[tex]-(x^2+24x+144)=12y+12...[/tex]Answer:
(-12, 8)
Step-by-step explanation:
The standard form of this parabola, the one we can use to determine the vertex coordinates and the value of p is:
[tex](x-h)^2=4p(y-k)[/tex]
where h and k are the coordinates of the vertex and p is the distance between the vertex and the focus. We need that p value to determine how far above the vertex the focus is. In this case, the focus will lie on the same x-coordinate as the focus, we just need to find how far that distance away is. That requires us to do some algebraic gymnastics on that original equation. Putting it into vertex form.
Begin by multiplying everything by 12 to get rid of the pesky fraction:
[tex]12y=-x^2-24x-12[/tex]
Now we need to complete the square. The easiest way to do this is to have just the x terms on one side of the equals sign and everything else on the other side, so we will add 12 to both sides:
[tex]-x^2-24x=12y+12[/tex]
The leading coefficient when you complete the square has to be a positive 1; ours is a negative 1, so factor out the negative:
[tex]-(x^2+24x)=12y+12[/tex]
The rules for completing the square are as follows: Take half the linear term (ours is a 24), square that half, then add it into the parenthesis.
Half of 24 is 12 so
[tex]-(x^2+24x+144)=12y+12[/tex]
BUT...since this is an equation, if we add something to one side we have to add it to the other side too. BUT we didn't just add in a 144, we have to take into account the -1 sitting outside the parenthesis that will not be ignored. So we didn't add in 144, we added in -1(144) which is -144.
[tex]-(x^2+24x+144)=12y+12-144[/tex]
What we have done on the left by completing the square is to create a perfect square binomial. Rewriting it as such and combining like terms on the right:
[tex]-(x+12)^2=12y-132[/tex]
Don't forget the purpose of this is to find the value of p. We're almost there. On the right, factor out a 12:
[tex]-(x+12)^2=12(y-11)[/tex]
From this we can determine the coordinates of the vertex and the value of p. The vertex sits at (-12, 11).
The equation for p is 4p = 12 so p = 3
That means that the focus is 3 units below the vertex on the same x coordinate. The focus then is at (-12, 8)
Determine the rate of change and what the rate of change represents in this situation.
Answer:
The first answer is the one you want
Step-by-step explanation:
The rate of change is the slope. Here it is represented by the dollar value/number of tickets sold. This will give you the 1:1 ratio, meaning it will give you the number of dollars generated by the sale of 1 ticket. That's what rate of change is.
Use the slope formula and 2 points on the table. I chose the points (225, 250) and (200, 0):
[tex]\frac{0-250}{200-225} =\frac{-250}{-25} =10[/tex]
That translates to $10 per ticket.
Mrs. Paulson bought chicken wire to enclose a rectangular garden. She is restricted to a width of no more than 30 feet. She would like to use at most 180 feet of chicken wire. This situation can be represented by a system of inequalities, where x = the width of the chicken wire and y = the length of the chicken wire. Identify two possible combinations for the width and length of the chicken wire in order to make her a rectangular garden. Create a system of linear inequalities.
Answer:
Let l = length and w = width of the rectangular garden
=> w < 30
and 2(l + w) ≤ 180
=> l + w ≤ 90
=> 0 < w < 30 and 60 ≤ l < 90.
Step-by-step explanation:
A wall map is 45 cm high and 27 cm wide. Ashley wants to proportionately shrink it so its height is 12 cm. How wide would it be then?
Answer:
To shrink the height it has to be shrunk by:
27 / x = 12
x = 2.25 times
The width would be 45 / 2.25 =
20 centimeters.
Step-by-step explanation:
Answer:
7.2 cm wide
Step-by-step explanation:
Set it up as a proportion.. (45/27) = (12/x)
Cross multiply and solve for x.
A snack food company packs 5 cookies in each box. Let b represent the number of boxes and c represent the total number of cookies.
Answer:
5*B=C
Step-by-step explanation:
Five cookies times the number of boxes will give you the total number of cookies
A principal of $3100 is invested at 5.5% interest, compounded annually. How much will the investment be worth after 7 years?
