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]
Consider the following pair of equations:
y = −2x + 8
y = x − 1
Explain how you will solve the pair of equations by substitution. Show all the steps and write the solution in (x, y) form.
Answer:
(3,2)
Step-by-step explanation:
y = -2x + 8
y = x-1
Substitute x-1 for y in the equation y = -2x + 8. x-1 = -2x + 8
Use the Additive Property to move the
constant terms to one side of the equation. x-1 = -2x + 8
+1 +1
x = -2x + 9
Now do the same to the variable terms. +2x +2x
3x = 9
Divide both sides of the equation by the
constant in the variable term. ÷3 = ÷3
x = 3
To find y, simply substitute the value of x into
one of the original equations. y = x-1
y = 3-1
Solve: y = 2
The solution is (3, 2).
According to the bar graph, which teacher gave 16 minutes of homework on Wednesday?
A 12 foot ladder is placed against the side of the building. The base of the ladder is placed at an angle of 68° with the ground for secure footing. How far is the base of the ladder from the side of the building? Round to the nearest tenth of a foot.
Answer:
Step-by-step explanation:
A
What number is represented in the expanded notation below?
90,000,000 + 500,000 + 2,000 + 80 + 4 + 0.1 + 0.09
Answer:
90,502,084.19
Step-by-step explanation:
Just add them together
What is the area of a triangle with a base of 6mm and a height of 9mm
X/-2 - 4 < 2
what is x?
Answer:
x>-12
Step-by-step explanation:
x/-2-4<2
x/-2<2+4
x/-2<6
x<6(-2)
x<-12
x>-12
Write and expression for the phrase, "8 less than k".
PLEASE PLEASE PLEASE PLEASE HELP HELP HELP ME ME ME ME ME!!
Answer: k - 8
Step-by-step explanation:
The length of a rectangle is five more than triple the width. If the perimeter is 154 inches, what are the dimensions
Answer:
Width = 18 inches
Length = 59 inches
Step-by-step explanation:
The length of a rectangle is five more than triple the width. If the perimeter is 154 inches, what are the dimensions
Perimeter of a rectangle = 2L + 2W
The length of a rectangle is five more than triple the width.
L = Length = 5 + 3W
W = Width
If the perimeter is 154 inches, what are the dimensions
Hence,
154 = 2L + 2W
154 = 2(5 + 3W ) + 2W
154 = 10 + 6W + 2W
Collect like terms
154 - 10 = 8W
144 = 8W
W = 144/8
W = 18 inches
Length = 5 + 3W
Length = 5 + 3(18)
Length = 5 + 54
Length = 59 inches
An expression is shown.1608 ÷ 268 What is the value of the expression? When entering the answer, use numbers only.
Answer:
6
Step-by-step explanation:
We are to divide 1608 by 268
1608/268
4 can divide both the numerator and denominator. So divide the numerator and denominator by 4
This gives 402 / 67
Th 67 multiplication table
67 x 1 = 67
67 x 2 = 134
67 x 3 = 201
67 x 4 = 268
67 x 5 = 335
67 x 6 = 402
the value of the expression would thus be 6
Algebraically, determine the exact value of x for the
equation 85(2)0.4x = 6000.
Answer: 14,575
Step-by-step explanation:
85(2) 0.4x = 6000 Multiply 85 times 2
170 + 0.4x = 6000 Subtract 170 from both sides
0.4x = 5830 Divide 5830 by 0.4
x = 14,575
I need help solving this worksheet for corrections, I’m stuck.
Answer:
wasting ur time like u wasted mine
Step-by-step explanation:
Given ON, what is mABC ?
Answer:
personally, I would say try using the formula you were taught because there are different ways people do things
Step-by-step explanation:
formula is key
Could you guys please help me solve this basic math. PLS OMG ITS DUE :(
State whether the sequence is Arithmetic, Geometric, or Neither: 40, 43, 46, 49, ......
Answer:
t2-t1 =43-40
Step-by-step explanation:
you subtract the second term by the first term to get the common difference
how do you solve fire x.
Answer:
x = 10
Step-by-step explanation:
Sum of all the angle of triangle = 180
71 + 50 + 6x - 1 = 180
71 + 50 -1 + 6x = 180
120 + 6x = 180
6x = 180 - 120
6x = 60
x = 60/6
x = 10
7. An example for using estimation would be:
O O O O
A) counting back change
B) depositing money in the bank
C) guessing the number of pennies in a jar
D) counting the number of dollar bills in your wallet.
Ahhhh I need help now!Will mark brainlest! Please help with the question below:
Find the number set which satisfies each of the problems. If 3 is added to the absolute value of the product of a number and -9, the result is 4.
Answer:
10
Step-by-step explanation:
Answer:
either 1/9 or -1/9, they both work
Step-by-step explanation:
the problem is |-9x| + 3 = 4
-9x = 1
x = -1/9
and since it is absolute value you use both the positive and the negative...
hope this helps
Please help me I've been stuck on this one question and I can't find the answer to it.
For four Fridays in June, Grace earned $21.50, $29.75, $32.00, and $15.25 from babysitting.
In July, she earned $8 more for babysitting four Fridays than she did in June.
What is the increase in the mean for July compared to June?
Round the answer to the nearest penny.
