what is the following product 3 sqrt 2(5 sqrt 6-7 sqrt 3)
Answers
Answer:
[tex](30\sqrt{3}-21\sqrt{6})[/tex]
Step-by-step explanation:
We have to find the product of the expressions.
[tex]3\sqrt{2}(5\sqrt{6}-7\sqrt{3})\\[/tex]
= [tex]3\sqrt{2}\times 5\sqrt{6}-7\sqrt{3}\times 3\sqrt{2}[/tex] ( By distributive rule)
= [tex]3\times 5(\sqrt{2})(\sqrt{6})-7\times 3(\sqrt{3})(\sqrt{2})[/tex]
= [tex]15\sqrt{12}-21\sqrt{6}[/tex]
= [tex]15\sqrt{4\times 3}-21\sqrt{6}[/tex]
= [tex]30\sqrt{3}-21\sqrt{6}[/tex]
= [tex](30\sqrt{3}-21\sqrt{6})[/tex]
Related Questions
Two mechanics worked on a car. the first mechanic worked for 10 hours, and the second worked for 15 hours. together they charged a total of 1800. what was the rate charged per hour by each mechanic if the sum of the two rates was $155 per hour ?
Answers
Total money earned:
[tex]10x + 15y = 1800[/tex]
Sum of hourly rates:
[tex]x + y = 155[/tex]
To solve this, you solve for one equation first, substitute it in the second, then input it back into the first equation. I will first be solving for x.
[tex]x + y = 155 \\ x = 155 - y[/tex]
Input this into the other equation to find y:
[tex]10(155 - y) + 15y = 1800 \\ 1550 - 10y + 15y = 1800 \\ 5y = 250 \\ y = 50[/tex]
Input y into the first equation:
[tex]x + 50 = 155 \\ x = 105[/tex]
The first mechanic charges $105 per hour and the second mechanic charges $50 per hour.
The rates charged per hour by the first and second mechanics will be $105 and $50.
What is a linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
It is given that two mechanics worked on a car, the first mechanic worked for 10 hours, and the second worked for 15 hours. together they charged a total of 1800.
Suppose x is the hourly rate for the first mechanic and y is the hourly rate of the second.
If the total money earned is 1800.
10x + 15 y = 1800 ---- (1)
If both have their individual rates the sum of the hourly rate is,
x+ y = 15 -------- (2)
Rearrange the equation,
x = 155- y
Substitute the value of x in equation 1 we get,
10x + 15 y = 1800
10(155-y) + 15y = 1800
1150 - 10y + 15y = 1800
5y = 250
y = 50
Substitute the value of y in equation 2 we get,
x = 155- y
x= 155 - 50
x = 105
Thus, the rates charged per hour by the first and second mechanics will be $105 and $50.
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I really need help with these questions:
Use basic identities to find the simplified expression-
1. (cos^2 x + sin^2 x) / (cot^2 x - csc^2 x)
2. cosine θ^2 / sine θ^2 + csc θ sin θ
Explanations would be greatly appreciated
Answers
(cos^2 x + sin^2 x) / (cot^2 x - csc^2 x)
The numerator becomes 1 since addition order matters not.
1 / (cot^2 x - csc^2 x)
If we factor the denominator out a negative
1 / -(csc^2 x - cot^2 x)
Consider sin^2 x + cos^2 x = 1. Divide both sides by sin^2 x to get
1 + cot^2 x = csc^2 x
Subtract both sides by cot^2 x to get 1 = csc^2 x - cot^2 x.