Answer:
$4501.51
Step-by-step explanation:
Because you're compounding annually, which is only once per year, you can use a simple formula:
[tex]A(t)=P(1+r)^{t}[/tex]
Filling in our info gives us
[tex]A(t)=3100(1+.055)^7[/tex]
Do the adding inside the parenthesis and then raise 1.055 to the 7th power to get
A(t)= 3100(1.454679161)
A(t)= $4509.51
Find the domain and range of the graph below:
Answer:
Domain is all real numbers; Range is all numbers less than or equal to 0
Step-by-step explanation:
Domain covers x values, range covers y values. The domain of an x^2 parabola, which is what this is, has a domain of all real numbers. Meaning that while the branches of the function keep going up and up and up or down and down and down, the values of x will never stop growing.
The range here is indicative of the lowest y value to the highest that the function covers. In this case, since the parabola is upside down, we have the highest to the lowest. The highest that the function goes up the y axis is right at the origin, where y = 0. Then it drops down into forever. So the range is all values of y less than or equal to 0
The Domain of [tex]f(x)[/tex] is all the real numbers, and
The Range of [tex]f(x)[/tex] is [tex]y\le0[/tex] or [tex]f(x)\le0[/tex]
The diagram shows the graph of the quadratic function
[tex]f(x)=-x^2[/tex]
The domain of [tex]f(x)[/tex] is all the real numbers, since the function has defined values for all real values of [tex]x[/tex].
The range of [tex]f(x)[/tex] is the set of values that [tex]f(x)[/tex] can assume. The square function has a range
[tex]\{y \text{ }|\text{ }y=x^2\text{ and }y\ge0\}[/tex] or the half-open interval [tex][0,\infty)[/tex].
This means that the negative of the square function will have the range
[tex]\{y \text{ }|\text{ }y=-x^2\text{ and }y\le0\}[/tex] or the half-open interval [tex](-\infty,0][/tex].
So, the domain of [tex]f(x)[/tex] is the open interval [tex](-\infty,\infty)[/tex] (all the real numbers), and the range of [tex]f(x)[/tex] is the half-open interval [tex](-\infty,0][/tex] (or, [tex]y\le0[/tex])
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Which is equivalent to 216^1/3
For this case we must find an expression equivalent to:
[tex]216 ^ {\frac {1} {3}[/tex]
So, we can rewrite the 216:
[tex]216 = 6 * 6 * 6 = 6 ^ 3[/tex]
Rewriting the expression:
[tex](6 ^ 3) ^ {\frac {1} {3}=[/tex]
By definition of power properties we have:
[tex](a ^ n) ^ m = a ^ {n * m}[/tex]
So:
[tex]6 ^ {\frac {3} {3}} =\\6 ^ 1 =\\6[/tex]
Answer:
[tex]216 ^ {\frac {1} {3}}= 6[/tex]
The equivalent expression of [tex]216^{1/3}[/tex] is determined as 6.
What is simplification of an expression?Simplification refers to the process of reducing an expression, equation, or fraction into its simplest or most concise form.
The equivalent expression is determined by converting the exponents into roots as follows.
The given expression;
[tex]216^{1/3}[/tex]
This expression is simplified as follows;
[tex]216^{1/3}[/tex] = ∛ 216
The final expression becomes;
6³ = 216
So 6 is the answer
Thus, the equivalent expression of [tex]216^{1/3}[/tex] is determined as 6.
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Solve the triangle.
A = 32°, a = 19, b = 14
Answer:
A = 32°, a = 19, b = 14, B=22.98°, C = 125.02°, c = 29.36
Step-by-step explanation:
We have two sides of the triangle and we have an angle.
A = 32 °, a = 19, b = 14
We use the sine theorem to find the angle B.
We know that according to the sine theorem it is true that:
[tex]\frac{sin(A)}{a}=\frac{sin(B)}{b}=\frac{sin(C)}{c}[/tex]
[tex]\frac{sin(32\°)}{19}=\frac{sin(B)}{14}[/tex]
[tex]sin(B)=14*\frac{sin(32\°)}{19}\\\\B=Arcsin(14*\frac{sin(32\°)}{19})\\\\B=22.98\°[/tex]
We know that the sum of the internal angles of a triangle is always equal to 180.
