Given:
It takes the boat 3 hours to travel 108 miles going upstream
Return trip = 2hours going downstream
Distance, d = 108 miles
Time going upstream = 3 hours
Time going downstream = 2 hours
Let's find the rate in still water and the current rate.
Let s represent the still rate
Let c represent the current rate.
Apply the distance formula:
Distance = Rate x Time
We have the set of equations:
(s - c) x 3 = 108.................................Equation 1
(s + c) x 2 = 108.................................Equation 2
Apply distributive property:
3s - 3c = 108
2s + 2c = 108
Let's solve both equations simultaneously using substitution method.
Rewrite the first equation for s:
3s - 3c = 108
Add 3c to both sides:
3s - 3c + 3c = 108 + 3c
3s = 108 + 3c
Divide all terms by 3:
[tex]\begin{gathered} \frac{3s}{3}=\frac{108}{3}+\frac{3c}{3} \\ \\ s=36+c \end{gathered}[/tex]Substitute s for (36 + c) in equation 2:
2s + 2c = 108
2(36 + c) + 2c = 108
72 + 2c + 2c = 108
72 + 4c = 108
Subtract 72 from both sides:
72 - 72 + 4c = 108 - 72
4c = 36
Divide both sides by 4:
[tex]\begin{gathered} \frac{4c}{4}=\frac{36}{4} \\ \\ c=9 \end{gathered}[/tex]Substitute c for 9 in either of thee equation.
Take the first equation:
3s - 3c = 108
3s - 3(9) = 108
3s - 27 = 108
Add 27 to both sides:
3s - 27 + 27 = 108 + 27
3s = 135
Divide both sides by 3:
[tex]\begin{gathered} \frac{3s}{3}=\frac{135}{3} \\ \\ s=45 \end{gathered}[/tex]Thus, we have the solutions:
c = 9
s = 45
The rate of boat in still water is 45 miles per hour
The rate of the current is 9 miles per hour
Therefore, we have:
Rate of boat in still water: 45 mi/h
Rate of current: 9 mi/h
Take the firs
ANSWER:
Rate of boat in still water: 45 mi/h
Rae of the current: 9 mi/h
I need a math wiz to explain this to me, are you a math wiz?
SOLUTION
The questions is outside scope
How to solve this problem? (the answer is 262 Hz). i want to know the step by step on how to solve the equation given. if it helps, i am a grade 10 student. (YES, this is a MATH problem)
The frequency of middle C = 262 Hz
Explanation:The formula for calculating the frequency, F hertz, of a note n seminotes above the concert pitch is:
[tex]F\text{ = 440(}\sqrt[12]{2})^n[/tex]This can be re-written as:
[tex]F=440(2^{\frac{n}{12}})[/tex]Middle C is 9 semitones below the concert pitch
That is, n = -9
To find the frequency of middle C, substitute n = -9 into the equation for F
[tex]\begin{gathered} F=440(2^{\frac{-9}{12}}) \\ F\text{ = 440(}0.5946) \\ F\text{ = }261.62\text{ Hz} \\ F\text{ = 262 Hz (to the nearest hertz)} \end{gathered}[/tex]The frequency of middle C = 262 Hz
Which transformations of quadrilateral PQRS would result in the imageof the quadrilateral being located only in the first quadrant of thecoordinate plane?
Given:
The quadrilateral PQRS is given.
The aim is to locate the given quadrilateral into first quadrant only.
The graph will be reflected across x=4 then the graph will not be located to the first quadrant.
Hi, could you help me figure out why I got 8 points off in this problem?
In triangle PQR
Construction: Draw PX perpendicular to QR where x lies on QR
Since:
PX perpendicular to QR
In the 2 triangles PXQ and PXR
given
proved up
PX = PX ------- common side in the 2 triangles
Triangle PXQ congruent to triangle PXR by the AAS theorem of congruency
As a result of congruency
PQ = PR ------- proved
Please I really need help. I just need the answer no steps
Explanation
The question wants us to obtain the margin of error
A margin of error tells you how many percentages points your results will differ from the real population value.
