A person in a rowboat two miles from the nearest point on a straight shoreline wishes to reach a house six miles farther down the shore. if the person can row at a rate of 3 mi/h and walk at a rate of 5 mi/h, find the least amount of time required to reach the house. how far from the house should the person land the rowboat?
Answers
And the answer for this question is: The person should walk 6 - 1.5 miles or 4.5 miles along the shore
Through a time, distance and speed optimization problem, we find the least amount of time to reach the house is approximately 2.11 hours. The person should land the boat 2 miles from the house.
Explanation:The problem given is a classic time, distance and speed optimization problem which can be solved with a bit of calculus and geometric reasoning. The first step is to define variables. Let's say the person decides to row to a point x miles down the shore, and then walk the remaining distance.
The time taken to row is the distance rowed divided by the rowing speed, and since the rowing distance is the hypotenuse of the right triangle formed, it is sqrt(4+[tex]x^2[/tex]) miles. Hence, time rowing is sqrt(4+[tex]x^2[/tex])/3 hours.
The distance walked is 6-x and hence, the time walking is (6-x)/5 hours.
The total time is then T(x)=sqrt(4+[tex]x^2[/tex])/3+(6-x)/5. The task is to find an x that minimizes T(x). To do so, take the derivative of T(x), set it equal to 0 and solve for x. After conducting these steps, you find that the minimum time is attained at x=2. Therefore, the least amount of time required would be T(2) which is approximately 2.11 hours. The person should land the rowboat 2 miles from the house.
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Related Questions
Antoine wants to convert millimeters into kilometers. Which operation should he choose?
Answers
1m = 100m
1000m = 1000 x 100cm
1000m = 100 000cm
1cm = 10 mm
100 000 cm = 100 000 mm x 10mm
100 000 cm = 1000 000 mm
therefore 1km = 1000 000mm
23. A glass box shaped like a rectangular prism has width 7 inches, length 9 inches, and height 12 inches. It is being shipped in a larger box with width 10 inches, length 10 inches, and height 15 inches. The space between the glass box and the shipping box will be filled with Styrofoam peanuts. How many cubic inches of space will contain Styrofoam peanuts? Show all work.
Answers
larger box volume = 10 * 10 * 15 = 1,500 cubic inches
Styrofoam peanut volume = 1,500 - 756 = 744 cubic inches
Provide a possible situation that the graph shown could represent
Answers
Given f(x)=3x^2+10x−8 and g(x)=3x^2−2x .
What is (fg)(x)
Answers
Answer:
[tex]\frac{x+4}{x} \hspace{8}where\hspace{8}x\neq0,\frac{2}{3}[/tex]
Step-by-step explanation:
The division between functions is defined as:
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)} ,\hspace{10}g(x)\neq0[/tex]
So:
[tex](\frac{f}{g} )(x)=\frac{3x^2+10x-8}{3x^2-2x}\\ \\Factor\hspace{3}x\hspace{3}out\hspace{3}the\hspace{3}denominator\\\\(\frac{f}{g} )(x)=\frac{3x^2+10x-8}{x(3x-2)}\\\\Factor\hspace{3}the\hspace{3}numerator\\\\(\frac{f}{g} )(x)=\frac{4(3x-2)+x(3x-2)}{x(3x-2)}\\\\Factor\hspace{3}3x-2\hspace{3}from\hspace{3}the\hspace{3}numerator\\\\(\frac{f}{g} )(x)=\frac{(3x-2)(x+4)}{x(3x-2)}=\frac{x+4}{x}[/tex]
Since [tex]g(x) \neq0[/tex] , let's find its roots:
[tex]3x^2-2x=0\\\\Factor\\\\x(3x-2)=0\\\\Split\hspace{3}into\hspace{3}two\hspace{3}equations:\\\\(1):x=0\\(2):3x-2=0[/tex]
For (2)
[tex]3x=2\rightarrow x=\frac{2}{3}[/tex]
Therefore the roots are:
[tex]x=0,\hspace{3}x=\frac{2}{3}[/tex]
Finally the complete answer is:
[tex](\frac{f}{g})(x)= \frac{x+4}{x} \hspace{8}where\hspace{8}x\neq0,\frac{2}{3}[/tex]
Give the domain and range a. Domain: {-3, 0, 2}, range: {3, 0, -2} b.
Answers
The domain of a function is the set of all possible input values, while the range is the set of possible output values. In the given example, the function's domain is {-3, 0, 2}, which means the input data will be either -3, 0, or 2. Similarly, the function's range is {3, 0, -2}, which means the output data will be either 3, 0, or -2.
