Proved that if the function f(x,y)=e^(ax+by) with a^2+b^2=1, then f''(xx)+f''(yy)=1
To find the second partial derivatives of f(x, y), we differentiate f(x, y) twice with respect to x and y:
f'(x, y) = ae^(ax + by) and f''(xx) = a^2e^(ax + by)
f'(x, y) = be^(ax + by) and f''(yy) = b^2e^(ax + by)
Adding these second partial derivatives, we get:
f''(xx) + f''(yy) = a^2e^(ax + by) + b^2e^(ax + by)
Since a^2 + b^2 = 1, we have:
f''(xx) + f''(yy) = e^(ax+by)
But f(x, y) = e^(ax+by), so:
f''(xx) + f''(yy) = f(x, y)
Therefore, we have proved that f''(xx) + f''(yy) = 1.
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The design of a digital box camera maximizes the volume while keeping the sum of the dimensions at 7 inches. If the length must be 1.1 times the height, what should each dimension be?
To optimise the volume while maintaining the total dimensions at 7 inches, the camera's dimensions should be roughly 3.01 inches besides 3.311 inches besides 0.679 inches.
What 3 dimensions do we have?
The homes we reside in and the items we use on a daily basis all have three dimensions: length, weigth, and breadth.
Let's start by assigning variables to the dimensions. Let x be the height of the camera, then the length must be 1.1 times the height, which gives us a length of 1.1x.
The width is not explicitly given, but we can express it in terms of x and 1.1x. Since the sum of the dimensions is 7 inches, we have:
x + 1.1x + w = 7
where w is the width of the camera. Simplifying this equation, we get:
2.1x + w = 7
w = 7 - 2.1x
Now we can express the volume of the camera in terms of x:
V = x(1.1x)(7 - 2.1x)
Simplifying this expression, we get:
V = 8.235x³ - 15.365x² + 7x
To find the maximum volume, we need to find the value of x that maximizes V. We can do this by taking the derivative of V with respect to x, and setting it equal to zero:
dV/dx = 24.705x² - 30.73x + 7 = 0
Using the quadratic formula, we can answer this quadratic equation:
x = (-b ± √(b² - 4ac)) / 2a
where a = 24.705, b = -30.73, and c = 7. Plugging in these values, we get:
x = 0.735 inches or x = 3.01 inches
Since x represents the height of the camera, we discard the smaller root and take x = 3.01 inches.
Then the length is 1.1 times the height, which gives us a length of 3.311 inches.
The width can be found using the equation w = 7 - 2.1x, which gives us w = 0.679 inches.
Therefore, the dimensions of the camera should be approximately 3.01 inches by 3.311 inches by 0.679 inches to maximize the volume while keeping the sum of the dimensions at 7 inches.
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there are black,green,yellow counters in a bag in the ratio 3:10:7
there are 105 yellow counters
how many black counters are there
Create a rational expression that simplifies to 2x/(x+1)
and that has the following restrictions on x:
x ≠ −1, 0, 2, 3. Write your expression here.
-Contains multiplication of two rational
expressions
-Contains division of two rational expressions
Answer:
One possible expression that meets the given requirements is:
(2x)/(x+1) = (2x/[(x-2)(x-3)]) / ([(x+1)/(x-2)(x-3)])
This expression simplifies to 2x/(x+1) when x is not equal to -1, 0, 2, or 3, as required.
Explanation: We can rewrite 2x/(x+1) as (2x/(x-2)(x-3)) * ((x-2)(x-3)/(x+1)). The first term in this expression is a division of two rational expressions, while the second term is a multiplication of two rational expressions. Then, we can simplify the first term by cancelling the (x-2)(x-3) terms in the numerator and denominator, which gives 2x/[(x-2)(x-3)]. We can also simplify the second term by expanding the denominator, which gives (x-2)(x-3)/(x-2)(x-3)(x+1) = 1/[(x-2)(x-3)] * 1/(x+1). Then, we can combine the two simplified terms to get the expression given above.
Step-by-step explanation:
Please ASAP Help
Will mark brainlest due at 12:00
Answer:
(0,-2)
Step-by-step explanation:
What is the best prediction for the number of times the spinner will not land on an elephant?
We might estimate that 0.8n is the approximate prediction number of times it won't land on the elephant.
