The speed of the boat in still water is 18 mph.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
We know distance = speed×time.
Therefore, time = distance/speed.
Let, The speed of the boat at still water be 'x'.
Therefore, From the information,
94/(x + 6) = 47/(x - 6).
94(x - 6) = 47(x + 6).
94x - 564 = 47x 282.
47x = 846.
x = 18.
So, The speed of the boat is 18 mph in still water.
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Find the maxima and minima of the following function:
[tex]\displaystyle f(x) = \frac{x^2 - x - 2}{x^2 - 6x + 9}[/tex]
To find the maxima and minima of the function, we need to calculate the derivative of the function. Note, before the denominator is a perfect square trinomial, so the function can be simplified as
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(x) = \frac{x^2 - x - 2}{(x - 3)^2}} \end{gathered}$}[/tex]
So the derivative is:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(2x - 1)(x - 3)^2 - 2(x - 3)(x^2 - x - 2)}{(x - 3)^4} } \end{gathered}$}[/tex]
Simplifying the numerator, we get:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(x - 3)(-5x + 7)}{(x - 3)^4} = \frac{-5x + 7}{(x - 3)^3} } \end{gathered}$}[/tex]
The function will have a maximum or minimum when f'(x) = 0, that is,
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{-5x + 7}{(x - 3)^3} = 0 } \end{gathered}$}[/tex]
which is true if -5x + 7 = 0. Then x = 7/5.
To determine whether x = 7/5 is a maximum, we can use the second derivative test or the first derivative test. In this case, it is easier to use the first derivative test to avoid calculating the second derivative. For this, we evaluate f'(x) at a point to the left of x = 7/5 and at a point to the right of it (as long as it is not greater than 3). Since 1 is to the left of 7/5, we evaluate:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(1) = \frac{-5 + 7}{(1 - 3)^3} = \frac{2}{-8} < 0} \end{gathered}$}[/tex]
Likewise, since 2 is to the right of 7/5, then we evaluate:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \displaystyle \bf{\frac{-10 + 7}{(2 - 3)^3} = \frac{-3}{-1} > 0} \end{gathered}$}[/tex]
Note that to the left of 7/5 the derivative is negative (the function decreases) and to the right of 7/5 the derivative is positive (the function increases).
The value of f(x) at 7/5 is:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f\left(\tfrac{7}{5}\right) = \frac{\tfrac{49}{25} - \tfrac{7}{5} - 2}{\tfrac{49}{25} - 6 \cdot \tfrac{7}{5} + 9} = -\frac{9}{16} } \end{gathered}$}[/tex]
This means that [tex]\bf{\left( \frac{7}{5}, -\frac{9}{16} \right)}[/tex] is a minimum (and the only extreme value of f(x)).
[tex]\huge \red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]
Answer:
[tex]\text{Minimum at }\left(\dfrac{7}{5},-\dfrac{9}{16}\right)[/tex]
Step-by-step explanation:
The local maximum and minimum points of a function are stationary points (turning points). Stationary points occur when the gradient of the function is zero. Differentiation is an algebraic process that finds the gradient of a curve.
To find the stationary points of a function:
Differentiate f(x)Set f'(x) = 0Solve f'(x) = 0 to find the x-valuesPut the x-values back into the original equation to find the y-values.[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]
[tex]\text{Given function}: \quad \text{f}(x)=\dfrac{x^2-x-2}{x^2-6x+9}[/tex]
Differentiate the function using the Quotient Rule:
[tex]\text{Let }u=x^2-x-2 \implies \dfrac{\text{d}u}{\text{d}x}=2x-1[/tex]
[tex]\text{Let }v=x^2-6x+9 \implies \dfrac{\text{d}v}{\text{d}x}=2x-6[/tex]
[tex]\begin{aligned}\implies \dfrac{\text{d}y}{\text{d}x} & =\dfrac{(x^2-6x+9)(2x-1)-(x^2-x-2)(2x-6)}{(x^2-6x+9)^2}\\\\& =\dfrac{(2x^3-13x^2+24x-9)-(2x^3-8x^2+2x+12)}{(x^2-6x+9)^2}\\\\\implies \text{f}\:'(x)& =\dfrac{-5x^2+22x-21}{(x^2-6x+9)^2}\\\\\end{aligned}[/tex]
Set the differentiated function to zero and solve for x:
[tex]\begin{aligned}\implies \text{f}\:'(x)& =0\\\\\implies \dfrac{-5x^2+22x-21}{(x^2-6x+9)^2} & = 0\\\\-5x^2+22x-21 & = 0\\\\-(5x-7)(x-3) & = 0\\\\\implies 5x-7 & = 0 \implies x=\dfrac{7}{5}\\\\\implies x-3 & = 0 \implies x=3\end{aligned}[/tex]
Put the x-values back into the original equation to find the y-values:
[tex]\implies \text{f}\left(\frac{7}{5}\right)=\dfrac{\left(\frac{7}{5}\right)^2-\left(\frac{7}{5}\right)-2}{\left(\frac{7}{5}\right)^2-6\left(\frac{7}{5}\right)+9}=-\dfrac{9}{16}[/tex]
[tex]\implies \text{f}(3)=\dfrac{\left(3\right)^2-\left(3\right)-2}{\left(3\right)^2-6\left(3\right)+9}=\dfrac{4}{0} \implies \text{unde}\text{fined}[/tex]
Therefore, there is a stationary point at:
[tex]\left(\dfrac{7}{5},-\dfrac{9}{16}\right)\:\text{only}[/tex]
To determine if it's a minimum or a maximum, find the second derivative of the function then input the x-value of the stationary point.
