Answers
Answer:
4733
Step-by-step explanation:
Please refer to the attached diagram.
Point A can be assigned x-coordinate "p". Then its y-coordinate is 6p^2. The slope at that point is y'(p) = 12p.
Point B can be assigned x-coordinate "r". Then its y-coordinate is 6r^2. The slope at that point is y'(r) = 12r.
We want the slopes at those points to have a product of -1 (so the tangents are perpendicular). This means ...
(12p)(12r) = -1
r = -1/(144p)
The slope of line AB in the diagram is the ratio of the differences of y- and x-coordinates:
slope AB = (ry -py)/(rx -px) = (6r^2 -6p^2)/(r -p) = 6(r+p) . . . . simplified
The slope of AB is also the tangent of the sum of these angles: the angle AC makes with the x-axis and angle CAB. The tangent of a sum of angles is given by ...
tan(α+β) = (tan(α) +tan(β))/1 -tan(α)·tan(β))
__
Of course the slope of a line is equal to the tangent of the angle it makes with the x-axis. The tangent of angle CAB is 2 (because the aspect ratio of the rectangle is 2). This means we can write ...
slope AB = ((slope AC) +2)/(1 -(slope AC)(2))
[tex]6(p+r)=\dfrac{12p+2}{1-(12p)(2)}\\\\3(p+r)(1-24p)=6p+1\qquad\text{multiply by $1-24p$}\\\\3\left(p-\dfrac{1}{144p}\right)(1-24p)=6p+1\qquad\text{use the value for r}\\\\3(144p^2-1)(1-24p)=144p(6p+1)\qquad\text{multiply by 144p}\\\\ 3456 p^3+ 144 p^2+ 24 p+1 =0\qquad\text{put in standard form}\\\\144p^2(24p+1)+(24p+1)=0\qquad\text{factor by pairs}\\\\(144p^2+1)(24p+1)=0\qquad\text{finish factoring}\\\\p=-\dfrac{1}{24}\qquad\text{only real solution}\\\\r=\dfrac{-1}{144p}=\dfrac{1}{6}[/tex]
So, now we can figure the coordinates of points A and B, and the distance between them. That distance is given by the Pythagorean theorem as ...
d^2 = (6r^2 -6p^2)^2 +(r -p)^2
d^2 = (6(1/6)^2 -6(-1/24)^2)^2 +(1/6 +1/24)^2 = 25/1024 +25/576 = 625/9216
Because of the aspect ratio of the rectangle, the area is 2/5 of this value, so we have ...
Rectangle Area = (2/5)(625/9216) = 125/4608 = a/b
Then a+b = 125 +4608 = 4733.
_____
Comment on the solution
The point of intersection of the tangent lines is a fairly messy expression, and that propagates through any distance formulas used to find rectangle side lengths. This seemed much cleaner, though maybe not so obvious at first.
Related Questions
. Francisco states the following: When two or more factors are multiplied and they have the same sign, the product is positive. List two examples that support Francisco's statement. Show their products.
Your answer
6B. List an example that refutes Fracisco's statement. Show its product. Explain why it refutes his statement.
Your answer
6C. How could Francisco rewrite his statement so it is always true?
pls Help me
Answers
Answer:
6A) i) 5 and 3 product is 15
ii) (-5) and (-3) product is 15
6B. The product of 5·i and 3·i is -15
6C. When two or more real factors are multiplied and they have the same sign, the product is positive
Step-by-step explanation:
6A. To examples that support Francisco's statement are;
Given the factors that have the same sign;
i) 5 and 3 both positive
ii) (-5) and (-3) both negative
The products of each of the two factors are therefore;
5 × 3 = 15 (Positive product)
(-5) × (-3) = 15 (Positive product)
6B. An example that refutes Francisco's claim is the product of two imaginary numbers 5·i and 3·i where i = √(-1) as follows;
5·i × 3·i = 15·i² = 15×i×i = 15 × √(-1) ×√(-1) = 15×(√(-1))² = 15 × (-1) = -15
Therefore;
5·i × 3·i = -15 which is a negative number and therefore refutes Francisco's claim
6C. Francisco could rewrite his statement as follows;
When two or more real factors are multiplied and they have the same sign, the product is positive
To make it always true.