Replace the denominator
1 / -(1) = -1
For cos^2 θ / sin^2 θ + csc θ sin θ, we use cscθ = 1/sinθ and cosθ/sinθ = cotθ so
= cos^2 θ / sin^2 θ + 1
= cot^2 θ + 1
We use 1 + cot^2 θ = csc^2 θ to simplify this to
= csc^2 θ
Answers: -1
csc^2 θ
[tex]\bf sin^2(\theta)+cos^2(\theta)=1 \qquad \qquad cot(\theta)=\cfrac{cos(\theta )}{sin(\theta)} \qquad csc(\theta)=\cfrac{1}{sin(\theta)}\\\\ -------------------------------\\\\ \cfrac{cos^2(x)+sin^2(x)}{cot^2(x)-csc^2(x)}\implies \cfrac{1}{\frac{cos^2(x)}{sin^2(x)}-\frac{1}{sin^2(x)}}\implies \cfrac{1}{\frac{cos^2(x)-1}{sin^2(x)}} \\\\\\ \cfrac{1}{\frac{-[1-cos^2(x)]}{sin^2(x)}}\implies \cfrac{1}{\frac{-[\underline{sin^2(x)}]}{\underline{sin^2(x)}}}\implies \cfrac{1}{-1}\implies -1[/tex]
2)
[tex]\bf \cfrac{cos^2(\theta )}{sin^2(\theta )}+csc(\theta )sin(\theta )\implies \cfrac{cos^2(\theta )}{sin^2(\theta )}+\cfrac{1}{sin(\theta )}\cdot sin(\theta ) \\\\\\ \cfrac{cos^2(\theta )}{sin^2(\theta )}+1\implies \cfrac{cos^2(\theta )+sin^2(\theta )}{sin^2(\theta )}\implies \cfrac{1}{sin^2(\theta )}\implies csc^2(\theta )[/tex]
Two lighthouses are located 40 miles from each other. A boat is spotted at an angle of 37 degrees northeast of the western lighthouse. At this point, the boat is 28 miles from the eastern lighthouse. How far is the boat from the western lighthouse?
A. 46.27 miles
B. 17.43 miles
C. Either 46.27 miles or 17.43 miles
D. There is no possible solution.
Answers
Using sine rule,
28/Sin 37 = 40/Sin B => Sin B = (40 Sin 37)/28 = 0.8597 => B = Sin ^-1(0.8597) = 59.287°
Therefore,
Angle E = 180 - (37+59.287) = 83.713°
Using the sine rule again;
BW/Sin E = 28/Sin 37 => BW = (28 Sin 83.713)/Sin 37 = 46.246 ≈ 46.27 miles.
The boat is 46.27 miles from the western lighthouse. The correct answer is A.
Answer:
C. Either 46.27 miles or 17.43 miles
In Miami the number of accidents increased by 20% over a four year period.how many accidents were there in 2013 if there were 5120 in 2009
Answers
On two examinations, you have grades of 83 and 89. there is an optional final examination, which counts as one grade. you decide to take the final in order to get a course grade of a, meaning a final average of at least 90
Answers
The average of all three examinations must be at least 90 to get an A.
[tex] \dfrac{x + 83 + 89}{3} \ge 90 [/tex]
[tex] x + 172 \ge 270 [/tex]
[tex] x \ge 98 [/tex]
In order to get an A, you must get at least a 98 on the third examination.
Find the surface area of a rectangular prism with a height of 6 inches, a width of 7 inches, and a length of 13 inches. A) 338 in2 B) 422 in2 C) 520 in2 D) 1364 in2
If you can, please explain. Thank you vv much! :D
Answers
Answer:
422 in.2
Step-by-step explanation:
Substitute the measurements of the rectangular prism into the formula SA = 2(lw) + 2 (lw) + 2(lh): 2(13)(7) + 2(6)(7) + 2(13)(6) = 422 in2.
4+3x-ax=9-7x+bx
PLEASE HELP
Answers
a=−bx+10x−5x
Answer:
[tex]x=\frac{5}{(10-a-b)}[/tex]
Step-by-step explanation:
4+3x-ax=9-7x+bx
take all variable terms at one side and all constant terms at one side,
3x+7x-bx-ax=9-4
10x-ax-bx=5
x(10-a-b)=5
[tex]x=\frac{5}{(10-a-b)}[/tex]
The measure of an angle is 27°. what is the measure of a complementary angle?
Answers
Because 27 + 63 = 90
The measure of a complementary angle to a 27 degree angle is 63 degrees because two angles are complementary if their measures add up to 90 degrees. This is calculated by subtracting the given angle from 90.
