola.
The equation for this parabola is y = -x2 + 36.
1. In the distance, an airplane is taking off. As it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. Create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow.
2. Analyze the two functions. Answer the following reflection questions in complete sentences. What is the domain and range of the rainbow? Explain what the domain and range represent. Do all of the values make sense in this situation? Why or why not? What are the x- and y-intercepts of the rainbow? Explain what each intercept represents. Is the linear function you created with your table positive or negative? Explain. What are the solutions or solution to the system of equations created? Explain what it or they represent.
3. Create your own piecewise function with at least two functions. Explain, using complete sentences, the steps for graphing the function. Graph the function by hand or using a graphing software of your choice (remember to submit the graph).
We know that after a dreary day of rain, the sun peeks through the clouds and a rainbow forms. So you notice the rainbow is the shape of a parabola as shown in the first figure. This parabola has the following equation:
[tex]y=-x^2+36[/tex]
On the other hand, in the distance you notice that an airplane is taking off. As it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. So let's choose two points on the graph of the parabola to build up the equation of the line:
[tex]\bullet \ If \ x=-5, then: \\ \\ y=-(-5)^2+36=11 \\ \\ So: \\ \\ P_{1}(-5,11) \\ \\ \\ \bullet \ If \ x=4, then: \\ \\ y=-(4)^2+36=20 \\ \\ So: \\ \\ P_{2}(4,20)[/tex]
Therefore, the equation of the line using these two points is:
[tex]y-20=\frac{20-11}{4-(-5)}(x-4) \\ \\ y-20=\frac{9}{9}(x-4) \\ \\ y-20=x-4 \\ \\ y=x+20-4 \\ \\ y=x+16[/tex]
FOR THE PARABOLA, THE TABLE IS:
[tex]Other \ two \ points: \\ \\ \bullet \ If \ x=0, then: \\ \\ y=-(0)^2+36=36 \\ \\ So: \\ \\ P_{3}(0,36) \\ \\ \\ \bullet \ If \ x=1, then: \\ \\ y=-(1)^2+36=35 \\ \\ So: \\ \\ P_{4}(1,35)[/tex]
So the table is:
[tex]\left[\begin{array}{cc}x & y\\-5 & 11\\4 & 20\\0 & 36\\1 & 35\end{array}\right][/tex]
FOR THE LINE, THE TABLE IS:
[tex]Other \ two \ points: \\ \\ \bullet \ If \ x=0, then: \\ \\ y=x+16=0+16=16 \\ \\ So: \\ \\ P_{3}(0,16) \\ \\ \\ \bullet \ If \ x=1, then: \\ \\ y=x+16=1+17=17 \\ \\ So: \\ \\ P_{4}(1,17)[/tex]
So the table is:
[tex]\left[\begin{array}{cc}x & y\\-5 & 11\\4 & 20\\0 & 16\\1 & 17\end{array}\right][/tex]
2. Analyzing the functions:2.1 Domain and Range:The Domain of both the quadratic function and the linear function is the set of all real numbers. The range of the quadratic function is [tex](-\infty,36)[/tex] while the range of the linear function is the set of all real numbers.The domain represents the horizontal distance while the range represents the height. In this situation, all values don't make sense. First, the rainbow starts in a point on the earth's crust. Also, the airplane starts at a point on the runway, so this doesn't include all the real numbers. A similar thing happens to the height since the airplane turns into an horizontal way at a moment of its trip and this doesn't include all the real numbers.
2.2 x and y intercepts of the rainbow.x-intercepts:
The x intercepts are the points at which the graph passes through the x-axis, that is, when [tex]y=0[/tex]. So:
[tex]-x^2 + 36=0 \\ \\ x^2=36 \\ \\ x=\pm\sqrt{36} \\ \\ x=\pm 6[/tex]
So the x intercepts are [tex]x_{1}=6 \ and \ x_{2}=-6[/tex]
y-intercepts:
The y-intercept is the point at which the graph passes through the y-axis, that is, when [tex]x=0[/tex]. So:
[tex]y=-x^2+36 \\ \\ y=-(0)^2+36 \\ \\ y=36[/tex]
So the y-intercept is [tex]b=36[/tex]
In conclusion, if we take the x-axis as the ground, the x-intercepts represent the beginning and the end of the rainbow while the y-intercept represent its maximum height.
2.3 Is the linear function positive or negativeThe equation of this line is:
[tex]y=x+16[/tex]
As you can see, the slope [tex]m=1[/tex] is positive, therefore the line I've created is positive. This means as x increases y increases, but how do x and y increase? Well, they increase as the airplane is taking off, so this means the horizontal distance and the height increase.
2.4 What are the solutions or solution to the system of equations created?The system of equations we created is:
[tex]\left\{ \begin{array}{c}y=x+16\\y=-x^{2}+36\end{array}\right.[/tex]
Remember that in Part 1, chose two points on the graph of the parabola to build up the equation of the line, so the solutions to this system is indeed those two points:
[tex]P_{1}(-5,11) \ and \ P_{2}(4,20)[/tex]
They represents that the airplane cuts through the rainbow at those two points.
3. Create a piecewise function.A piecesewise function is a function defined by two or more equations over a domain. Our piecewise function is shown in the second figure below and is defined by:
[tex]f(x)=\left\{ \begin{array}{c}x+16\quad if\quad-16\le x\le-5\\-x^{2}+36\quad if\quad-5\le x\le4\\x+16\ \quad if\quad4\le x\le24\\40\quad if\quad x\ge24\end{array}\right.[/tex]
To graph this function, let's follows these steps:
STEP 1: Graph [tex]y=x+16[/tex] from -16 to -5. This represents the graph of the trajectory of the airplane before touching the rainbow.
STEP 2: Graph [tex]y=-x^2+36[/tex] from -5 to 4. This is represents the graph of the rainbow as the airplane passes under its path.
STEP 3: Graph [tex]y=x+16[/tex] from 4 to 24. This is represents the graph of the trajectory of the airplane after touching the second point of the rainbow.
STEP 4: Graph [tex]y=x+16[/tex] from 24 and forward. This is represents the graph of the trajectory of the airplane when turning into an horizontal way.
Write an equation. Let x be the unknown number.
nine is seventeen more than two times a number
Answer:
Unknown number, x=-4
Step-by-step explanation:
Forming the equation from the information provided above.
9=2x+17
If we solve for the unknown number, we first collect like terms together.
2x=9-17
2x=-8
Divide both sides of the equal sign by 2 the coefficient of x
x=-4
Answer:
x = -4
Explanation:
We are given the following statement which we are to translate into a mathematical equation and then solve it:
'nine is seventeen more than two times a number'
Assuming [tex] x [/tex] to be the unknown number, this can be written as:
[tex] 9 = 2 x + 1 7 [/tex]
Solving for x:
[tex] 2 x = 9 - 1 7 [/tex]
[tex] 2 x = - 8 [/tex]
[tex]x=\frac{-8}{2}[/tex]
[tex]x=-4[/tex]
Therefore, the unknown number is -4.
Which transformation is not a rigid motion?
A. Translate 3 units left.
B. Rotate 360° counterclockwise around the origin.
C. Reflect over the x-axis.
D. Dilate by a factor of 2.