Find the value of b. Round your answer to the nearest tenth.
The figure shows acute triangle A B C. The measure of angle B is 40 degrees. The length of side A B is 10. The length of side B C is 12. The length of side C A is b.
Answers
Answer:
Side CA = 7.8
Step-by-step explanation:
Given:
Acute angled [tex]\triangle ABC[/tex].
[tex]\angle B =40^\circ[/tex]
AB = 10
BC = 12
We can use cosine rule here to find the side AC = b
Formula for cosine rule:
[tex]cos B = \dfrac{a^{2}+c^{2}-b^{2}}{2ac}[/tex]
Where
a is the side opposite to [tex]\angle A[/tex]
b is the side opposite to [tex]\angle B[/tex]
c is the side opposite to [tex]\angle C[/tex]
[tex]cos 40 = \dfrac{12^{2}+10^{2}-b^{2}}{2\times 12\times 10}\\\Rightarrow cos 40 = \dfrac{144+100-b^{2}}{240}\\\Rightarrow 0.77 = \dfrac{244-b^{2}}{240}\\\Rightarrow 244-b^{2} = 0.77 \times 240\\\Rightarrow 244-b^{2} = 183.85\\\Rightarrow 244-183.85 = b^{2}\\\Rightarrow b^2 = 60.15\\\Rightarrow b = 7.76[/tex]
To the nearest tenth b = 7.8
n chemistry, the pH of a solution is defined by: pH space equals space minus log space left square bracket straight H plus right square bracket, where H+ is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a pH of approximately 7. A solution with a pH under 7 is called an acid, and one with a pH over 7 is called a base. Tomato juice has a hydrogen ion concentration of 10 to the power of negative 4.1 end exponent mole per liter. Find the pH of tomato juice. pH space equals space 4.9 pH space equals space 2.8 pH space equals space 3.5 pH space equals space 4.1
Answers
Answer:
The correct option is;
pH = 4.1
Step-by-step explanation:
The given information are;
The hydrogen ion [H⁺] concentration in tomato juice = [tex]10 ^{-4.1}[/tex]
The pH of a solution = -㏒{H⁺] = -㏒( [tex]10 ^{-4.1}[/tex]) = -(-4.1) × ㏒(10) = 4.1
The pH of the of the tomato juice = 4.1
The pH is the term used to refer to the acidity of water with the range of pH values ranging from 0 to 14 where pH values of a solution lesser than 7 are considered acidic while pH values above 7 indicate a basic solution.
) We throw 9 identical balls into 7 bins. How many different ways are there to distribute these 9 balls among the 7 bins such that no bin is empty? Assume the bins are distinguishable (e.g., numbered 1 through 7).
Answers
Answer:
28 ways
Step-by-step explanation:
After placing 1 ball in each of the seven bins, there are two balls left.
If we place both balls in a single bin, there are 7 different ways to place the balls (place both on bins 1 through 7).
If we place each of the remaining balls in a different bin, the number of ways to place the balls is:
[tex]n_2=\frac{7!}{(7-2)!2!}=7*3=21[/tex]
The total number of ways to distribute those balls is 21 + 7 = 28 ways.
What is the period of the function y= 3/2 cot(3/5x) +5?
A. Pi/5 units
B. 3pi/5 units
C. 2pi/3
D. 5pi/3
Answers
Answer:
D; 5pi/3 units
Step-by-step explanation:
Here, we want to find the period of the function;
y = 3/2 cot (3/5x) + 5
By definition, the period of a function is the interval between two matching points in the function.
Let’s say it is the distance between two peaks, crests, etc on a function.
To find the value of the period. We shall standardize the function.
What this means is that we shall be writing the function in the standard form.
The standard form is as follows;
f(x) = A trig(Bx -C) + D
Where trig refers to the accompanying trigonometric function in question.