If f''(x) > 0 then its a minimum.If f''(x) < 0 then its a maximum.Differentiate f'(x) using the Quotient Rule:
Simplify f'(x) before differentiating:
[tex]\begin{aligned}\text{f}\:'(x) & =\dfrac{-5x^2+22x-21}{(x^2-6x+9)^2}\\\\& = \dfrac{-(5x-7)(x-3)}{\left((x-3)^2\right)^2}\\\\& = \dfrac{-(5x-7)(x-3)}{(x-3)^4}\\\\& = -\dfrac{(5x-7)}{(x-3)^3}\\\\\end{aligned}[/tex]
[tex]\text{Let }u=-(5x-7) \implies \dfrac{\text{d}u}{\text{d}x}=-5[/tex]
[tex]\text{Let }v=(x-3)^3 \implies \dfrac{\text{d}v}{\text{d}x}=3(x-3)^2[/tex]
[tex]\begin{aligned}\implies \dfrac{\text{d}^2y}{\text{d}x^2} & =\dfrac{-5(x-3)^3+3(5x-7)(x-3)^2}{(x-3)^6}\\\\& =\dfrac{-5(x-3)+3(5x-7)}{(x-3)^4}\\\\\implies \text{f}\:''(x)& =\dfrac{10x-6}{(x-3)^4}\end{aligned}[/tex]
Therefore:
[tex]\text{f}\:''\left(\dfrac{7}{5}\right)=\dfrac{625}{512} > 0 \implies \text{minimum}[/tex]
Convert 203 yards into meters.
Hint: 1 yard=0.91 meters
203 yards=[blank]−−−−−− meters
Enter your answer as a number that correctly fills in the blank.
Round the answer to two decimal places, like this: 45.53
Answer:
184.73
Step-by-step explanation:
1 yard = 0.91m
203 yard=?
(203x0.91)/1=184.73.
+1(415) 450-6164
The largest bubblegum bubble ever blown was 23 inches in diameter. What was its circumference?.
Answer:
[tex]C=23\pi[/tex] inches
or as a decimal
72.26 inches
Step-by-step explanation:
Formula for circumference: [tex]C=2\pi r[/tex]
Radius = [tex]\frac{d}{2}[/tex]
Altogether we get: [tex]C=2*\pi*\frac{d}{2}[/tex]
Which simplifies to: [tex]C=\pi d[/tex]
So now we can plug in 23 for [tex]d[/tex].
[tex]C=\pi*23[/tex]
Our final answer is:
[tex]C=23\pi[/tex] inches
or as a decimal
72.26 inches
Find the missing side of each triangle. Leave your answers in simplest radical form.
2√3 m
A) √ 19 m
c) √5 m
O a
a
Ob b
Oc
√7m
C
Od d
B) √17 m
D) √√2 m
Answer:
C [tex]\sqrt{5}[/tex]
Step-by-step explanation:
Use pythagorean theorem.
[tex]x^{2}[/tex] +[tex]\sqrt{7} ^{2}[/tex] = (2[tex]\sqrt{3}) ^{2}[/tex]
[tex]x^{2}[/tex] + 7 = 12 Subtract 7 from both sides
[tex]x^{2}[/tex] = 5 Take the square root of both sides to solve
x = [tex]\sqrt{5}[/tex]
what is 3x+2y=6 ordered pair?
The ordered pair of the equation 3x +2y = 6 will be ( 0,3) and ( 2 , 0).
What is an equation?The equation in mathematics is the relationship between the variables and the number and establishes the relationship between the two or more variables.