Explanation:The measure of an angle is given as 27°, and you are asked to find the measure of a complementary angle. In mathematics, two angles are considered complementary if their measures add up to 90 degrees. Therefore, to find the measure of a complementary angle, you subtract the given angle from 90 degrees.
So, to solve this problem, you would simply subtract 27 from 90.
The calculation would look like this:
90-27 = 63. Therefore, the measure of the complementary angle is 63°.
A footbridge is in the shape of an arc of a circle. the bridge is 10 ft tall and 27 ft long, horizontally. what is the radius of the circle that contains the bridge? round your answer to the nearest tenth.
Answers
The radius of the circle that includes the footbridge, given the bridge's arc length and height, is 14.1 ft.
Explanation:The problem relates to the geometry of a circle, specifically the concept of the arc length and radius. Given the arc length (L) and the height of the arc (h), we can use these to find the radius (r) of the circle using the formula r = L²/(8h) + h/2.
Here, the arc length is the horizontal length of the footbridge, which is 27 ft and the height h is the height of the bridge, 10 ft.
Substituting these values into the formula we get:
r = (27² ft)/(8*10 ft) + 10 ft/2 = 729/80 + 5 = 14.1 ft
So, to the nearest tenth of a foot, the radius of the circle that includes the footbridge is 14.1 ft.
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Brand x copier has improved its copier so that it produces 10% more copies than its old model. if the old model ran 420 copies per hour, how many copies would the new model run? round your answer to the nearest whole number.
a. 221 copies per hour
c. 462 copies per hour
b. 435 copies per hour
d. 447 copies per hour
Answers
to solve this, do the following equation:
x=brand x copier
y=old version
x=y+(y*.1) or x=y*1.1
using either of these equations, the math I will show will get you the correct answer
462=420+(420*.1) or 462=420*1.1
your final answer is: the new copier produces 462 copies per hour
I hope this helps, and have a nice day!
Answer:
462
Step-by-step explanation:
Simplify the expression -2x^2(4x − 3).
-8x3 − 6x2
-8x3 + 6x2
8x3 − 6x2
8x3 + 6x2
Answers
We can infer from our problem that [tex]a=-2x^2[/tex], [tex]b=4x[/tex], and [tex]c=-3[/tex]. So lets apply the property to simplify our expression:
[tex]-2x^2(4x-3)=(-2x^2)(4x)+(-2x^2)(-3)[/tex]
[tex](-2x^2)(4x)+(-2x^2)(-3)=-8x^3+6x^2[/tex]
We can conclude that the correct answer is -8x3 + 6x2
1.) How is the graph of y equals negative 2 x squared minus 5 different from the graph of y equals negative 2 x squared?
2.)
Answers
The transformation on the graph of y = -2x²- 5 is described as 5 units downward.
What is transformation on the graphs?Let the functions f(x) and g(x) be two real functions.
And g (x) = f (x) + k, where k is real numbers.
The function can be sketched by shifting f (x), k units vertically.
The direction of shift can be found by the value of k:
if k > 0, the base graph shifts k units up, and
if k < 0, the base graph shifts k units down.
As per the given information:
The graphs are let f(x) = y = -2x²- 5 and g(x) = y = -2x².
As per the definition, the transformation on the graph can be described as,
-5 < 0.
The graph shifts downward.
Therefore, the graph of y = -2x²- 5 is 5 units down than y = -2x².
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PLZ help
which inequality does the graph represent
Answers
For -2 on the number line, we can see a hollow circle. This means that -2 is not included in the interval, eliminating the possibilities of greater and equal than or less and equal than (≥ and ≤).
The graph will have to use the original greater than symbol (>) or less than symbol (<).
The graph represents all numbers less than -2. Thus, the final inequality will be
x < -2.
The inequality the graph represent is y > x - 2.
What is inequality?Inequality is defined as the relation which makes a non-equal comparison between two given functions. An inequality is a mathematical statement that compares two values or expressions using a relational operator, such as less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), or not equal to (≠). Inequalities are used to describe relationships between quantities or to express constraints or conditions.
We are given that;
The number line
Now,
The slope of the line is 1 and the y-intercept is -2,
so we can write y = x - 2 for the equation of the line.
