Question:
Find the point (,) on the curve [tex]y = \sqrt x[/tex] that is closest to the point (3,0).
[To do this, first find the distance function between (,) and (3,0) and minimize it.]
Answer:
[tex](x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})[/tex]
Step-by-step explanation:
[tex]y = \sqrt x[/tex] can be represented as: [tex](x,y)[/tex]
Substitute [tex]\sqrt x[/tex] for [tex]y[/tex]
[tex](x,y) = (x,\sqrt x)[/tex]
So, next:
Calculate the distance between [tex](x,\sqrt x)[/tex] and [tex](3,0)[/tex]
Distance is calculated as:
[tex]d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}[/tex]
So:
[tex]d = \sqrt{(x-3)^2 + (\sqrt x - 0)^2}[/tex]
[tex]d = \sqrt{(x-3)^2 + (\sqrt x)^2}[/tex]
Evaluate all exponents
[tex]d = \sqrt{x^2 - 6x +9 + x}[/tex]
Rewrite as:
[tex]d = \sqrt{x^2 + x- 6x +9 }[/tex]
[tex]d = \sqrt{x^2 - 5x +9 }[/tex]
Differentiate using chain rule:
Let
[tex]u = x^2 - 5x +9[/tex]
[tex]\frac{du}{dx} = 2x - 5[/tex]
So:
[tex]d = \sqrt u[/tex]
[tex]d = u^\frac{1}{2}[/tex]
[tex]\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}[/tex]
Chain Rule:
[tex]d' = \frac{du}{dx} * \frac{dd}{du}[/tex]
[tex]d' = (2x-5) * \frac{1}{2}u^{-\frac{1}{2}}[/tex]
[tex]d' = (2x - 5) * \frac{1}{2u^{\frac{1}{2}}}[/tex]
[tex]d' = \frac{2x - 5}{2\sqrt u}[/tex]
Substitute: [tex]u = x^2 - 5x +9[/tex]
[tex]d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}[/tex]
Next, is to minimize (by equating d' to 0)
[tex]\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0[/tex]
Cross Multiply
[tex]2x - 5 = 0[/tex]
Solve for x
[tex]2x =5[/tex]
[tex]x = \frac{5}{2}[/tex]
Substitute [tex]x = \frac{5}{2}[/tex] in [tex]y = \sqrt x[/tex]
[tex]y = \sqrt{\frac{5}{2}}[/tex]
Split
[tex]y = \frac{\sqrt 5}{\sqrt 2}[/tex]
Rationalize
[tex]y = \frac{\sqrt 5}{\sqrt 2} * \frac{\sqrt 2}{\sqrt 2}[/tex]
[tex]y = \frac{\sqrt {10}}{\sqrt 4}[/tex]
[tex]y = \frac{\sqrt {10}}{2}[/tex]
Hence:
[tex](x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})[/tex]
Rachel has scored 98, 83, and 68 on her previous three tests. What score does she need on her next test so that her average (mean) is 82?
Answer:Write the equation of this geometric sequence and find the value of the 9th term.
1/3, 1, 3, 9...
F(n)=
F(9)=
Step-by-step explanation:
You can put this solution on YOUR website!
The total of Jenny's 5 tests = 375
If she wants an average of 77 then
her 6 tests must total 6 x 77 = 462
462 - 375 = 87 which is what Jenny
must score on the sixth test for an
average of 77.
:-)
Answer:
79
Step-by-step explanation:
Sum of 4 test for Average of 82 must be 82 x 4 which is 328.
Subtract previous 3 scores from 328:
328 - (98 + 83 + 68)
328 - 249 = 79
She needs a 79
I need help with my math homework it’s overdue someone please please help me
Answer: 2.5833
Step-by-step explanation:
K = 1/4 or .25
3/5 = .6
3 - 0.25 / .6 = 2.5833
What is the first step Anika made from the original equation.
Answer:
She factored out a 4 from the equation.
Step-by-step explanation:
So our original equation is 12x + 4 = 20x - 12
Anika's first steep of her equation is 3x + 1 = 5x - 3
We know that the terms were not moved around, so the first step was likely multiplication or division.
We also see that a 4 has been factored out of the entire equation, like this:
12x + 4 = 20x - 12
4(3x + 4) = 4(5x - 3)
And these 4's cancel each other out.
So, I conclude that Anika's first step was dividing the entire equation by 4 or factoring out a 4 from the equation.
WILL GIVE BRAINLIEST!