We have an equation 3x + 2y = 6 now the ordered pair of the equation will be found by the graph will be ( 0,3) and ( 2, 0). The graph is also attached with the answer below.
Therefore the ordered pair of the equation 3x +2y = 6 will be ( 0,3) and ( 2 , 0).
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Find y
y
A.4
B. 4√2
C. 8
D. 2√2
X
30°
4√3
Step-by-step explanation:
z = 4√3
y = ?
Z° = 90° - 30° = 60°
Y° = 30°
• Find y.
y/sin(Y) = z/sin(Z)
y/sin(30°) = 4√3/sin(60°)
y/(1/2) = 4√3/(1/2 √3)
y . 1/2 √3 = 4√3 . 1/2
y . 1/2 √3 = 2√3
y = 2√3 . 2/√3
y = 4√3/√3
y = 4
The answer is A.
what is the square root of 36/196
√36/169
Answer:
The square root of 36/196 is 3/7.
The square root of 36/169 is 6/13.
Please help me
Find the value of x
Answer:
52=x-15(being vertically opposite angle)
52+15=x
x=67
Which set of graphs can be used to find the solution set to 3e* > >-x?m
The green-colored area of the graph attached below shows the solution set to 3eˣ>(-1/2)x.
Inequality is the relationship between two expressions showing the relationships like greater than, lesser than, lesser than equals to, and greater than equals to.
Here assume f is a function which is given by f(x)=3eˣ
and g be another function which is given by g(x)= (-1/2)x
Here we have to find the solution graph showing the relationship
f(x) > g(x)
Let at point (a,b) the f(x) and g(x) meet each other.
After that intersection point, g(x) will decrease as g(x) is a decreasing function. and f(x) will increase.
So in the solution set will contain the area where g(x) is larger than f(x).
Therefore the solution graph of this inequality is the green-colored area as shown below,
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Tara has a box of 908 beads for making bracelets. She wants to put 15 beads on each bracelet she makes. What is the greatest number of bracelets Tara can make with these beads? Tara has a box of 908 beads for making bracelets . She wants to put 15 beads on each bracelet she makes . What is the greatest number of bracelets Tara can make with these beads ?
Answer:
Tara can make 60 bracelets.
Step-by-step explanation:
908 / 15 = 60.5
Which congruence theorem can be used to prove ?
Answer:
ASA
Step-by-step explanation:
by ASA congruence theorem
4. Angles A and B are supplementary. If mA = 67, find mB
Answer:
113 degrees
Step-by-step explanation:
Two angles are called supplementary when they add up to 180 degrees
[tex]180 = x + y \\ 180 = 67 + y \\ y = 180 - 67 \\ y = 113 \\ therefore \: mB \: = 113[/tex]
Select all numbers that are a solution of the inequality.
Every month a salesperson adds 7 new accounts. The algebraic expression that represents the number of new accounts that he will add in m months is _____.
The algebraic expression that represents the number of new accounts that he will add in m months is x ≥ 7m.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
Every month, a salesperson adds 7 new accounts.
Let m be the number of the month and x be the number of the accounts.
Then the inequality equation will be
x ≥ 7m
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a list of 8 numbers has a mean of 6. work out the total of the numbers.
Answer:
48
Step-by-step explanation:
The mean of a number is the sum of all the numbers divided by how many numbers there are. We can set up this formula: [tex]mean = \frac{sum}{amountnumbers}[/tex].
When we plug our values into this equation, we get: [tex]6 = \frac{sum}{8}[/tex]. Solving this equation gets us sum = 6 * 8 = 48
Work out (6 × 10²) ÷ (3 × 105)
Give your answer in standard form.
Answer: 40/21
Step-by-step explanation:
[tex]3 \times 105=315\\\\6 \times 10^{2}=600\\\\\implies \frac{6 \times 10^{2}}{3 \times 105}=\frac{600}{305}=\boxed{\frac{40}{21}}[/tex]
Find x and y.
help please ty:)
Please, make sure you understand the solution:
We can notice that x + (x - 28) = 90
2x - 28 = 90
2x = 118
x = 59
Now for y:
We can notice that y + 90 + x = 180
y + 90 + 59 = 180
y + 149 = 180
y = 180 - 149
y = 31
What is the value of y in the equation 2(2y - 12) = 0?
04
06
07
08
Hey there!
2(2y - 12) = 0
2(2y) + 2(-12) = 0
4y - 24 = 0
ADD 24 to BOTH SIDES
4y - 24 + 24 = 0 + 24
SIMPLIFY IT!