We need to know the likelihood that the spinner will land on an elephant as well as the total number of spins in order to anticipate the number of times it won't.
Assume the spinner contains five equal parts, one of which is decorated with an elephant. The likelihood of the spinner touching the elephant is then 1/5, or 0.2.
If we know the total number of spins, we can use the likelihood that the spinner will fall on an elephant to calculate the likelihood that it won't. 1 - 0.2 = 0.8 is the probability that the spinner will miss the elephant.
Hence, if we suppose that the spinner is spun n times, we may estimate that 0.8n is the approximate number of times it won't land on the elephant. The prediction of probability of the spinner landing on the elephant is assumed to be constant for each spin and that each spin is independent of the others.
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Which segment is perpendicular to DE?
C. DF
a. AB
d. EF
b. CF
The required perpendicular segment to DE is EF.
What is perpendicular?In simple geometry, two geometric objects are perpendicular if the intersection at the place of intersection known as a foot results in right angles. The perpendicular symbol can be used to graphically depict the condition of perpendicularity.
According to question:In the given diagram we can see that segment DE and EF.
So, we can say that,
Segment DE is perpendicular to segment EF.
Thus, required perpendicular segment to DE is EF.
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Complete question:
how many integers are between 48 and 172
The number of integers between two given integers is 124.
The two given integers are 48 and 172.
We know that, number of integers between two integers = Difference of two integers
Here, number of integers = 172-48
= 124
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Rebecca buys some scarves that cost $5 each and 2 purses that cost $12 each. The cost of Rebecca’s total purchase is $39. What equation can be used to find n, the number of scarves that Rebecca buys?
1.5 + 24n = 39
2.5n + 24 = 39
3.(24 + 5) n = 39
4.24 + n = 39
The equation [tex]24 + n = 39[/tex] may be used to get n, the quantity of scarves Rebecca purchases.
What sort of equation would that be?The concept of an equation in algebra is a logical statement that demonstrates the equality of two mathematical equations. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
What does a basic equation mean?A formula that describes how contain a copy on either sides of a symbol are connected. It typically has an equal sign and one variable. The variable x is included in the preceding example.
The cost of one scarf is $[tex]5[/tex]
The cost of one purse is $[tex]12[/tex]
Rebecca buys 2 purses, so the cost of the purses is [tex]2*12=24[/tex]
The total cost of the purchase is $[tex]39[/tex]
[tex]Total cost = Cost of scarves + Cost of purses[/tex]
[tex]39 = 5n + 24[/tex]
Now we can solve n,
[tex]39 - 24 = 5n[/tex]
[tex]15 = 5n[/tex]
[tex]n = \frac{15}{5}[/tex]
[tex]n=3[/tex]
Therefore, Rebecca buys [tex]3[/tex] scarves.
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Nancy has twice as many apples as jay. Jay has 3 more apples than Ava. Nancy has 22 apples. How many apples dose Ava have?
Answer:
Step-by-step explanation:
nancy: 22
jay: 11 cause nancy has twice as many apples as jay.
ava has 8 apples cause jay has 3 more apples than ava :)
HEEEELLLLLLLPPPPPP MEEEEEEEEEEEE PLS
1. 10-7 13. -4+(-7)
2. 5-(-9) 14. -5+5
3. -14-9 15. 13+(-1)
4. -2+(-1) 16. 14+(-1)
5. 12-(-12) 17. 9+(-5)
6. 9+(-3) 18. -3+4
7. 7-5 19. 7+(+2)
8. 9-(-7) 20. 15+(-7)
9. -16-(-8) 21. -5+(-5)
10. 3-5 22. -11. 4
11. -6+13 23. 26-(-1)
12. 8+(-3) 24. 7-(-4)
Answer:
Step-by-step explanation:
Simplify 10−7 to 3
3,13.−4−7
simplify 13.- 4 13. - 4
3,13.−11
You drive 180 miles and your friend drives 150 miles in the same amount of time. Your
average speed is 10 miles per hour faster than your friend's speed. Write and use a
rational model to find each speed
Your speed is 60 miles per hour, when your friends average speed is 50 miles.