Given that ℓ1∥ℓ2 and the measure of ∠c = 44°, find the measures of angles a and b.
m∠a =
°
m∠b =
°
Answer:
<a = 44 degrees
<b = 136degrees
Step-by-step explanation:
From the given diagram;
<a = <c (alternate interior angle)
Since <c = 44 degrees, hence <a = 44 degrees
Also the sum of a and b is supplementary hence;
a + b = 180
44 + b = 180
b = 180 - 44
b = 136degrees
Hence m<b =136degrees
PLSSSSS HELPPPPPPPPPPPP THIS WAS SUPPOSED TO BE FOR
YESTERDAY PLSSSS SOMEONE URGENT!!!!!!!!!!!!!!!plssssssssssssssss
Answer:
C
Step-by-step explanation:
Can yall help me with this
WILL GIVE BRAINLIEST
Answer:
19 guests
Step-by-step explanation:
26 dollars = 13 guests
26/13 = 2
2*19 = 38
Final Answer: 19 guests
Hope this helps. :)
describe the following number patterns in your own words and state the constant difference or constant ratio if there is one
Step-by-step explanation:
awq-pzcq-mje
girls
come
to
see
my
22cm
sperm
fast
girl
Can you help? I'll give brainliest
Answer:
w=2 3/4
Step-by-step explanation:
If x-1= k and k=3, what is the value of x
3
Answer:
x-1/3 = k where k=3
We must be happy enough to find the value of only one valuable here, that is, x.
therefore, x-1/3=3
To find what is x-1, all we need to do is,
Take the denominator of the fraction to the answer (3)and multiply them. (that's how you'll find the values in the numerator)
So, you'll get,
x-1=3*3
x-1=9
To find the value of x,
The other number (-1) should be taken to the answer (9)
(Remember that the sign changes when the number crosses the equals sign)
therefore,
x= 9 + 1 (-1 will turn to +1 on crossing the equal sign)
Therefore , by addition, normally,
you'll get x=10
Hope this helps:)
can someone help me with these 2 questions. angle sum theorem
Answer: D. 8
Step-by-step explanation:
Total sum of angles in a triangle is 180
I'll do the first one, you can try the second
9x + 5 + 78 + 25 = 180
9x + 108 = 180
9x = 72
x = 8
what is 20 divided by 10 but to the power or 2
Which function does the graph represent?
3, 15)
Of(x) = (x + 3)(x + 7)
O f(x) = (x - 3)(x - 7)
O f(x) = 3(x – 3)(x – 7)
Of(x) = 11(x + 3)(x + 7)
Answer: imgurl=http%3A%2F%2Fwww.rasmus.is%2FIS%2FT%2FF%2FSu52k03_m01.gif&imgrefurl=http%3A%2F%2Fwww.rasmus.is%2Fuk%2Ft%2FF%2FSu52k03.htm&tbnid=0mYNOM8hE-8CkM&vet=12ahUKEwj4y9mb3NrvAhXNtFMKHZ9kCDEQMygIegUIARCjAQ..i&docid=Iv4G7-Ojzx0JrM&w=290&h=247&q=Which%20function%20does%20the%20graph%20represent%3F3%2C%2015)Of(x)%20%3D%20(x%20%2B%203)(x%20%2B%207)O%20f(x)%20%3D%20(x%20-%203)(x%20-%207)O%20f(x)%20%3D%203(x%20%E2%80%93%203)(x%20%E2%80%93%207)Of(x)%20%3D%2011(x%20%2B%203)(x%20%2B%207)&safe=active&ved=2ahUKEwj4y9mb3NrvAhXNtFMKHZ9kCDEQMygIegUIARCjAQ
what is s
7
+ 46
= 49
Answer:
s 7 + 46 = 49 is s = 7 √ 3 or s = 1.16993081 …
Step-by-step explanation:
Exact Form:
s = 7 √ 3
Decimal Form:
s = 1.16993081 …
What is the proportion for x
Which equation can be used to solve for c? Triangle A B C is shown. Angle A C B is 90 degrees and angle A B C is 35 degrees. The length of C B is 5 inches, the length of A C is b, and the length of B A is c.
Answer:
c = 5/Cos 35
Step-by-step explanation:
We are given the following
m<ACB = 90degrees
m<ABC = 35degrees
The side opposite to m<ABC is AC
BA = hypotenuse
CB = adjacent = 5inches
Required
Equation to solve BA which is c
Using the Pythagoras theorem;
Cos theta = Adjacent/Hypotenuse
Cos m<ABC = CB/BA
Cos m<ABC = 5/c
c = 5/cosm<ABC
c = 5/Cos 35
Hence the equation that can be used to solve for c is c = 5/Cos 35
Does anybody know the answer to this? (Also could I give the link to someone so they could help with the rest?)
Answer:
2x+168=166
2x= -2
x=-2
LMT= 94
Step-by-step explanation:
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!
Answer:
Its 268.08 hope this helps
Step-by-step explanation:
The correct answer and volume to this is 268.08 CM
The ages of the members of a volunteer group are shown below.
13, 14, 14, 14, 15, 15, 15, 16, 16, 21, 23
Which box and whisker plot best represents these data?