The formula to be used is
To do so, we will have to list out the parameters to be used
[tex]\begin{gathered} standard\text{ deviation=}\sigma=13.8 \\ sample\text{ size=n=18} \\ confidence\text{ level=}\gamma=80\text{ \%} \end{gathered}[/tex]The next step will be to find the z-score value for a confidence level of 80%.
From the statistical table, we have
[tex]Z_{\gamma}=1.28[/tex]So, we can input the given data obtained into the formula
So we will have
[tex]\begin{gathered} MOE=1.28\times\sqrt{\frac{13.8^2}{18}} \\ \\ MOE=1.28\times\frac{13.8}{\sqrt{18}} \\ \\ MOE=1.28\times3.2527 \\ \\ MOE=4.16344 \end{gathered}[/tex]So the margin of error (M.E.) = 4.163 (To 3 decimal places)
Use the formula for compound amount:$14,800 at 6% compounded semiannually for 4 years
SOLUTION
Given the question in the question tab, the following are the solution steps to answer the question.
STEP 1: Write the formula for calculating compound amount
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]where
A = final compounded amount
P = initial principal balance
r = interest rate
n = number of times interest applied per time period
t = number of time periods elapsed
STEP 2: Write the given data
Semiannually means that n will be 2
[tex]P=14,800,r=\frac{6}{100}=0.06,n=2,t=4[/tex]STEP 3: Calculate the compound amount
[tex]\begin{gathered} A=14800(1+\frac{0.06}{2})^{2\times4}\Rightarrow A=14800(1+0.03)^{2\times4} \\ A=14800(1.03)^8 \\ A=14800\times1.266770081 \\ A=\text{\$}18,748.1972 \end{gathered}[/tex]Hence, the compounded amount after 4 years is $18,748.1972
Passes through (8,8) with slope 11/6
Given:
point (8,8).
slope 11/6
The slope intercept form is,
[tex]y=mx+b[/tex]where m is the slope and b is the y-intercept.
we know that m=11/6 so subistute in the equation.
[tex]y=\frac{11}{6}x+b[/tex]Now, let us plug in the point in the equation to find the value of b that is the y-intercept.
[tex]undefined[/tex]Let the graph of f(x) be given below. Find the formula of f(x), a polynomial function, of least degree.
To detrmine the formula of the polynomial, we check for the roots on the graph:
when y = 0, x = -2
when y = 0, x = 4
We have two roots.
x = -2
x + 2 = 0
x = 4
x - 4 = 0
3rd factor is x = 0
Hence, we have two factors: x(x + 2) and (x - 4)
The polynomial function using the factors:
[tex]f(x)\text{ = ax(x + 2)(x - 4)}[/tex]Next, we find the value of a:
To get a , we pick a point on the graph. let the point be (0, -4)
substitute the point in the function above:
[tex]\begin{gathered} f(x)\text{ = y = -4, x = 0} \\ -4\text{ = a(0 + 2) (0 - 4)} \\ -4\text{ = a(2)(-4)} \\ -4\text{ = -8a} \\ a\text{ = -4/-8} \\ a\text{ = 1/2} \end{gathered}[/tex]The formula of the polynomial becomes:
[tex]f(x)\text{ = }\frac{1}{2}x(x\text{ }+2)(x-4)[/tex]Solve the following system of linear equations using elimination. x-y=5 -x-y=-11
Elimination Method : In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
The given system of equation :
x - y = 5 ( 1 )
- x - y = - 11 ( 2 )
Add the equation ( 1 ) & ( 2 )
x - y + ( -x - y ) = 5 + ( -11 )
x - y -x - y = 5 - 11
x - x - y - y = -6
0 - 2y = - 6
y = -6/( -2)
y = 3
Substitute the value of y = 3 in the equation ( 1)
x - y = 5
x - 3 = 5
x = 5 + 3
x = 8
Answer : x = 8, y = 3
URGENT!! ILL GIVE
BRAINLIEST!!!!! AND 100
POINTS!!!!!
the rriangle is 3 4 5 triangle so 5×5 is 25
P is inversely proportional to Q. If P = 24 when Q = 3, then write the inverse variation equation that relates P and Q.