4y = 0 + 24
4y = 24
4y/4 = 24/4
SIMPLIFY IT!
y = 24/4
y = 6
Therefore, your answer should be:
y = 06 (Option B.)
Good luck on your assignment and enjoy your day!
~Amphitrite1040:)
Answer:
y=6 (06) should be your answer
Step-by-step explanation:
4y-24=0
After you simply the first equation in the parenthesis, you then move the 24 to the 0,
4y=24
The answer becomes positive 24 because when you move things across the equal sign you switch the sign
y=6
Then divide 24 by 4 to isolate y, which then gives you the answer 6.
find the arc length of a sector with a radius of 4 feet and a central angle of 6°
Answer:
24 ft.
Step-by-step explanation:
Radius = 4 ft
Central angle = 6°
Arc length = radius × central angle
= 4 × 6°
= 24 ft.
My square patio is tiled with square tiles, all the same size. All the tiles are gray, except the tiles along the two diagonals, which are all yellow. (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.) If there are $89$ yellow tiles, how many gray tiles are there
Answer:
1936
Step-by-step explanation:
We can start by removing the center tile, because all four diagonals over lap on that tile. We'll add it back later. 88/4=22. which means there are 22 tiles per half, 44+1(the center tile) total tiles per side, which means there are a total of 1936 tiles.
AOPS ANSWER:
Suppose that we have a square with dimensions [tex]$e \times e$[/tex]. The diagonals each have [tex]$e$\\[/tex] tiles. If [tex]$e$\\[/tex] is even, the number of yellow tiles is [tex]$2e$[/tex], because the [tex]2[/tex] diagonals don't intersect. If [tex]$e$[/tex] is odd, then the number of yellow tiles is [tex]$e+e-1=2e-1$[/tex]. (We have to subtract [tex]1[/tex] because the center tile is counted [tex]2[/tex] times). We know that there are [tex]89[/tex] yellow tiles, which is an odd number, so [tex]$89=2e-1$[/tex]. This implies that [tex]$e = 45$[/tex]. So the dimensions of the floor are [tex]$45 \times 45$[/tex], or [tex]2025\\[/tex] square units. Since all of the other tiles are gray, there are [tex]$2025-89=\boxed{1936}$[/tex] gray tiles.
someone please help me w this standard form question
Answer:
2.4 × 10⁶
Step-by-step explanation:
Given:
[tex]\sf x=3.8 \times 10^5[/tex][tex]\sf y=5.9 \times 10^4[/tex]To calculate the value of R, substitute the given values into the given formula:
[tex]\begin{aligned}\sf R & = \sf \dfrac{x^2}{y}\\\\ \implies \sf R& = \sf \dfrac{(3.8 \times 10^5)^2}{5.9 \times 10^4}\\\\& = \sf \dfrac{3.8^2 \times (10^5)^2}{5.9 \times 10^4}\\\\& = \sf \dfrac{14.44 \times 10^{10}}{5.9 \times 10^4}\\\\& = \sf \dfrac{14.44}{5.9} \times \dfrac{10^{10}}{10^4}\\\\& = \sf 2.4474... \times 10^{(10-4)}\\\\& = \sf 2.4474... \times 10^6\\\\\end{aligned}[/tex]
Therefore, the value of R to 1 decimal place is 2.4 × 10⁶
--------------------------------------------------------------------------
Exponent Rules used
[tex](a^b)^c=a^{bc}[/tex]
[tex]{a^b}{a^c}=a^{b-c}[/tex]
[tex]R = \frac{ {x}^{2} }{ y} \\ \\ R = \frac{ {(3.8 \times {10}^{5})}^{2} }{5.9 \times {10}^{4} } \\ \\ R = \frac{ ({3.8})^{2} \times ( { {10}^{5} )}^{2} }{5.9 \times {10}^{4} } \\ \\ R = \frac{14.44 \times {10}^{10} }{5.9 \times {10}^{4} } \\ \\ R = \frac{14.44 \times {10}^{10 - 4} }{5.9} \\ \\ R = 2.45 \times {10}^{6} [/tex]
Given the image below DY EY FY are perpendicular bisectors of triangle ABC
The value of FY in the triangle will be 30.11
How to calculate the value?From the information given, the value of BE will be:
= [tex]\sqrt{64.2}[/tex]² - [tex]\sqrt{51.2}[/tex]²
= 38.7
From the triangle, DY will be
= [tex]\sqrt{64.22}[/tex]² - [tex]\sqrt{61.7}[/tex]²
= 17.7
AY will be:
= [tex]\sqrt{61.7}[/tex]² + [tex]\sqrt{17.7}[/tex]²
= 64.2
FY will now be:
= [tex]\sqrt{64.2 - 56.7}[/tex]²
= 30.11
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!!! TIME SENSITIVE !!! Determine the period.