The graph of an equation is shown below:
Answer:
D, or (-3,2)
Step-by-step explanation:
Because the point (-3,2) is on the line of the equation shown, it is a solution. You can also notice that none of the other options are on the line and use the process of elimination.
Find the value of x degrees.
Answer:
X=25°
Step-by-step explanation:
[tex]x = \frac{1}{2} (t2 - 22) [/tex]
[tex] = \frac{1}{2} (50)[/tex]
[tex] = 25[/tex]
[tex]x = 25[/tex]
hope it helps u
Melanie invested $8,600 in an account paying an interest rate of 2\tfrac{7}{8}2 8 7 % compounded monthly. Oliver invested $8,600 in an account paying an interest rate of 2\tfrac{3}{4}2 4 3 % compounded daily. After 10 years, how much more money would Melanie have in her account than Oliver, to the nearest dollar?
Answer:
$139
Step-by-step explanation:
The formula for compound interest =
A = P(1 + r/n)^nt
Where
P = Principal
r = Interest rate
n = Compounding frequency
t = time in years
A = Amount after t years
For Melanie
Melanie invested $8,600 in an account paying an interest rate of 2 7/8% compounded monthly.
Rate = 2 7/8% = 23/8% = 2.875%
Time = 10 years
n = monthly = 12
First, convert R percent to r a decimal
r = R/100
r = 2.875%/100
r = 0.02875 per year,
Then, solve our equation for A
A = P(1 + r/n)^nt
A = 8,600.00(1 + 0.02875/12)^(12)(10
A = 8,600.00(1 + 0.002395833)^(120)
A = $11,460.64
For Oliver
Oliver invested $8,600 in an account paying an interest rate of 2 3/4 % compounded daily.
Rate = 2 3/4% = 11/4% = 2.75%
Time = 10 years
n = daily = 360 days
First, convert R percent to r a decimal
r = R/100
r = 1.375%/100
r = 0.01375 per year,
Then, solve our equation for A
First, convert R percent to r a decimal
r = R/100
r = 2.75%/100
r = 0.0275 per year,
Then, solve our equation for A
A = P(1 + r/n)^nt
A = 8,600.00(1 + 0.0275/360)^(360)(10)
A = 8,600.00(1 + 7.6389E-5)^(3600)
A = $11,322.05
After 10 years, how much more money would Melanie have in her account than Oliver, to the nearest dollar?
This is calculated as
$11,460.64 - $11,322.05
=$ 138.59
Approximately to the nearest dollar = $139
Answer:
139
Step-by-step explanation:
it’s right
2. FIND THE AREA OF THE SHADED
REGION. help me plzz !.!.!.
HELP ASAP PLAESE HELPP
Answer:
Ur right it's C.448
Step-by-step explanation:
I did it on goo.gle lol
hope this helps luv
Can someone help me please
Answer:
A. Point T
Step-by-step explanation:
Answer:
A- T (-4.5, 3)
Step-by-step explanation:
A- T (-4.5, 3)
Can someone help?
Solve the following equation for all values of x. Be sure to check for extraneous roots.
Answer:
[tex]x=2[/tex]
Step-by-step explanation:
First, let's multiply both sides by [tex]\sqrt{x+11}[/tex]. If we do this, we would have:
[tex]x-1=\frac{1}{\sqrt{x+11}}\cdot\sqrt{x+11}[/tex]
The [tex]\sqrt{x+11}[/tex] would cancel out and we would be left with:
[tex]x-1=1[/tex]
If we add 1 on both sides, we would get:
[tex]x=2[/tex]
Hope this helps!!!
Marcus mowed 1/2 of his parents yard in 2/3 of an hour. At this same rate, how much of the yard will he mow in an hour ?
Answer:0.5
Step-by-step explanation:
1/2 devide by 3 = 0.16666666666
0.16666666666 times 3 = 0.5
Given that x and y are positive integers and (x*2)+x(2^2)+(y*2^3)+(y*2^4)=42, what is the value of xy? PLEASE HELP!!!!!!
Note: Answer is not x=0, y=7 or y=0, x=7
Answer:
x = 3, y = 1
Step-by-step explanation:
If your question is this: 2x + 4x + 8y + 16y = 42
6x + 24y = 42
3x + 12y = 21
For x and y are positive integers, there is only 1 set of x = 3 and y = 1 is the solution (because x = 7 and y = 0 is excluded)
check: 6 + 12 + 8 + 16 = 42
7/15=x/45 find the missing value
Answer:
the answer to your question is 21
A bag contains all the letters in the word MATHEMATICS. You draw two letters, one at a time. What is the probability of drawing a T, replacing it, then drawing a C?
EXTRA POINTS -- Find the measure of the angles marked a, b, c, etc. in each diagram below