Explanation:In mathematics, the domain of a function refers to the set of all possible input values (often represented as 'x' values) that the function can accept without producing an undefined result. Similarly, the range of a function refers to the set of all possible output values (often represented as 'y' values) that the function can produce.
To address your examples: for a function with domain {-3, 0, 2} and range {3, 0, -2}, it means all input data ('x' values) will be either -3, 0, or 2; and all output data ('y' values) will be either 3, 0, or -2.
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1. Use an inequality symbol (<, >, ≥, =) to compare −26 ____ 54
Answers
Fernando evaluated the expression below. What was Fernando’s error?
Answers
(should be +10 instead of -10)
Answer:
C. Fernando incorrectly found the product of -2 and -5.
Step-by-step explanation:
We have been given an image, which represents work of Fernando's evaluation of an expression. We are asked to find the error in Fernando's work.
[tex]\frac{5(9-5)}{2}+(-2)(-5)+(-3)^2[/tex]
Let us evaluate our given expression using order of operations (PEMDAS).
[tex]\frac{5(4)}{2}+(-2*-5)+9[/tex]
[tex]\frac{20}{2}+10+9[/tex]
[tex]10+10+9[/tex]
[tex]29[/tex]
Since we know that product of two negative numbers is always positive, therefore, the product of negative 2 and negative 5 will be positive 10.
Therefore, Fernando incorrectly found the product of -2 and -5 and option C is the correct choice.
An advertisement on a billboard measures 22 ft long and 8 ft high. If the ad is transferred to the side of a bus and is 30 in. Long, how tall is the new ad, to the nearest inch
Answers
the new ad is 10 ft and 11 inches tall, to the nearest inch.
What is Scale?The ratio between a distance on a map and its actual distance on the ground is known as the map's scale. Since scale must vary throughout a map due to the curvature of the Earth's surface, this straightforward idea is made more difficult. This change makes the idea of size significant in two different ways.
Given, An advertisement on a billboard measures 22 ft long and 8 ft high. If the ad is transferred to the side of a bus and is 30 ft. Long.
when the advertisement is 22 ft long then the height is 8 ft.
So,
When the advertisement is 1 ft long then the height is = 8/22 ft.
the advertisement is 1 ft long then the height is = 4/11 ft.
thus,
the advertisement is 30 ft long then the height is = 4/11 * 30 ft.
the advertisement is 30 ft long then the height is = 120/11 ft.
the advertisement is 30 ft long then the height is = 10 ft 11 inches.
Therefore, When the advertisement is 30 ft long then the height is 10 ft 11 inches.
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How much will it cost to mail a first-class letter that weighs 2.8 oz?
The rate is $.22 for the first ounce and $.17 for each additional
ounce or fraction of an ounce.
a. $.56
b. $5.60
c. $.53
d. $.39
Answers
Final answer:
The total cost to mail a first-class letter that weighs 2.8 oz is $0.56, which includes $0.22 for the first ounce and $0.34 for the additional weight.
Explanation:
To calculate the cost of mailing a first-class letter that weighs 2.8 oz, you need to know the initial rate for the first ounce and the rate for each additional ounce. According to the given rates, the cost for the first ounce is $0.22, and each additional ounce (or fraction thereof) costs $0.17.
Since the letter weighs 2.8 oz, you have 1.8 oz over the first ounce. Thus, you need to pay for 2 additional ounces.
The total cost = cost for first ounce + cost for additional ounces = $0.22 + 2 x $0.17 = $0.22 + $0.34 = $0.56. Therefore, the correct answer is (a) $.56.
Add the opposite number of 1 1/5 to the sum of the numbers (−8 3/4 ) and (−2 5/6 ).
Answers
-8 3/4 = -8 9/12 (multiply 4 by 3 to get 12, so multiply the numerator, 3, by 3 as well)
-2 5/6 = -2 10/12 (multiply 6 by 2 to get 12, so multiply the numerator, 5, by 2 as well)
-8 9/12 + -2 10/12 = -10 19/12
We must convert the improper fraction. 12 goes into 19 1 time with 7 left over, so 19/12 = 1 7/12
This means that -10 19/12 = -11 7/12
The opposite of 1 1/5 is -1 1/5. Add this to -11 7/12. Again find the LCD; in this case it is 60.
-1 1/5 = -1 12/60
-11 7/12 = -11 35/60
Adding these two we get -12 47/60.