The line is dashed, so we know that y is not equal to x - 2.
The region above the line is shaded, so we know that y is greater than x - 2.
Therefore, by the inequality the answer will be y > x - 2.
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The table below shows the relative frequencies of the color of cars bought last month by males and females. Which percentage represents the car bought most often?
Options:
20%
31%
43% is wrong
57%
(I already did my other questions, this is the only one I needed help with)
Answers
For the white car we have:
Male = 0.11
Female = 0.20
Total = 0.31
The percentage is given by:
(0.31) * (100) = 31%
Answer:
The percentage that represents the car bought most often is:
31%
Find the surface area of a sphere with radius, r = 11 in.
Surface Area = ________ in^2
(Type an exact answer in terms of pi.)
Answers
Sphere Surface Area = 484 PI
A number x, rounded to 1 decimal place is 12.3 write down the error interval for x
Answers
The error interval for a number x rounded to 12.3 is 12.25 ≤ x < 12.35, indicating the range that x could be before rounding to one decimal place.
Explanation:If a number x, rounded to 1 decimal place, is 12.3, the error interval for x defines the range of values that x could be in order to round to 12.3. Since the value is rounded to one decimal place, we consider half of a tenth (0.05) on either side of 12.3 to find the error interval. Hence, the lower bound of the error interval would be just below 12.3 (12.25), and the upper bound would be just before it would round up to the next tenth (12.35).
The error interval for x can be written as: 12.25 ≤ x < 12.35.
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Error interval for [tex]\( x \)[/tex] rounded to 1 decimal place as 12.3 is from 12.25 to 12.35.
To find the error interval for the number [tex]\( x[/tex], rounded to 1 decimal place as 12.3, we need to consider the possible range of values that [tex]\( x \)[/tex] could have been before rounding.
1. Lower Bound: To find the lower bound, we need to consider the smallest value [tex]\( x \)[/tex] could have been that would round to 12.3 when rounded to 1 decimal place. The smallest value that would round up to 12.3 is 12.25.
2. Upper Bound: Similarly, to find the upper bound, we need to consider the largest value [tex]\( x \)[/tex] could have been that would round to 12.3 when rounded to 1 decimal place. The largest value that would round down to 12.3 is 12.35.
So, the error interval for [tex]\( x \)[/tex] is from 12.25 to 12.35.
Suppose the diameter of a circle is \color{green}{10}10 start color green, 10, end color green. what is its radius
Answers
The diameter is always the twice of radius. If d represents the diameter and r represents the radius, we can write:
d = 2r
or
r = d/2
For d=10,
r = 10/2 = 5 units
So, the radius of the circle will be 5 units
If a person uses $700 in credit with an interest of 14%,what is that person going to owe after three billing cycles with simple interest
Answers
14%
This boils down to ----------- interest per month: a monthly rate of 0.0117.
12 mo
Thus, after 3 months, the amount owed, including the original $700, would be $700 + 3($700)(0.0117) = $724.57.
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Simplify (2√5 + 3√7)² Show ALL your work.
Answers
(a+b)² = a² + 2ab + b² = (2√5)² + 2(2√5)(3√7) + (3√7)²
= 4*5 + 12√35 + 9*7
= 83 + 12√35
Write the equation in standard form using integers y=4x+5
Answers
True or false ???????????????please help
Answers
2. False
3. True because 8000 ^1/3 can be written as 100^1/3 * 8^1/3
4 False
5 True because they are both = 15
If you have a scientific calculator you can check these out yourself
If the pattern below follows the rule "Starting with four, every consecutive line has a number one more than the previous line," how many marbles must be in the seventh line?
Answers
3 x 1 = 3.
Basically you have to add 3 to the number that is in the 4th row. This should be 10, because 7 + 3 = 10
The number of balls in 7th line would be 10.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a pattern as shown in the image.
An arithmetic sequence is of the form -a, a + d, a + 2d, a + 3d, ......,
{d} is common difference and {a} is first term.We can write the pattern as -
4, 5, 6, 7, .....