A reasonable model is what?A rational function, which is a function that can be represented as the ratio of two polynomials, is a function that may be used in a rational model, which is a mathematical model. Rates of change, growth or decay, and proportions are only a few examples of the many various kinds of real-world events that may be represented using rational models. In order to describe complicated systems or processes, they are frequently employed in disciplines like economics, physics, and engineering. To determine the values of the variables that make the equation true, rational models can be solved using algebraic techniques including factoring, simplification, and cross-multiplication.
Given that, average speed is 10 miles per hour faster than other person.
Then, your speed is = s + 10.
For 180 miles, and 150 miles for friend the equation can be set as:
180/(s+10) = 150/s
We can cross-multiply to simplify:
180s = 150(s+10)
180s = 150s + 1500
30s = 1500
s = 50
Substituting the value in s + 10 = 50 + 10 = 60.
Hence. your speed is 60 miles per hour.
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If a vendor sells single roses for R1,20 each, he can sell 350 roses. However, he will sell three fewer roses for each 20c increase in the price. If the price is increased to R1,80 per rose, then the vendor's revenue (to the nearest cent) will be equal to
The vendor's revenue at a price of R1.80 per rose is R9.72
Let's assume that the vendor sells x roses at a price of p per rose.
According to the given condition, if he sells roses at R1.20 per rose, he can sell 350 roses. Hence, we can write:
350 = x - 3((p - 1.20)/0.20)
Simplifying the above equation, we get:
x = 3(p - 0.60) + 350
Now, if the price is increased to R1.80 per rose, the vendor's revenue will be:
Revenue = number of roses sold * price per rose
= (x - 3((1.80 - 1.20)/0.20)) * 1.80
= (x - 15) * 1.80
Substituting the value of x from the first equation, we get:
Revenue = (3(p - 0.60) + 350 - 15) * 1.80
= (3p - 0.30) * 1.80
Simplifying the above equation, we get:
Revenue = 5.4p - 0.54
Therefore, the vendor's revenue at a price of R1.80 per rose is given by the equation 5.4p - 0.54, where p is the price per rose in rands.
To find the vendor's revenue to the nearest cent, we need to substitute p = 1.80 in the above equation:
Revenue = 5.4(1.80) - 0.54
= R9.72 (approx.)
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The function h is defined by the following rule.
h(x) = 4x+5
Complete the function table.
X
-5
0
3
4
5
X
0
0
0
0
5
Answer:
-15, 5, 17, 21, 25
Step-by-step explanation:
See attached worksheet.
Calculate each of the values of x on the table by using the number in place of the x in the equation.
h(x) = 4x + 5
h(-5) = 4*(-5) + 5
h(-5) = -15
===
h(3) = 4*(3)+5
h(3) = 12 + 5 or 17
==
Do the other values of x in the same manner to produce the attached table.
Select the correct answer. What is this expression in simplified form? √32 . 5√2
Answer: The answer is 40
Explanation: The picture
A triangle has an angle that measures 35°. The other two angles are in a ratio of 13:16. What are the measures of those two angles?
Answer:
Let's denote the measures of the other two angles by 13x and 16x, where x is a constant.
Since the sum of the angles in a triangle is always 180 degrees, we can write an equation:
35 + 13x + 16x = 180
Simplifying this equation, we get:
29x = 145
x = 5
Therefore, the measures of the other two angles are:
13x = 13(5) = 65 degrees
16x = 16(5) = 80 degrees
So, the measures of the other two angles are 65 degrees and 80 degrees.
Step-by-step explanation:
an urn consists of 20 red balls and 30 green balls. we choose 10 balls at random from the urn (without replacement). what is the probability that there will be exactly 4 red balls among the chosen balls?
The probability of selecting exactly 4 red balls from an urn with 20 red balls and 30 green balls is given by the formula P(X=4) = (20C4)(30C6) / (50C10) where X is the number of red balls chosen. This simplifies to P(X=4) = 0.2032.
if x:y=3:4 and y:z=1:3 find x:z
The value of x:z = 1:4.
To find the ratio of x to z, we need to have a common term between x, y, and z. Since we have y in both ratios, we can use it as the common term.
From the given ratios:
x:y = 3:4
y:z = 1:3
We can see that the y in the first ratio and the y in the second ratio are the same.