The result of the sum of the three numbers is: [tex]\frac{-767}{60}[/tex]
The numbers are given as:
Number 1: 1 1/5Number 2: -8 3/4Number 3: -2 5/6Start by calculating the sum of numbers 2 and 3 as follows:
[tex]Sum = -8\frac 34 -2\frac 56[/tex]
Express the numbers as improper fraction
[tex]Sum = -\frac{35}4 -\frac{17}6[/tex]
Take LCM
[tex]Sum = \frac{-3 \times 35 - 2 \times 17}{12}[/tex]
[tex]Sum = \frac{-139}{12}[/tex]
The opposite of number 1 is: -1 1/5
So, the overall sum is:
[tex]Sum = -1\frac 15 - \frac{139}{12}[/tex]
Express as improper fraction
[tex]Sum = -\frac 65 - \frac{139}{12}[/tex]
Take LCM
[tex]Sum = \frac{-72 - 695}{60}[/tex]
[tex]Sum = \frac{-767}{60}[/tex]
Hence, the result of the sum is: [tex]\frac{-767}{60}[/tex]
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What is the area of a sector with a central angle of 10π/7 radians and a radius of 18.4 m? Use 3.14 for π and round your final answer to the nearest hundredth. Enter your answer as a decimal in the box.
Answers
Area of sector= \frac{\Theta}{360}\times \Pi \times r^{2}
Area =\frac{10\Pi }{7 }\times \frac{1}{360}\times \Pi \times (18.4)^{2}
Area = \frac{1800 }{7 }\times \frac{1}{360}\times (3.14) \times (18.4)^{2}
So, Area of the given sector= 759.34 square meters.
Answer:
759.34 m²
Step-by-step explanation:
Joselyn is a manager at a sign-painting company. She has two painters, Allen and Brianne. Allen can complete a large project in 16 hours. Brianne can complete the project in 18 hours. Joselyn wants to know how long it will take them to complete the project together.
Write an equation and solve for the time it takes Allen and Brianne to complete the project together. Explain each step.
Answers
Brianne can take 18 hours;
Therefore, in 1 hour Allen will complete a fraction of 1/16, while
in 1 hour Brianne will complete a fraction of 1/18
When working together, in 1 hour they will complete a fraction of (1/18+1/16)
= 17/144,
Therefore; to complete the whole project working together, they will take;
144/17 = 8.47 hours
Use two variables k and total to write a for loop to compute the sum of the squares of the first 50 counting numbers, and store this value in total. thus, your code should put 1*1 2*2 3*3 ... 49*49 50*50 into total. use no variables other than k and total.
Answers
With the sens of "summing" or "accumulating", this is one way how it can be coded:
int total=0;
for k=1 to 50 {
total+=k^2;
};
print(total);
In GHI, HI = IHG =3x + 4, and IGH = 2x - 24. What is HIG?
Answers
<IGH = <HIG = 2x - 24
as you know sum of interior angles in a triangle = 180
so
2(2x - 24) + 3x + 4 = 180
4x - 48 + 3x + 4 = 180
7x - 44 = 180
7x = 224
x = 32
so
<HIG = <IGH = 2(32) - 24 = 64 - 24 = 40
answer
m<HIG = 40 degrees
Burt & ernie went to the toy store & saw a total of 51 rubber duckies. they saw twice as many yellow as orange. let y = the # of yellow duckies & r = the # of orange duckies. write a system that could be used to find the total of each color & then solve.
Answers
y = 2r
There are 34 yellow rubber ducks.
There are 17 orange rubber ducks.
I need help! Really struggling with understanding how to do this
Answers
y>=x+1
y>=4-x
using a graph tool
see the attached figure
the solution is the intersection of both inequalities
solve f(1) for f(x)=1-5x
f(1)= [?]
Answers
________________________________________
" f(x) = - 4 " .
________________________________________
Explanation:
________________________________________
Given: " f(x) = 1 – 5x " ; Find: " f(1) " ;
Substitute "1" for values of "x" ;
→ f(1) = 1 – 5(1) = 1 – 5 = - 4 .
The answer is: "- 4 " .
________________________________________
" f(x) = - 4 " .
________________________________________
The 12 members in Dante's hiking club shared 176 ounces of trail mix equally. How many ounces of trail mix did each member get?