Inverse proportionality is when the value of one quantity increases with respect to a decrease in another, they behave opposite in nature.
It is represented by the following expression:
[tex]P=\frac{k}{Q}[/tex]Since P=24 when Q=3, we can substitute and solve for the constant k:
[tex]\begin{gathered} 24=\frac{k}{3} \\ k=24\cdot3 \\ k=72 \end{gathered}[/tex]Then, the equation that represents the inverse variation would be:
[tex]P=\frac{72}{Q}[/tex]find the value of x
For supplementary angles, we can do the following equality
[tex]3x+4=x+70[/tex]What we have to do, is to clear "x" to find its value.
[tex]\begin{gathered} 3x-x=70-4 \\ 2x=66 \\ x=\frac{66}{2} \\ x=33 \end{gathered}[/tex]In conclusion, the value of x is 33
Original cost $21.99 Markup 5%. What's the new price?
Explanation:
We have to find 5% of the original cost first:
[tex]21.99\times\frac{5}{100}=21.99\times0.05=1.0995[/tex]And then add it to the original price:
[tex]21.99+1.0995=23.0895[/tex]Since it's a price, we have to round this result to the nearest hundredth
Answer:
The new price is $23.09
what are the three terms and 4x - 2y + 3
Solution
We have the following expression:
[tex]4x-2y+3[/tex]Here we have 3 terms:
[tex]4x,\text{ -2y and 3}[/tex]Variable terms:
[tex]4x,-2y[/tex]Constant term
[tex]3[/tex]Simplify [tex]{({4e}^{ - 8x})}^{0.5} [/tex]with no negative exponents. thanks!
Explanation
Given the following expression
[tex]\begin{gathered} \text{Simplify (4 }e^{-8x})^{\frac{1}{2}} \\ \text{This expression can be written as} \\ (4\cdot\text{ }e^{-8x})^{\frac{1}{2}} \\ \text{Splitting the expression, we can have the below expression} \\ (4)^{\frac{1}{2}}\cdot(^{}e^{-8x})^{\frac{1}{2}} \\ \text{According to the law of indicies} \\ x^{\frac{1}{2}}\text{ = }\sqrt[]{x} \\ \text{Hence, we have the following expression} \\ \sqrt[]{4\text{ }}\cdot\text{ (}e^{-8x\cdot\text{ }\frac{1}{2}}) \\ 2\cdot\text{ }e^{-4x} \\ 2e^{-4x} \\ \text{Therefore, the simplified form is 2}e^{-4x} \\ \frac{2}{e^{4x}} \end{gathered}[/tex]could someone please help :(
Given from the number line:
D = -2 and F = 13
So, the distance DF = 13 - (-2) = 13 + 2 = 15
1) find E such that, DE : EF = 2 : 1
so,
so, x : (15 - x) = 2 : 1
x = 30 - 2x
3x = 30
x = 10
So, E = -2 + 10 = 8
=========================================================================
2) E is 4/5 of the distance from F to D
So, the distance from F = 4/5 * 15 = 12
So, E = 13 - 12 = 1
=====================================================================
3) the ratio of DE : EF = 2 : 3
So,
3x = 2 ( 15 - x)
3x = 30 - 2x
5x = 30
x = 30/5 = 6
E = -2 + 6 = 4
=================================================
4) E is 1/3 of the distance from D to F
So, the distance DE = 1/3 * 15 = 5
So, E = -2 + 5 = 3
=====================================================
As a summery:
1) E = 8
2) E = 1
3) E = 4
4) E = 3
The diameter of a circle has endpoints P(-12, -4) and Q(6, 12).
ANSWER
[tex](x+3)^{2}+(y-4)^{2}=145[/tex]EXPLANATION
The equation of a circle is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h, k) = center of the circle
r = radius of the circle
The center of a circle is the midpoint of the endpoints of the diameter of the circle. Hence, to find the center of the circle, we have to find the midpoint of the diameter:
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]where (x1, y1) and (x2, y2) are the endpoints of the diameter.