To find the ratio of x to z, we can "connect" the two ratios by canceling out the y:
x:y = 3:4
y:z = 1:3
x:y × y:z = 3:4 × 1:3
x:z = 3:12
Simplifying the above ratio by dividing both terms by 3, we get:
x:z = 1:4
Alex wants to fence in an area for a dog park. he has plotted three sides of the fenced area at the points e (3, 5), f (6, 5), and g (9, 1). he has 22 units of fencing. where could alex place point h so that he does not have to buy more fencing? (0, 0) (−1, 0) (0, −3) (0, 3)
We found that he can only use approximately 7.39 units of fencing for the fourth side. The possible locations for point H are (4.5, 12.39) and (4.5, -2.39).
To determine where Alex could place point H, we need to first calculate the length of the three sides of the fenced area that he has already plotted. We can do this using the distance formula:
Distance [tex]= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
From point E to point F:
Distance = [tex]\sqrt{(6-3)^2 + (5-5)^2}[/tex]
= 3 units
From point F to point G:
distance = [tex]\sqrt{(9-6)^2 + (1-5)^2)}[/tex]
= [tex]\sqrt{(9 + 16)}[/tex]
= 5 units
From point G to point E:
Distance = [tex]\sqrt{(3-9)^2 + (5-1)^2}[/tex]
= [tex]\sqrt{(36 + 16)}[/tex]
= [tex]\sqrt{(52)[/tex]
= [tex]2 \sqrt{(13)}[/tex] units
Therefore, the total length of fencing that Alex has already plotted is 3 + 5 + 2√13 = approximately 14.61 units.
To find the possible x-coordinate(s) of point H, we can use the formula for the distance between two points on the coordinate plane:
distance = [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
For point H to be 7.39 units away from the midpoint of side EF (4.5, 5), we have:
[tex]\sqrt{(x - 4.5)^2 + (y - 5)^2}[/tex]
= 7.39
Squaring both sides and simplifying gives:
(x - 4.5)² + (y - 5)²= 7.39²
Since we know that x=4.5, we can substitute this value into the equation and solve for y:
(4.5 - 4.5)²+ (y - 5)² = 7.39²
(y - 5)² = 7.39²
y - 5 = ±7.39
y = 5 ± 7.39
So the possible locations for point H are (4.5, 5+7.39) and (4.5, 5-7.39), which correspond to the points (4.5, 12.39)
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3. A yard plan includes a rectangular garden that is surrounded by bricks. In the drawing, the garden is 7 inches
by 4 inches. The length and width of the actual garden will be 35 times larger than the length and width in the
drawing.
(a) What is the perimeter of the drawing? Show your work.
(b) What is the perimeter of the actual garden? Show your work.
(c) What is the effect on the perimeter of the garden with the dimensions are multiplied by 35? Show
your work.
For the rectangle, the answers will be a. Perimeter=52 inches, b. Perimeter=770 inches and c. Perimeter will be multiplied by 35 also.
What exactly is a rectangle?
A rectangle is a four-sided flat shape with opposite sides that are parallel and equal in length. It is a type of quadrilateral, a polygon with four sides.
Now,
(a) The perimeter of the drawing is the sum of the lengths of all sides of the rectangular garden plus the lengths of the two bricks on the top and bottom and the lengths of the two bricks on the left and right.
The length of the garden in the drawing is 7 inches, and the width is 4 inches. Thus, the perimeter of the garden in the drawing is:
P = 2(7 inches) + 2(4 inches) = 14 inches + 8 inches = 22 inches
Since the garden is surrounded by bricks, we need to add the lengths of the two bricks on the top and bottom and the lengths of the two bricks on the left and right. Each brick has a length of 1 inch, so the total length of the bricks is:
2(1 inch + 7 inches + 1 inch) + 2(1 inch + 4 inches + 1 inch) = 2(9 inches) + 2(6 inches) = 18 inches + 12 inches = 30 inches
Therefore, the perimeter of the drawing is:
22 inches + 30 inches = 52 inches
(b) The actual garden is 35 times larger than the drawing, so its length is 35 × 7 inches = 245 inches, and its width is 35 × 4 inches = 140 inches. Thus, the perimeter of the actual garden is:
P = 2(245 inches) + 2(140 inches) = 490 inches + 280 inches = 770 inches
(c) When the dimensions of the garden are multiplied by 35, the perimeter of the garden is also multiplied by 35. This is because the perimeter is a linear function of the length and width of the garden. Therefore, the effect on the perimeter of the garden is to increase it by a factor of 35.