Answer:
A wave period is the measure of the time it takes for the wave cycle to complete.
The period of the wave is 2 seconds.
Determine the Angle Measures
The angles are p=67°, q=15°, x=82°, y=81°, s=28° and m=72.5°.
For the given figures we need to find the unknown angles.
In the first figure to find angle p first apply the angle sum property to the triangle.
That is, 82°+31°+p=180° (∵ Sum of interior angles of a triangle is 180°)
⇒p=67°.
p=67° is a vertically opposite angle to the other triangle.
To find q:
67°+98°+q=180°
⇒q=15°.
In the second figure to find angles x and y apply the angle sum property to the triangle.
That is, 28°+(53°+x)+17°=180°
⇒98°+x=180°
⇒x=82°.
To find y:
x+y+17°=180°
⇒82°+y+17°=180°
⇒y=81°
In the third figure to find angles x and y apply the angle sum property to the triangle.
That is, 100-x+30+4x+7x=180°
⇒10x=80°⇒x=10°.
Now the angles of the triangle are 100-10=90°, 30+4x=70°
y=90°+70°=160° (∵Exterior angle is equal to the sum of two opposite interior angles).
In the 4th figure, to find angle 's' use the isosceles triangle concept.
That is, s+76°+76°=180° (∵In isosceles triangle base angles are equal)
⇒s=28°
In the 5th figure, to find angle 'm' use the isosceles triangle concept.
That is, 35°+m+m=180° (∵In isosceles triangle base angles are equal)
⇒2m=145°
⇒m=72.5°
Hence, the angles are p=67°, q=15°, x=82°, y=81°, s=28° and m=72.5°.
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Is this pattern a net for the three-dimensional figure?
no
yes
Answer:
yes
Step-by-step explanation:
2 bases 3 rectangles, folds properly.
A swimming pool is 50 metres long and 25 metres wide. How many lengths of the pool will a competitor in a 1500 metre race have to swim?
Reason:
I'm assuming that the swimmer travels along the 50 meter side
Divide 1500 over 50 to get 1500/50 = 30, which tells us that the swimmer must make 30 trips.
18 A student has 273 matchsticks with which to make a pattern of nested triangles. Fig. 18.9 shows the first three triangles. Fig. 18.9 If all the matchsticks are used, how many matchsticks will each side of the biggest triangle contain? 18 A student has 273 matchsticks with which to make a pattern of nested triangles . Fig . 18.9 shows the first three triangles . Fig . 18.9 If all the matchsticks are used , how many matchsticks will each side of the biggest triangle contain ?
The number of matchsticks that each side of the biggest triangle contains is; 137 matchsticks
How to find the biggest side of a triangle?We know that for a diagram to be a triangle, then two smallest sides must not be greater than the third side.
Now, if there are 273 matchsticks, then the greatest side of the triangle must not be less than 273/2 = 136.5 ≈ 137
Thus, the largest side will have at least 137 matchsticks
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What are the zeros of this function?
Can someone answer this as soon as possible.
What remainder is represented by the synthetic division below?
A vertical line and horizontal line combine to make a L shape. There are two rows of entries within the shape. Row 1 has entries 2, 10, 1, 5. Row 2 has entries blank, negative 10, 0, negative 5. Entry negative 5 is on the outside to the left of the shape, and a third row of entries is outside and below the shape. Row 3 has entries 2, 0, 1, 0.
The remainder represented by synthetic division is -5.
What is synthetic division?
Synthetic division can be defined as a simplified way of dividing a polynomial with another polynomial equation of degree 1.
Given
A vertical line and a horizontal line combine to make an L shape.
There are two rows of entries within the shape.
Row 1 has entries 2, 10, 1, 5.
Row 2 has entries blank, -10, 0, -5.
Entry -5 is on the outside to the left of the shape. The entry represents the zero of the divisor. This entry to the variable (assume the variable is x) is;
x + 5 = 0
x = -5
Hence, the divisor of the synthetic division is -5.
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Answer:
its a) -5
Step-by-step explanation:
decrease 2155.45 by 18.5 round 2dp
The solution is 1756.69
What is number system?A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner.
We have to find the 18.5% of 2155.45.
So,
2155.45 * 0.185 = 398.75825
Now, subtract
21.55.45 - 398.75825
= 1756.69175
Hence, the value is 1756.69
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