So:
[tex]C=180-32-22.98\\\\C=125.02\°[/tex]
Finally we find the c side
[tex]\frac{sin(A)}{a}=\frac{sin(C)}{c}[/tex]
[tex]\frac{sin(32\°)}{19}=\frac{sin(125.02)}{c}[/tex]
[tex]0.02789=\frac{sin(125.02)}{c}[/tex]
[tex]c=\frac{sin(125.02)}{0.02789}\\\\c=29.36[/tex]
You drop a ball from a height of 10ft. Each bounce the ball gains only 90% of its height back. How high does the ball bounce on its 6th bounce? Starting amount: decay rate: equation: answer:
Answer:
6th bounce: 40%
Starting amount: 100%
Decay rate: 10%
Equation: 100% - 10% - 10% - 10% - 10% - 10% - 10% =
Step-by-step explanation:
100% - 10% - 10% - 10% - 10% - 10% - 10% = 40%. 40% is the answer.
Using an exponential function, it is found that on the 6th bounce, the ball bounces up 5.31 ft.
What is an exponential function?A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
A(0) is the initial value.r is the decay rate, as a decimal.In this problem:
You drop a ball from a height of 10ft, hence A(0) = 10.Each bounce the ball gains only 90% of its height back, hence r = 0.1.Then, the height after the nth bounce is given by:
[tex]A[n] = 10(0.9)^n[/tex]
After the 6th bounce, we have that:
[tex]A[6] = 10(0.9)^6 = 5.31[/tex]
On the 6th bounce, the ball bounces up 5.31 ft.
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I am running out of points!! Please help me!!
ANSWER
[tex]165.6 \degree[/tex]
EXPLANATION
The measure of arc AC corresponds to 22% + 24%=46%
The complete angle of the circle is 360°.
Therefore the measure of arc AC is
[tex] = \frac{46}{100} \times 360[/tex]
This simplifies to
[tex]165.6 \degree[/tex]
The correct answer is A.
In ⊙L, m∠NMO=9x−3 and m∠NPO=4x+12. Find mNO.
Answer:
arc NO has measure 48
Step-by-step explanation:
We assume all measures are in consistent units (degrees or something similar). The two inscribed angles intercept the same arc, so are congruent:
9x -3 = 4x +12
5x = 15 . . . . . . . add 3-4x
x = 3 . . . . . . . . . divide by 5
The measure of the inscribed angle is then ...
4x +12 = 4(3) +12 = 24
That is half the measure of the arc, so the measure of arc NO is ...
arc NO = 2·24 = 48
Answer:
48°
Step-by-step explanation:
I'm actually just assuming that you mean arc NO. Proceeding with that...
Angle NMO is an inscribed angle which intercepts arc NO. Angle NPO is also an inscribed angle that intercepts arc NO. Because they both intercept the same arc, both inscribed angles have the same measure. Therefore,
9x - 3 = 4x + 12
Solving for x:
5x = 15
x = 3. Plug 3 in for x in either one of the equations to get that angles NMO and NPO measure
9(3) - 3 = 24°
The rule is that inscribed angles measure HALF of the arcs they intercept, so the measure of arc NO is 48°
if m AD=98 and m CD=120 Whats is m
Answer:
The measure of arc BC is 44°
Step-by-step explanation:
we know that
m arc AD+m arc DC+m arc BC+m arc AB=360° ----> by complete circle
substitute the given values and solve for m arc BC
98°+120°+m arc BC+98°=360°
m arc BC+316°=360°
m arc BC=360°-316°=44°
Answer:
109Step-by-step explanation:
Please help me with this!!!!!!!!!!
Answer:
True
Step-by-step explanation:
The incentre is equally distant from the triangles 3 sides and like centroids is always inside the circle.
Two rectangles have the same width. One is 12 units long and the other is 8 units long. The area of the first rectangle is 320 square units more than the area of the second rectangle. Find the width of each rectangle.
Answer:
80
Step-by-step explanation:
let w = the width of the rectangles
then
12w = the area of one rectangle
8w = the area of other
:
12w - 8w = 320
4w = 320
w = 320/4
w = 80 units wide
;
:
Check
12 * 80 = 960
8 * 80 = 640
--------------
difference: 320
We know that both rectangles have the same width, which is 80 units.
How to find the width of each rectangle?
We know that both rectangles have the same width, so we can say that both have the width W.
One of the rectangles has a length of 12 units, and the other of 8 units, so the areas are:
A = 12*W
A' = 8*W
And the largest area is 320 square units more than the other area, so we have:
12*W = 8*W + 320
Solving for W we get:
12*W - 8*W = 320
4*W = 320
W = 320/4 = 80
Then the width of each rectangle is 80 units.