Comparing this standard form with our question, we can see that;
A is 3/2
B is 3/5
C is 0
D is 5
Now for cot and tan functions, we shall need to divide pi by the absolute value of B
Thus we have; pi divided by 3/5 which gives 5pi/3 units
Answer:
D. 5pi/3, period is the distance between the repetition of a function.
Step-by-step explanation:
How might you describe the y values for this section of x values: -1
Answers
Answer:
Answer is below
Step-by-step explanation:
The y-values are all real numbers because they reach every possible y-value from -∞ to ∞ on the y-axis.
Please help me I will give BRAINLIST and a like
I rlly need help
Answers
Answer:
?=3
Step-by-step explanation:
a= 3/5 b - 7
Answer:
the number that should be in the green box is 3
Step-by-step explanation: since it is three-fifths, in numerical way, it should be written as 3/5..
7 less than three fifths of b is a gives 3/5b-7=a
Sam bakes a variety of biscuits. 1/3
are peanut, 12 are raisin and the
remaining 12 were oat. How many
biscuits are baked?
Answers
Answer: 36
Step-by-step explanation:
Let the no. of biscuits be x
Peanuts will be 1/3x
So,
1/3x + 12 + 12 = x
1/3x + 24 = x
x + 72 = 3x
x-3x = -72
x = 72/2 = 36
Peanut biscuits will be 1/3 * 36 = 12
Total no. of biscuits baked were 12+12+12 = 36
15 POINTS!!!!! suppose f(x)=x find the graph of f(x+2) please include what the graph would look like
Answers
Answer:
The graph f(x+2) would make the graph go left by 2 units.
Step-by-step explanation:
Using graph transformations, adding two would make the graph go left 2 and subtracting would make it go right 2
(a) Which of the following has only one subset? (1)
(i) {2} (ii) { } (iii) {Ǿ} (iv) None of these
Answers
Answer:
Option (i)
Step-by-step explanation:
{2} has only 1 subset i.e. {2} and no other subset. While { } or ∅ has no subset.
the length of a rectangle is twice its width, the perimeter of it is 36cm. what is the are of it
Answers
Answer: 72 cm²
Step-by-step explanation: lets take the width as x and length as 2x as length is twice the width= perimeter of rectangle is 2*(l+b)=36cm
So, 2×(2x+x)=36cm
2×3x=36cm
6x=36cm
X=36/6:6cm( width, since width was taken as x)
Length= 2x: 2×6=12cm
So area of the rectangle is( l*b):
12×6= 72cm²
someone help me asap please/math 10
Answers
Answer:
Step-by-step explanation:
4. a) tan x=17/12=1.416
x=54.8≈55
b)sin b=78/132=0.59
b=36.2≈36
cos 28=x/82
x=82*cos 28=82*0.9=73.8
HELP ME PLEASE HELP ME HELP ME PLEASE
Answers
Answer:
no, because the remainder is 126Step-by-step explanation:
if x+3 is a factor, then -3 is a root of expression, and the remainder would be 0
calculating remainder:
[tex]-3(-3)^3+6(-3)^2+6(-3)+9=-3\cdot(-27)+6\cdot9-18+9=\\\\=81+54-9=126[/tex]
I really need help with this question, it's confusing. It's geometry btw
Answers
Answer:
Two regular nonagons and nine congruent rectangle
Step-by-step explanation:
Two regular nonegon=on the surface and base..
Nine congruent rectangle=look at beside it will be look like 9 rectangle.. I hope it will be right.
11. Three men can build a wall in 10 hours. How
many men would be needed to build the wall in
7 hours?
Answers
Answer:
Man-hours to build the wall: 3 * 10 = 30
:
let m = no. of men required to do it 7.5 hrs
7.5m = 30
m = [tex]\frac{30}{7.5}[/tex]
m = 4 men to build it in 7.5 hrs
Step-by-step explanation:
help help meeeee pls
Answers
When x= 12, y= 144. (a) Express y in terms of x. (b) Find the value of y when x= 7.