This is a arithmetic progression. We can write the nth term as -
a{n} = a + (n - 1)d
a{7} = 4 + (7 - 1) x 1
a{7} = 4 + 6
a{7} = 10
Therefore, the number of balls in 7th line would be 10.
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Find one pair of real numbers, $(x,y),$ such that $x + y = 6$ and $x^3 + y^3 = 144.$
Answers
[tex]\[\{(3 + \sqrt{5}, 3 - \sqrt{5})}\][/tex] is the one pair of real numbers [tex]\((x, y)\)[/tex] that satisfies the given equations.
Given the system of equations:
[tex]\[x + y = 6\][/tex]
[tex]\[x^3 + y^3 = 144\][/tex]
We can use the identity for the sum of cubes:
[tex]\[x^3 + y^3 = (x + y)(x^2 - xy + y^2)\][/tex]
Substitute [tex]\(x + y = 6\):[/tex]
[tex]\[x^3 + y^3 = 6(x^2 - xy + y^2)\][/tex]
Given [tex]\(x^3 + y^3 = 144\)[/tex], we substitute this into the equation:
[tex]\[144 = 6(x^2 - xy + y^2)\][/tex]
Divide both sides by 6:
[tex]\[24 = x^2 - xy + y^2\][/tex]
Next, we use the square of the sum:
[tex]\[(x + y)^2 = x^2 + 2xy + y^2\][/tex]
Substitute [tex]\(x + y = 6\):[/tex]
[tex]\[6^2 = x^2 + 2xy + y^2\][/tex]
[tex]\[36 = x^2 + 2xy + y^2\][/tex]
We already have [tex]\(x^2 - xy + y^2 = 24\)[/tex]. Let’s denote the first and second equations:
1. [tex]\(x^2 - xy + y^2 = 24\)[/tex]
2. [tex]\(x^2 + 2xy + y^2 = 36\)[/tex]
Subtract the first equation from the second equation:
[tex]\[(x^2 + 2xy + y^2) - (x^2 - xy + y^2) = 36 - 24\][/tex]
[tex]\[3xy = 12\][/tex]
[tex]\[xy = 4\][/tex]
Now, we have two equations:
1. [tex]\(x + y = 6\)[/tex]
2. [tex]\(xy = 4\)[/tex]
We solve these equations using the quadratic formula. Consider the quadratic equation:
[tex]\[t^2 - (x+y)t + xy = 0\][/tex]
[tex]\[t^2 - 6t + 4 = 0\][/tex]
Solve for [tex]\(t\)[/tex] using the quadratic formula [tex]\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):[/tex]
[tex]\[t = \frac{6 \pm \sqrt{36 - 16}}{2}\][/tex]
[tex]\[t = \frac{6 \pm \sqrt{20}}{2}\][/tex]
[tex]\[t = \frac{6 \pm 2\sqrt{5}}{2}\][/tex]
[tex]\[t = 3 \pm \sqrt{5}\][/tex]
Thus, the solutions for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[x = 3 + \sqrt{5}, \quad y = 3 - \sqrt{5}\][/tex]
or
[tex]\[x = 3 - \sqrt{5}, \quad y = 3 + \sqrt{5}\][/tex]
PLS ANSWER! THE WORLD HATES ME!!
0.29999999999 As a fraction
Answers
Here is your answer:
All you need to do is keep the 29 and then add a hundred because of how many 9's there are.
Also, you could see that:
[tex] .29999999999 \neq \frac{29999999999}{100000000000} [/tex]
Which would also mean that:
[tex] \frac{29999999999}{100000000000} \neq \frac{29}{100} [/tex]
Therefore the answer is "29/100."
Hope this helps!
Answer:
13/46
Step-by-step explanation:
Write an expression for the total Surface Area (SA) of a rectangular prism whose base is a square with side of length x and height of y.