Answer:
$2
Step-by-step explanation:
Its actually quite simple.
first find the mean for june.
add $21.50 + $29.75 + $32.00 + $15.25
that is equal to 98.5
then divide by 4.
that is 24.625
do not round to the nearest penny yet.
now you need to find the mean for july.
just add $8 to the total amount you made in june to find the total amount you made in july.
that is 106.5
then divide that by four because you worked four fridays in july as well.
that is 26.625.
finally you need to find the difference between 26.625 and 24.625.
26.625-24.625= 2
and ta-da!!!
the answer us $2.
your welcome. tell me if it is correct.
A magician's assistant makes $10.60 an hour. If he has already saved $53.70, how many hours must he work to save at least $165?
Answer:
Step-by-step explanation:
[tex]10.6h+53.7\ge 165\\ \\ 10.6h\ge 111.3\\ \\ h\ge 10.5[/tex]
A student is working a part time job delivering pizzas The student earns $40 a shift and earns $1.25 per pizza delivered. One shift, the student earned $58.75. How many pizzas did they deliver that shift?
Answer:
14
hope this helped
Answer:
i believe the student delivered 40 pizzas but i could be very wrong
Step-by-step explanation:
What kind of triangle is 8,14 and 12 and 25,16,20
A.not a triangle
B.right
C.acute
D. Obtuse
asapppppppppppppppppp
Answer: A not a triangle
Step-by-step explanation:
Find the solution to this equation - 3x + 32 - 7x = -2 (5x + 10)
Remember the steps
1. Distribute
2. Combine like terms on the left side
3 Pick sides variables = constants
4 Move terms that don't belong by doing the opposite operations
5 Decide what the statement means If it's always true write "infinitely many solutions." If it's always false, write "no solutions."
Answer:
[tex]x =\frac{-26}{3}[/tex]
Step-by-step explanation:
Step 1: Simplify both sides of the equation.
3x+32−7x=−2(5x+10)
3x+32+−7x=(−2)(5x)+(−2)(10)(Distribute)
3x+32+−7x=−10x+−20
(3x+−7x)+(32)=−10x−20(Combine Like Terms)
−4x+32=−10x−20
−4x+32=−10x−20
Step 2: Add 10x to both sides.
−4x+32+10x=−10x−20+10x
6x+32=−20
Step 3: Subtract 32 from both sides.
6x+32−32=−20−32
6x=−52
Step 4: Divide both sides by 6.
6x
6
=
−52
6
x=
−26
3
20 POINTS Find the diameter?? Pls help I dont know what to do..
Answer:
Diameter = 19.33
Step-by-step explanation:
Imaging a radius line from O to the endpoints of the 7. call this line R.
Label the part of the vertical line from O to the 90 degree intersection y.
Now you have a right triangle.
Using the Pythagorean theorem:
R² = 7² + y²
also
y = R - 3
substitute for y:
R² = 49 + (R-3)²
R² = 49 + R² - 6R + 9
simplify:
0 = 58 - 6R
6R = 58
R = 9.6667
Diameter = 2(9.6667) = 19.33
Lincoln is saving $360 to buy a new bike. he already has $85 and can earn $12 per hour at work. which of the following equation describes the number of hours, h, for which Lincoln must work to earn enough money to buy the new bike?
Please help
Answer:
its c
Step-by-step explanation:
I WILL GIVE BRAINLIEST!!!
NO SENDING ME LINKS!!!
Answer:
N(t)=52(9/7)^t
Step-by-step explanation:
hope it's helpful..
have a great day
Khaya is delivering groceries to his mother who stays 8km from the shop. How long will it take him to cover this distance if he drives at an average speed of 65km/h? Give your answer rounded to the nearest minute.
Answer:
7.385 minutes
Step-by-step explanation:
Given that :
Average speed = 65km/hr
The distance covered = 8km
The time taken Given the information above :
Recall:
Speed = distance / time
Time taken = distance / speed
Time taken = 8km / 65km/hr
Time taken = 0.1230769 hr
Time taken = 0.1230769 * 60 = 7.385 minutes
The time taken to cover this distance is 7.3848 minutes
What is speed?Speed is the change of an object's distance within a specified period of time.
Mathematically, speed can be calculated by using the relation:
Speed = distance/timeFrom the given information:
The average speed = 65 km/hThe distance = 8 kmFrom the above formula, we can make the time the subject of the formula:
Speed = distance/timeBy cross multiplying
Time = distance/speedTime = 8 km / 65 km/hTime = 0.12308 hourWe all know that:
60 seconds = 1 minute60 minutes = 1 hourThus, time(t) = (0.12308 × 60) minutes
Time(t) = 7.3848 minutes
Learn more about speed here:
https://brainly.com/question/4931057
Which point is on the graph of f(x) = 5^x
Answer:
B (1,5)
Step-by-step explanation:
A right rectangular prism has a length of 9, a width of 4, and a height of 5. How many unit cubes can fit in the prism?
Answer:
The answer is 54 cubes.
Step-by-step explanation:
I need this for bonus points !
1. What is the solution of the system
{3x - y = 14
{4x + y = 21 ?
A (−5, 3)
B (4, −2)
C (5, 1)
Answer:
C
Step-by-step explanation:
3x-y=14
4x+y=21
add
7x=35
x=35/7=5
4×5+y=21
y=21-20=1
so (5,1 ) is the solution.