Answers
Answer:
see explanation
Step-by-step explanation:
Given x = 12, y = 144
We can see that y is 12 times the value of x, thus
(a)
y = 12x
(b)
when x = 7, then
y = 12 × 7 = 84
A soccer player gets 2 points for a goal and 1 point for an assist. If the combined number of goals and assists that a player has is 24, and the total number of points that the player has is 42, which system of equations can be used to determine the number of goals and assists the player has? Assume g represents the number of goals and a represents the number of assists.
A)g + a = 24. g + 2 a = 42.
B)g + a = 24. 2 g + a = 42.
C)g + a = 42. g + 2 a = 24.
D)g + a = 42. 2 g + a = 24.
Answers
Answer:
B
Step-by-step explanation:
The soccer player has g goals and a assists.
For every goal, he/she gets 2 points and for every assist, he/she earns 1 point. This means that if we multiply g by 2, we get the number of points the player receives from g goals and if we multiply a by 1, we get the number of points the player receives from a assists.
And, that total number of points is equal to 42, so we can write:
2 * g + 1 * a = 42
2g + a = 42
Now, we understand that g and a are the number of goals and assists the player has made, and we also know that the total number of such kicks is 24, so we can write:
g + a = 24
Thus, our system is:
g + a = 24
2g + a = 42
The answer is B.
~ an aesthetics lover
Answer:
The correct answer is B. g + a = 24. 2 g + a = 42
4.8x10^-3 as an ordinary number
Answers
Answer:
0.008.
Step-by-step explanation:
10^-3 basically tells you to move the decimal point to the left by three digits.
Right now, you have 8. If the decimal point were to move left by three digits, you can visualize 8 as 0008. Move left by three digits, and you get 0.008.
Hope this kind of helps!
Answer:
0.008.
hope this helps :)
Help please solve this
Answers
RU is a diameter, so the two equations make half a circle which is 180 degrees.
24x+5 + 10x+5 = 180
Combine like terms:
34x + 10 = 180
Subtract 10 from both sides:
34x = 170
Divide both sides by 34
X= 5
RS= UV
RS = 10x +5
Replace x with 5
RS = 10(5) + 5 = 55
Answers:
X = 5
RS = 55
In a sale normal prices are reduced by 10% Angelina bought a pair of shoes in the sale for £54 what was the original pricee of the shoes?
Answers
Answer:
£60
Step-by-step explanation:
Given that
The cost after reduction = £54
So, 90% = £54
We have to find 100%
To do so, find 10% and ,multiply that by 10
90% = £54
10% = 54/9 = £6
1005 = 6x10 = £60
Hope this helps,.
Please mark brainiest
In the diagram, the measure of angle 6 is 98°. what is the measure of angle 7?
Answers
Answer:
7∠82°
Step-by-step explanation:
Well angle 6 and 7 are complementary angles, meaning they both add up to 180°.
So we do 180 - 98 which is 82°.
Answer: The measure of angle 7 is 82 degrees.
Step-by-step explanation:
Angle 6 and 7 lies on a straight line so they will add up to 180 degrees.
So if Angle 6 is 98 degrees then an a number plus 98 has to equal 180.
so we could generate an equation as x + 98 = 180
x + 98 = 180 solve for x
-98 -98
x= 82
A group of 4 friends have a bag of 49 sweets. They divide the sweets equally between them. a) How many sweets does each friend get? b) How many sweets are left over
Answers
Answer:
a) Each friend gets 12 sweets
b) There is 1 sweet left
Step-by-step explanation:
When you divide 49 by 4 you will get 12 with a remainder 1.
49 ÷ 4 = 12 R1
The remainder is the left over while the total result is how much each friend gets.
Answer:
A) 12
B) 1
Step-by-step explanation:
49 cannot be divided equally among the 4 friends but 48 can be divided.
A) So,
48/4 = 12 sweets
Each friend gets 12 sweets
B) 1 sweet is left
A page in an average newspaper has 8 columns of print. Each column consists of 160 lines and each line averages 6 words. What's the average number of words on a full page?
Answers
Answer:
7,680
Step-by-step explanation:
A page has 8 columnsEach column consists of 160 lines.Each line averages 6 words.The average number of words on a full page
=Number of columns X Number of Lines X Number of words per line
=8 X 160 X 6
=7680
The average number of words on a full page is 7,680.
a line has a gradient of 4 and passes through the post (1,7). what is the equation?