Answers
S.A = 2x ^ 2 + 2x *y + 2x * y
Rewriting we have:
S.A = 2x ^ 2 + 4x *y
S.A = 2 (x ^ 2 + 2x * y)
Where,
x: side length of a square base
y: height of the prism
Answer:
an expression for the total Surface Area (SA) of a rectangular prism is:
S.A = 2 (x ^ 2 + 2x * y)
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ruth was hauling a wide load out of portland. at 7:00 am she headed north from exit 102 at an average speed of 35 pmh. michael headed north from the same exit at 9:00 am in his pontiac. by 3:00 pm the same day michael was 20 miles ahead of ruth. what was michael's average speed
Answers
so at 9 am, she was 70 mile ahead of michael
but michael was 20 miles ahead by 3 pm.
so in 6 hours, he went past ruth by 90 miles which is 90/6 = 15mph
michael speed = ruth speed + 15
= 35 + 15
= 50mph
Answer:
Ruth was out of Portland at 35mph at 7:00 am
so at 9 am, she was 70 miles ahead of Michael
but Michael was 20 miles ahead by 3 pm.
so in 6 hours, he went past Ruth by 90 miles which is 90/6 = 15mph
Michael speed = ruth speed + 15. = 35 + 15 = 50mph
what is the value of x, if the volume of the cone is 12piem^3
Answers
12π=πx²h/3
12=x²h/3
36=x²h
36/h=x²
x=6/√h=(6√h)/h
How do science and math relate to each other?
Answers
Final answer:
Mathematics and science are foundational to engineering, providing the tools and understanding necessary for the engineering design process. They enable engineers to create models, solve design problems, and innovate, thus influencing education and daily life through engineered products.
Explanation:
Science and mathematics are intimately related, especially in fields such as engineering. Mathematics provides the language and tools for abstract reasoning and quantitative analysis, which are fundamental in engineering. Math is used to create mathematical models that describe physical phenomena and to evaluate the merit of different possible solutions. Science, on the other hand, offers understanding of natural phenomena and constraints that inform engineering solutions. Together, math and science are essential throughout the engineering design process, from conceptualization to detailed design implementation, and in team decision-making.
In high school and college, a strong foundation in math and science courses is crucial for success in engineering education. For instance, courses like thermodynamics, circuits, and fluids—commonly referred to as engineering science courses—are grounded in these basic disciplines. The role of science and mathematics in engineering extends beyond educational prerequisites; they are operational tools that engineers use in their work every day to innovate and create products that improve our lives, such as the pocket radio, cell phones, and computers.
The connections between math, science, and engineering are dynamic and multidimensional. They enable engineers to navigate through the iterative steps of the engineering design process, which include analyzing problems, developing solutions, and optimizing products, often moving nonlinearly as real-world complexities dictate adjustments and refinements.
Which is least expensive per oz rounded to the nearest cent: 20 oz for $1.79 or 2 lb 2 oz for $2.59?
A.) 20 oz
B.) 2 lb 2 oz
C.) Neither
Answers
11.17 = 11.2
2 lb 2 oz / 2.59 = 0.82
0.82 = 0.8
So the answer is B) 2 lb 2 oz.
Hope this Helps!
Answer:
B.) 2 lb 2 oz
Step-by-step explanation:
$1.79 for 20 ounces gives us
1.79/20 = $0.0895 ≈ $0.09 per ounce.
1 lb = 16 oz.; this means that 2 lb = 2(16) = 32 oz.
$2.59 for 2 lb 2 oz then gives us
2.59/34 = $0.07617647059 ≈ $0.08 per ounce.
The second choice is cheaper by 1 cent.
A cruise ship can cover 17 nautical miles in 374 minutes. How many nautical miles will it travel in 330 minutes?
Answers
330/22= 15 nautical miles in 10 minutes
To check your work:
17/374=.04545 nautical miles per minute
.04545*330= 14.999 or 15.
So your answer is that the cruise ship covered 15 nautical miles in 330 minutes. I hope that helped!
The ship's speed in deep water is 0.045 nautical miles per minute. In shallow water, the ship's maximum speed typically decreases due to increased drag and turbulence.
To find the speed of the ship in very deep water:
Calculate speed = Distance / Time = 17 nmi / 374 min = 0.045 nautical miles per minute.For the ship's speed in shallow water:It depends on the water resistance; usually, in shallow water, the ship's maximum speed decreases due to increased drag and turbulence.