Hence, the center of the circle is:
[tex]\begin{gathered} M=(\frac{-12+6}{2},\frac{-4+12}{2}) \\ M=(\frac{-6}{2},\frac{8}{2}) \\ M=(-3,4) \end{gathered}[/tex]To find the radius of the circle, we have to find the distance between any endpoint of the circle and the center of the circle.
To do this apply the formula for distance between two points:
[tex]r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Therefore, the radius of the circle is:
[tex]\begin{gathered} r=\sqrt{(6-(-3))^2+(12-4)^2}=\sqrt{9^2+8^2} \\ r=\sqrt{81+64}=\sqrt{145} \end{gathered}[/tex]Hence, the equation of the circle is:
[tex]\begin{gathered} (x+3)^2+(y-4)^2=(\sqrt{145})^2 \\ (x+3)^2+(y-4)^2=145 \end{gathered}[/tex]Solve the following system of equation using substitution4x + 2y = 10x - y= 13What is the solution for y?
ANSWER
y = -7
EXPLANATION
To solve using the substitution method we have to clear x from one of the equations as a function of y. For example, for equation 2:
[tex]x=13+y[/tex]Then replace x in the first equation by this expression:
[tex]4(13+y)+2y=10[/tex]And solve for y:
[tex]\begin{gathered} 4\cdot13+4y+2y=10 \\ 52+6y=10 \\ 6y=10-52 \\ 6y=-42 \\ y=\frac{-42}{6} \\ y=-7 \end{gathered}[/tex]Garret is removing a hem from a skirt. It takes
Garret 5 min to remove 4 in. of the hem. He wants to
know how long it will take to remove 5 ft of the hem if
he continues to work at the same rate.
Lavar
How can Garret determine how long it will take to remove 5 ft of the hem?
Choose one option from each drop-down menu to answer the question.
It takes Garret Choose... min to remove 1 ft of hem.
He should multiply the number of minutes by Choose... to determine the number of minutes it will take to
remove 5 ft of hem.
It will take Choose.... min to remove 5 ft of hem.
It takes Garret Choose min to remove 1 ft of hem. He should multiply the number of minutes by Choose... to determine the number of minutes it will take to remove 5 ft of hem.
What is the unitary method?The unitary method is a technique used to determine the value of a single unit from the value of many units and the value of multiple units from the value of a single unit. We typically utilize it for math calculations. This approach will come in handy for topics involving ratio and proportion, algebra, geometry, etc. In the unitary technique, we always count the value of a unit or one quantity first before figuring out the values of more or fewer quantities. This method is referred to as the "unitary method" for this purpose.
There are two types of unitary methods because they result in two types of variations and those are given below:
Direct VariationIndirect VariationTo know more about the unitary method ,visit:
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Solve the system by graphing:2x – y= -14x - 2y = 6Solution(s):
To find the solution of the system by graphing we need to plot each line in the plane and look for the intersection.
First we need to write both equations in terms of y:
[tex]\begin{gathered} y=2x+1 \\ y=2x-3 \end{gathered}[/tex]now we need to find two points for each of this lines. To do this we give values to the variable x and find y.
For the equation 2x-y=-1, if x=0 then:
[tex]y=1[/tex]so we have the point (0,1).
If x=1, then:
[tex]y=3[/tex]so we have the point (1,3).
Now we plot this points on the plane and join them with a straight line.
Now we look for two points of the second equation.
If x=0, then:
[tex]y=-3[/tex]so we have the point (0,-3)
If x=1, then:
[tex]y=-1[/tex]so we have the point (1,-1).
We plot the points and join them wiith a line, then we have:
once we have both lines in the plane we look for the intersection. In this case we notice that the lines are parallel; this means that they wont intersect.
Therefore the system of equations has no solutions.