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I legit hate Word problems they are so stupid, so if anyone can help me that would be great, thank you
Suppose that a man standing at the edge of a cliff near the North Rim of the Grand Canyon is looking downward towards a campground inside the canyon. The elevation of the North Rim is 5389 ft and the elevation of the campground is 2405 ft. The man's range finder indicates that his line of sight distance to the campground is 3044 ft.
What is the angle of depression of the man's line of sight to the campground?
Express your answer in degrees rounded to the nearest hundredth.
Enter your answer in the box.
To find the angle of depression, we can use trigonometry. The elevation of the North Rim and the campground, along with the line of sight distance, can be used to calculate the angle using the tangent function.
Explanation:To find the angle of depression of the man's line of sight to the campground, we can use trigonometry. The angle of depression is the angle between the line of sight and a horizontal line. In this case, the elevation of the North Rim is 5389 ft and the elevation of the campground is 2405 ft. The line of sight distance to the campground is 3044 ft.
We can use the tangent function to find the angle of depression. Tangent(theta) = opposite/adjacent. The opposite side is the difference in elevation between the North Rim and the campground (5389 ft - 2405 ft) and the adjacent side is the line of sight distance (3044 ft). So, tangent(theta) = (5389 ft - 2405 ft) / 3044 ft.
Using a scientific calculator, we can find the value of theta by taking the inverse tangent (or arctan) of the ratio: theta = arctan((5389 ft - 2405 ft) / 3044 ft). The answer is approximately 57.38 degrees.
Please help! Will mark brainiest!!!!
Answer:
This is the alternate exterior angle theorem
Step-by-step explanation:
This is because angle 2 and angle 7 are in the outer part of each side of opposite lines
Suppose that a laser light, positioned 100 ft from the base of a flag pole, illuminates a flag that is 85 ft above the ground.
What is the angle of inclination (angle of elevation) of the light beam?
Express your answer in degrees rounded to the nearest hundredth.
Enter your answer in the box.
Answer:
The angle of elevation is [tex]40.36\°[/tex]
Step-by-step explanation:
Let
[tex]\theta[/tex] ----> the angle of elevation
we know that
The tangent of an angle is equal to divide the opposite side to the angle by the adjacent side to the angle
we have that
[tex]tan(\theta)=\frac{85}{100}[/tex]
[tex]\theta=arctan(\frac{85}{100})=40.36\°[/tex]
Please please help me
Answer:
a) Acute
Step-by-step explanation:
To clasify the triangle as Acute, Right, or Obtuse, we use the converse of the pythagorean theorem and see if c^2 is greater, less than, or equal to b^2+a^2.
So 16^2 ? 11^2 + 12^2
Since 11^2 +12^2 = 265 and 16^2 is 256, c^2 is less than b^2+a^2. So the triangle is acute.
Answer:
a) Acute
Step-by-step explanation:
If 11 and 12 were the legs of a right triangle, the length of the hypotenuse would be ...
√(11² +12²) = √(121 +144) = √265 ≈ 16.3
The actual length of the longest side is shorter, so the opposite (largest) angle measure will be less than 90°. The triangle is acute.
PLEASE HELP SHOW ALL YOUR WORKING OUT MARK BRAINLIEST
Answer:
y=1/1 x+6
Step-by-step explanation:
y=mx+b
m=slope
b=y-intercept
the slope from one point to another is rise up one and go the the right one, therefore the slope of the line is 1/1.
The y-intercept is 6 which is shown on the graph
Two consecutive perfect squares have a difference of $99$. what is the value of the larger perfect square?
Answer:
2500
Step-by-step explanation:
Let the first square number be [tex]x^2[/tex] then the next square number is [tex](x+1)^2[/tex].
The difference between these two consecutive numbers is 99.
This implies that:
[tex](x+1)^2-x^2=99[/tex]
We expand to get:
[tex]x^2+2x+1-x^2=99[/tex]
Simplify:
[tex]2x=99-1[/tex]
[tex]2x=98[/tex]
Divide both sides by 2
[tex]x=49[/tex]
Therefore the value of the larger perfect number is
[tex](49+1)^2=50^2=2500[/tex].
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
What is the inverse of h?