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The volume of the cylinder?
Step-by-step explanation:
Volume of a cylinder = πr²h
[tex]\pi = 3.14[/tex]
[tex]r = \frac{d}{2} [/tex]
[tex]d = 14[/tex]
[tex]r = \frac{14}{2} [/tex]
[tex]r = 7[/tex]
[tex] {r}^{2} = 49[/tex]
[tex]h = 12[/tex]
substitute the formular with the values above
[tex]3.14 \times 7 \times7 \times 12 = 1846.32[/tex]
To one decimal place
[tex] = 1846.3[/tex]
Answer[tex]1846.3[/tex]
Many students from Europe come to the United States for their college education.
From 1980 through 1990, the number S(in thousands), of European students
attending a college or university in the U. S. Can be modeled by
S=0. 05(t3 - 11ť + 45t +277), where t = Ocorresponds to 1980.
In what year were there 31. 35 thousand European students attended a U. S. College
or university?
we can use the cubic equation formula to solve for t. The answer is t=1988, which means that there were 31.35 thousand European students in 1988.
To solve this problem, we need to find the value of t that corresponds to 31.35 thousand European students. We can use the given equation S=0.05(t3-11t+45t+277) and solve it for t. To do this, we can first subtract 277 from both sides to get S-277=0.05(t3-11t+45t). Then, we can divide both sides by 0.05 to get (S-277)/0.05=t3-11t+45t. Finally, we can use the cubic equation formula to solve for t. The answer is t=1988, which means that there were 31.35 thousand European students in 1988.
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(write the slope-intercept form of the equation of each line) helllpppppp- pls-
Answer:
(-1,0) and (0, 4)
Step-by-step explanation:
The x and y intercept of a function (say the x intercept), is when y = 0 and vice versa.
The x intercept in this function is when y = 0
Thus,
4x-0=-4
4x = -4
x = -1.
So the x- intercept point = (-1, 0)
Similarly,
4(0)-y= -4
(x is equated to 0 to find the y - intercept)
-y = -4
∴ y = 4
y- intercept point = (0, 4)
Hope this helps! :)
Sid mows two laws the first one is 60 feet long and 59 feet wide the area of the second lawn is 3 times as big but has the same width as the first want is the length of the second lawn
The length of the second lawn that Sid mows is 60 feet. We simply calculated the area of lawn here.
The area of the first lawn is given by:
A1 = 60 x 50 = 3000 square feet
Let L2 be the length of the second lawn. Since the width of the second lawn is the same as the first lawn, the area of the second lawn can be expressed as:
A2 = L2 x 150
Since the area of the second lawn is 3 times as big as the first lawn, we have:
A2 = 3A1
Substituting A1 and simplifying, we get:
L2 x 150 = 3 x 3000
L2 x 150 = 9000
L2 = 60 feet
Therefore, the length of the second lawn is 60 feet.
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helppp please
HELP
[tex]5555[/tex]
Answer: C
Step-by-step explanation:
For A, look at the slope of the graph: 70/2 = 35
For B, slope calculation: m = y-y / x-x = 315-105 / 9-3 = 35
=> C
Slope models the direction and steepness of a line, while the y-intercept defines the starting point. Explain what following equation of a line represents y= -2/3x + 6.
can someone please answer and explain this problem to me?
The equation y = (-2/3)x + 6 represents a line that starts at the point (0, 6) on the y-axis and slopes downwards from left to right at a rate of 2 units down for every 3 units to the right.
What is co-ordinate geometry ?Coordinate geometry is a branch of mathematics that deals with the study of geometry using the principles of algebra. It involves using algebraic equations and geometric concepts to analyze shapes and figures in a plane or in higher dimensions.
According to given information:In this equation, -2/3 is the slope of the line, which tells us how steep the line is and in what direction it's heading. Specifically, a slope of -2/3 means that for every increase of 3 units in the x-direction, the y-value decreases by 2 units. So the line slopes downwards from left to right.