Using the completing-the-square method, rewrite f(x) = x2 − 8x + 3 in vertex form. (2 points)
A) f(x) = (x − 8)2
B) f(x) = (x − 4)2 − 13
C) f(x) = (x − 4)2 + 3
D) f(x) = (x − 4)2 + 16
By using the completing the square method, f(x) = x² − 8x + 3 in vertex form is: B. f(x) = (x − 4)² − 13.
The vertex form of a quadratic equation.In this exercise, you're required to rewrite the given function in vertex form by using the completing the square method. Mathematically, the vertex form of a quadratic equation is given by this formula:
y = a(x - h)² + k
Where:
h and k represents the vertex of the graph.
In order to complete the square, we would have to add (half the coefficient of the x-term)² to both sides of the quadratic equation as follows:
f(x) = x² − 8x + 3
f(x) = x² − 8x + (8/2)² - 13
f(x) = x² − 8x + (4)² - 13
f(x) = x² − 8x + 16 - 13
f(x) = (x² − 8x + 16) - 13
f(x) = (x − 4)² − 13
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Write the slope-intercept form of the equation of the line with the given characteristics. Perpendicular to y = -5x + 2 and passing through (3,-1).
The slope intercept form of a line can be expressed as,
[tex]y=mx+c[/tex]Here, m is the slope of the line and c is the y intercept.
Comparing the above equation with the given equation of a line y=-5x+2, we get
m=-5.
The slope of a line perpendicular to line with slope m is -1/m.
Hence, the slope of line perpendicular to y=-5x+2 is,
[tex]m_1=\frac{-1}{m}=\frac{-1}{-5}=\frac{1}{5}[/tex]The new line is given to be passing through point with coordinates (x1, y1)=(3, -1).
The point slope form of a line passing through point with coordinates (x1, y1)=(3, -1) and having slope m1 is,
[tex]\begin{gathered} y-y_1=m_1(x-x_1) \\ y-(-1)=\frac{1}{5}(x-3) \\ y+1=\frac{1}{5}x-\frac{3}{5} \\ y=\frac{1}{5}x-\frac{3}{5}-1 \\ y=\frac{1}{5}x-\frac{3-5}{5} \\ y=\frac{1}{5}x-\frac{8}{5} \end{gathered}[/tex]Therefore, the slope-intercept form of the equation of the line perpendicular to y = -5x + 2 and passing through (3,-1) is,
[tex]y=\frac{1}{5}x-\frac{8}{5}[/tex]a company loses $5,400 as the result of manufacturing defect. each of the 8 owners have agreed to pay an equal amount, x, to pay for the loss. How much each owner paid?
Explanation:
If 'x' is the amount each owner will pay, there are 8 owners and the total amount to pay is $5,400 the equation to solve is:
[tex]8x=5,400[/tex]Solving for x:
[tex]x=\frac{5,400}{8}=675[/tex]Answer:
Each owner has to pay $675
Given the graph of f (x), determine the domain of f –1(x).
Radical function f of x that increases from the point negative 3 comma negative 2 and passes through the points 1 comma 0 and 6 comma 1
The domain of the function f(x) that has a range of [-2, ∞) is [-2, ∞)
What is the inverse of a function?The inverse of a function that maps x into y, maps y into x.
The given coordinates of the points on the radical function, f(x) are; (-3, -2), (1, 0), (6, 1)
To determine the domain of
[tex] {f}^{ - 1}( x)[/tex]
The graph of the inverse of a function is given by the reflection of the graph of the function across the line y = x
The reflection of the point (x, y) across the line y = x, gives the point (y, x)
The points on the graph of the inverse of the function, f(x), [tex] {f}^{ - 1} (x)[/tex] are therefore;
[tex]( - 3, \: - 2) \: \underrightarrow{R_{(y=x)}} \: ( - 2, \: - 3)[/tex]
[tex]( 1, \: 0) \: \underrightarrow{R_{(y=x)}} \: ( 0, \: 1)[/tex]
( 6, \: 1) \: \underrightarrow{R_{(y=x)}} \: ( 1, \: 6)
The coordinates of the points on the graph of the inverse of the function, f(x) are; (-2, -3), (1, 0), (1, 6)
Given that the coordinate of point (x, y) on the image of the inverse function is (y, x), and that the graph of the function, f(x) starts at the point (-3, -2) and is increasing to infinity, (∞, ∞), such that the range of y–values is [-2, ∞) the inverse function, [tex] {f}^{ - 1}( x)[/tex], which starts at the point (-2, -3) continues to infinity, has a domain that is the same as the range of f(x), which gives;
The domain of the inverse of the function, [tex] {f}^{ - 1}( x)[/tex], using interval notation is; [-2, ∞)
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in a public opinion poll 624 people from a sample of 1100 indicated they would vote for specific candidate how many votes can the candidate expect to receive from a population of 40000
Hello!