Answer: [tex]\bold{D)\quad h^{-1}(x)=\dfrac{1}{6}(x-1)}[/tex]
Step-by-step explanation:
Inverse is when you swap the x's and y's and then solve for y
y = 6x + 1
x = 6y + 1 swapped the x's and y's
x - 1 = 6y subtracted 1 from both sides
[tex]\dfrac{1}{6}[/tex] (x - 1) = y divided both sides by 6
What is the value of s if 8.25s - 2.375 = 10 ?
Answer:
s = 1.5
Step-by-step explanation:
10 + 2.375 = 12.375 = 8.85
12.375/8.25 = 8.25s/8.25
s = 1.5
What is the solution of log((3x + 4))4096 = 4? ( the 3x+4 is like an exponent but lower
[tex]\bf \textit{exponential form of a logarithm} \\\\ \log_a b=y \implies a^y= b\qquad\qquad a^y= b\implies \log_a b=y \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \log_{3x+4}(4096)=4\implies \stackrel{\textit{exponential form}}{(3x+4)^4=4096}\implies (3x+4)^4=2^{12} \\\\\\ \stackrel{~\hfill \textit{same exponents, the bases must be the same}}{(3x+4)^4=2^{3\cdot 4}\implies (3x+4)^4=(2^3)^4}\implies 3x+4=2^3\implies 3x+4=8 \\\\\\ 3x=4\implies x=\cfrac{4}{3}[/tex]
The double number lines show the ratio of minutes to days. How many minutes are in 222 days? minutes
2880
Step-by-step explanation:
Final answer:
To find out how many minutes are in 222 days, multiply the number of days (222) by the number of hours in a day (24) and then by the number of minutes in an hour (60). The result is 319,680 minutes.
Explanation:
The student is asking how many minutes are in 222 days. We begin the conversion using the provided information:
There are 24 hours in 1 day
There are 60 minutes in 1 hour
Now, we just need to multiply the number of days by the number of hours in a day, and then by the number of minutes in an hour, to get our answer.
222 days × 24 hours/day × 60 minutes/hour = 319,680 minutes
There are 319,680 minutes in 222 days.
A ladder leans against a building. The angle of elevation of the ladder is 70 degrees. The top of the ladder is 25ft from the ground.
To the nearest tenth of a foot, how long is the ladder?
Using law of sines:
Sin(angle) = opposite leg (height) / hypotenuse ( length of ladder)
Sin(70) = 25/x
x = 25 * sin(70)
x = 26.6 feet
Find function of f(-1)
f(-1) means the x value is -1 and you need to find what the Y value is.
Find -1 in the set of parenthesis and see what the Y value is.
When x = -1, y = 3
The answer would be 3.
A manager samples the receipts of every fifth person who goes through the line. Out of 50 people, 6 had a mispriced item. If 1,600 people go to this store each day, how many people would you expect to have a mispriced item?
Answer:
48 people
Step-by-step explanation:
1. 4/50 = x/600 do 4x600 and 50 times x
2.50x=2400
3. divide both sides by 50 to get a result of x=48.
write the expression in complete factored form 5u(n+6)+x(n+6)
Answer:
[tex](n+6)(5u+x)[/tex]
Step-by-step explanation:
The given expression is;
[tex]5u(n+6)+x(n+6)[/tex]
The greatest common factor is (n+6).
Let us factor the GCF to obtain:
[tex](n+6)(5u+x)[/tex]
Therefore the completely factored form of
[tex]5u(n+6)+x(n+6)[/tex]
is
[tex](n+6)(5u+x)[/tex]
Help please I don’t know and I really need to finish
Answer:
2) First option
3) Second option
Step-by-step explanation:
5² = 625
13 × 8 × 5 = 520
Subtract (5x + 3) from (9x – 7).
A. 4x – 4
B. 4x + 10
C. 4x – 10
D. –4x + 10
D is the correct answer
the result of subtracting (5x + 3) from (9x - 7) is 4x - 10. Therefore, the correct option is C: 4x - 10.
To subtract (5x + 3) from (9x - 7), we need to distribute the negative sign to all the terms in (5x + 3) and then perform the subtraction term by term.
(9x - 7) - (5x + 3)
= 9x - 7 - 5x - 3
Now, combine like terms:
= (9x - 5x) + (-7 - 3)
= 4x - 10
So, the result of subtracting (5x + 3) from (9x - 7) is 4x - 10. Therefore, the correct option is C: 4x - 10.
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