The 6 in the equation is the y-intercept of the line, which tells us where the line intersects the y-axis. Specifically, the y-intercept is the point (0, 6) on the line. This means that when x = 0, y = 6, so the line starts at the point (0, 6) on the y-axis.
Therefore, the equation y = (-2/3)x + 6 represents a line that starts at the point (0, 6) on the y-axis and slopes downwards from left to right at a rate of 2 units down for every 3 units to the right.
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In a race, 29 out of the 50 swimmers finished in less than 39 minutes. What percent of swimmers finished the race in less than 39 minutes? Write an equivalent fraction to find the percent.
Answer:
percentage = (part/whole) x 100
where the "part" is the number of swimmers who finished in less than 39 minutes, and the "whole" is the total number of swimmers.
So, if 29 out of 50 swimmers finished in less than 39 minutes:
percentage = (29/50) x 100
percentage = 58
Therefore, 58% of swimmers finished the race in less than 39 minutes.
An equivalent fraction to represent 58% is 29/50.
Step-by-step explanation:
6(t+2) I need work to be shown
Answer:
6t + 12
Step-by-step explanation:
6(t+2)
= 6t + 12
So, the answer is 6t + 12
1. Find the exact value of each of the following: a. cos(sin−1(1)) b. tan(cos−1(−23 )) 2. Find the exact value of each of the following: a. sin(cos−1(32)) b. csc(tan−1(−51)) 3. Find the exact value, in terms of a, of each of the following. (HINT: Don't let the variable a scare you! The same strategy you used in Problem 2 still works - draw a point on a circle, assign it coordinates, and find the appropriate trig function's value.) a. sin(tan−1(a2)) b. cot(cos−1(a))
The cot(cos−1(a))= a/√(1-a^2).
Value of cos(sin−1(1)): Let's assume that θ= sin−1(1). It means that sinθ= 1. The value of θ is π/2, therefore we can say that cos(sin−1(1))=cos(π/2)= 0b) Value of tan(cos−1(-23)): Let's suppose that θ= cos−1(-23). It implies that cosθ= -23/25. As we know that tanθ= sinθ/cosθ, Therefore;tan(cos−1(−23))= sin(θ)cos(θ) = -24/25.2. a) Value of sin(cos−1(3/2)): Let's suppose that θ= cos−1(3/2). It implies that cosθ= 3/2. As we know that sin^2θ= 1- cos^2θ, Therefore;sin(cos−1(3/2))= √(1- (3/2)^2)= √(1/4)= 1/2.b) Value of csc(tan−1(-5/1)): Let's assume that θ= tan−1(-5/1). It implies that tanθ= -5/1. As we know that cscθ= 1/sinθ, Therefore;csc(tan−1(−51))= 1/sin(θ)= 1/√(1+tan^2θ) = 1/√(1+25)= -1/√26.3. a) Value of sin(tan−1(a^2)): Let's suppose that θ= tan−1(a^2). It implies that tanθ= a^2. As we know that sinθ= tanθ/√(1+tan^2θ), Therefore;sin(tan−1(a2))= (a^2)/√(1+(a^4)).b) Value of cot(cos−1(a)): Let's suppose that θ= cos−1(a). It implies that cosθ= a. As we know that cotθ= cosθ/sinθ, Therefore;cot(cos−1(a))= a/√(1-a^2).
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What is the minimum number of rows of bricks needed to build a wall that is at least 4 feet tall if each brick is 2.25 inches tall?
Answer:
22 rows
Step-by-step explanation:
Divide 48 by 2.25 and round up
What is a counter example to the conditional statement? If an odd number is greater than 1 and less than 10, then it has no other factors than 1 and its self
The factors of 9 are 1 and 9.. To determine this, one can use the formula to find the factors of a number, which is n = a × b.
A counter example to the conditional statement "If an odd number is greater than 1 and less than 10, then it has no other factors than 1 and its self" would be the number 9. 9 is an odd number that is greater than 1 and less than 10, but it has other factors than 1 and itself. The factors of 9 are 1, 3, and 9. To determine this, one can use the formula to find the factors of a number, which is n = a × b, where n is the number, a is the first factor, and b is the second factor. In this case, n = 9, a = 1, and b = 9. Therefore, the factors of 9 are 1 and 9.
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