In a sample of 1100 people, the specific candidate got 624 votes. So, we can write it as 624/1100.
And if the total of voters is 40,000, how many votes this specific candidate will receive? We can write it as x/40,000.
Now, let's equal both fractions look:
[tex]\begin{gathered} \frac{624}{1100}=\frac{x}{40000} \\ \\ 1100x=624\times40000 \\ 1100x=24960000 \\ x=\frac{24960000}{1100} \\ \\ x\cong22691 \end{gathered}[/tex]Answer:Approximately 22691 votes.
The figure shows rectangle PQRS in the first quadrant of the coordinate plane?
The quadrants of a coordinate plane are:
Then, we can say that the rectangle PQRS is in the first quadrant.
The measures of the angles of a triangle are shown in the figure below. Solve for x.
44°
61°
(8x+11)°
Ninas math classroom is 6 and 4/5 meters long and 1 and 3/8 meters wide. What is the area of the classroom?
The most appropriate choice for area of rectangle will be given by -
Area of classroom = [tex]4\frac{27}{40}[/tex] [tex]m^2[/tex]
What is area of rectangle?
Rectangle is a four sided figure whose parallel sides are equal and whose every angle is 90°
The total space taken by the rectangle is called area of the rectangle.
If the length of the rectangle be l and the breadth of the rectangle be b, then area of the rectangle is given by
Area = [tex]l \times b[/tex]
Here,
Length of classroom = [tex]6\frac{4}{5}[/tex] m = [tex]\frac{34}{5}[/tex] m
Width of classroom = [tex]1\frac{3}{8}[/tex] m = [tex]\frac{11}{8}[/tex] m
Area of classroom = [tex]\frac{34}{5} \times \frac{11}{8}[/tex]
= [tex]\frac{187}{40}[/tex]
= [tex]4\frac{27}{40}[/tex] [tex]m^2[/tex]
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Which expression is equivalent to 8 - (-5) ?O 8+50 -8 +(-5)O 8+-5O -5 +8
Answer:
The first option is correct
[tex]8+5[/tex]Explanation:
[tex]\begin{gathered} 8--5 \\ \\ 8+5 \\ \end{gathered}[/tex]Two negatives makes a positive.
the length of a rectangle is 11 yd more than twice the width and the area of the rectangle is 63 yd squared. find the dimensions of the rectangle.
the length of a rectangle is 11 yd more than twice the width and the area of the rectangle is 63 yd squared. find the dimensions of the rectangle.
Let
L ------> the lenght
W ----> the width
we know that
the area of rectangle is
A=L*W
A=63 yd2
63=L*W -------> equation 1
and
L=2W+11 ------> equation 2
substitute equation 2 in equation 1
63=(2W+11)*w
2W^2+11w-63=0
solve the quadratic equation using the formula
a=2
b=11
c=-63
substitute
[tex]w=\frac{-11\pm\sqrt[]{11^2-4(2)(-63)}}{2(2)}[/tex][tex]\begin{gathered} w=\frac{-11\pm\sqrt[]{625}}{4} \\ \\ w=\frac{-11\pm25}{4} \\ \end{gathered}[/tex]the solutions for W are
w=3.5 and w=-9 (is not a solution, because is negative)
so
Find the value of L
L=2W+11 -------> L=2(3.5)+11
L=18
therefore
the dimensions are
Length is 18 yardsWidth is 3.5 yards