Answers
Answer:
y = 4x + 3
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Here m = 4, thus
y = 4x + c ← is the partial equation
To find c substitute (1, 7) into the partial equation
7 = 4 + c ⇒ c = 7 - 4 = 3
y = 4x + 3 ← equation of line
Answer:
The equation is y = 4x+3
Step-by-step explanation:
The slope intercept form of a line is
y = mx+b where m is the slope and b is the y intercept
y = 4x + b
Substitute the x and y value into the equation
7 = 4(1)+b
7 = 4+b
Subtract 4
7-4 =b
3=b
The equation is y = 4x+3
−6(x+7)=−4x−2 how many solutions
Answers
Answer: There is one solution, -20
Step-by-step explanation:
[tex]-6(x+7)=-4x-2\\\\Distribute\\\\-6x-42=-4x-2\\\\Add(6x)\\\\-42=2x-2\\\\Add(2)\\\\-40=2x\\\\x=-20[/tex]
There is one solution, -20
Hope it helps <3
Answer:
Only one solutionStep-by-step explanation:
Solution,
[tex] - 6(x + 7) = - 4x - 2[/tex]
Distribute -6 through the parentheses
[tex] - 6x - 42 = - 4x - 2[/tex]
Move variable to L.H.S and change its sign
[tex] - 6x + 4x - 42 = - 2[/tex]
Move constant to R.H.S and change its sign
[tex] - 6x + 4x = - 2 + 42[/tex]
Collect like terms
[tex] - 2x = - 2 + 42[/tex]
Calculate the sum
[tex] - 2 x = 40[/tex]
Divide both sides of the equation by - 2
[tex] \frac{ - 2x}{ - 2} = \frac{40}{ - 2} [/tex]
Calculate
[tex] x = - 20[/tex]
Hope this helps..
Good luck on your assignment..
NEED MATH HELP NOW. Need help find the vertex and y intercept. Please show work.
Answers
Answer:
Vertex: ( 1 , 9 )
Y-intercept: ( 0 , 8 )
Step-by-step explanation:
y = - (x+2) (x-4)
y = -x² + 2x + 8
Find the vertex.
x = -b/2a
x = -2/2(-1)
x = -2/-2
x = 1
y = -(1)² + 2(1) + 8
y = -1 + 2 + 8
y = 9
Find the y-intercept.
Put x as 0.
y = -(0)² + 2(0) + 8
y = 8
Here were 87 sunflowers at the flower shop in the morning. There were 56 sunflowers left at the end of the day. How many sunflowers were sold? Explain a way to solve the problem.
Answers
Answer:
31
Step-by-step explanation:
We just have to calculate 87 - 56 which is 31 so the answer is 31 sunflowers.
Answer:
31
Step-by-step explanation:
Since we know that we started with a higher number than we ended with, it is obvious that this is a subtraction problem. Then, we simply have to find the difference by subtracting 87 by 56 (87 - 56 = x). After the calculation, we see that the answer is 31 (87 - 56 = 31).
-3.42, 10.26, -30.78, 92.34, ___,... what is the next number
Answers
Answer:
-277.02
Step-by-step explanation:
Given series:
-3.42, 10.26, -30.78, 92.34,...We can see the series is a GP with common ratio of -3, as:
10.26/-3.42= -30.78/10.26= 92.34/-30.78= -3So the next term will be:
92.34 × (-3)= -277.02cube of -1/2 is equal to-----------. i need urgent please.I will mark brainest
Answers
Answer:
-1/8
Step-by-step explanation:
(-1/2)(-1/2)(-1/2)=-1/8
Answer:
[tex] \frac{ - 1}{8} [/tex]solution,
[tex] (\frac{ - 1}{2} ) ^{3} \\ = \frac{( { - 1)}^{3} }{( {2}^{3} )} \\ = \frac{ - 1 \times ( - 1) \times ( - 1)}{2 \times 2 \times 2} \\ = - \frac{ 1}{8} [/tex]
hope this helps...
Good luck on your assignment