Answers
so each member receives 14 2/3 ounces of trail mix
Line segment XY is tangent to circle Z at point U. If the measure of UV is 84, what is the measure of YUV. A. 42 B. 84 C. 96 D. 168
Answers
Answer:
The measure of angle YUV is equal to [tex]m<YUV=42\°[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we have that
[tex]arc\ UV=84\°[/tex] -----> given problem
so
[tex]m<UZV=84\°[/tex] ------> by central angle
we know that
The triangle UZV is an isosceles triangle
because
[tex]ZU=ZV=radius[/tex]
so
[tex]m<ZUV=m<ZVU[/tex] -----> bases angle of the isosceles triangle
Remember that
The sum of the internal angles of a triangle is equal to [tex]180\°[/tex]
so
[tex]m<ZUV+m<ZVU+m<UZV=180\°[/tex]
[tex]2m<ZUV+m<UZV=180\°[/tex]
substitute and solve for m<ZUV
[tex]2m<ZUV+84\°=180\°[/tex]
[tex]m<ZUV=(180\°-84\°)/2=48\°[/tex]
[tex]m<ZUV+m<YUV=90\°[/tex] ------> by complementary angles
solve for m<YUZ
[tex]48\°+m<YUV=90\°[/tex]
[tex]m<YUV=90\°-48\°=42\°[/tex]
Answer: A. 42
Step-by-step explanation: trust i got a 100% on the quiz
What is the volume of the cone with diameter 7in and height 9in? Round to the nearest cubic inch
Answers
radius = 7÷2 = 3.5
Volume of cone = 1/3 [tex] \pi r^{2} h[/tex]
Volume of cone = 1/3 [tex] \pi 3.5^{2} (9)[/tex] = 115 cubic inches (Answer B)
Which statement describes the translation of y = −one half (x − 2)2 − 2 from standard position?
Answers
[tex]y = a(x-h)^{2} +k[/tex]
a represents reflection over the x-axis, and a vertical stretch or compress
- is reflection and a fraction (1/2) represents a compression.
-h represents a shift of that many units to the right (-2 shifts to the right two units)
k represents a shift up or down (-2 is shifting down 2 units_
Reflected over the x-axis, Vertically compressed by a factor of 1/2, shifted 2 units to the right, and shifted 2 units down
Frank is constructing the circumscribed circle for △XYZ. He has already used his compass and straightedge to complete the part of the construction shown in the figure.
Which construction could be his next step?
Place the point of the compass on point Y and draw an arc intersecting the two arcs.
Place the point of the compass on point X and draw an arc intersecting the two arcs.
Place the point of the compass on point Z and draw an arc through points X and Y.
Place the point of the compass on one of the arcs and draw an arc through point X.
Answers
the diagram already has arcs marked from point Y, with the aim of drawing a perpendicular bisector to line XY.
Therefore the next step is to place the compass on point X and draw two arcs which cross the arcs already drawn (second option is the correct answer)
The points at which the 2 arcs meet are connected to each other making a perpendicular bisector that would cut XY perpendicularly.
Next step is to place the compass at Z and draw 2 arcs between Z and Y points, similarly place the compass at Y and draw 2 arcs that would cut the arcs already drawn.
Then connect the two points where the arcs meet and thats the perpendicular bisector of ZY.
The point at which perpendicular bisectors of XY and YZ meet is known as the circumcenter. Place the compass on this point with the radius any of the three points X,Y or Z and draw the circle
Answer:
Place the point of the compass on point X and draw an arc intersecting the two arcs.
Step-by step explanation:
This is to complete the angle bisector.
What is 7200 minutes, in days? (Remember, there are 60 minutes in an hour and 24 hours in a day.)
Answers
Answer:
5 Days is the answer
Which algebraic expression is equivalent to the expression below?
9(8x + 10) + 9(9 - x)
A. 63x - 171
B. 81x + 171
C. 63x + 9
D. 63x + 171
Answers
9* 10 = 90
so far, 72x + 90 + 9(9-x)
9 * 9 = 81
9 * -x = -9x
now, we have: 72x + 90 + 81 -9x
Now we can simplify by combining like terms
72x - 9x = 63x
90 + 81 = 171
63x + 171 is our answer
The answer is B.
Hope I helped!
Let me know if you need anything else!
~ Zoe
What is the line of symmetry for the parabola whose equation is y = x2 + 10x + 25? x = -10 x = -5 x = 5
Answers
Answer:
[tex]y=x^2 + 10x + 25\\\\y=(x+5)^2[/tex]
Used the identity, [tex](a+b)^2=a^2+2ab+b^2[/tex]
The given parabola is of the form , [tex](x+a)^2=y[/tex], having vertex at ,which can be obtained by
x+a=0
x= -a
(-a,0).
So, vertex of the given parabola is , at (-5,0).
The Meaning of term line of symmetry,is that line which divides the parabola in two equal halves.
Drawing the parabola,and finding the line of symmetry,which can be obtained by drawing a line parallel to y axis, passing through (-5,0).
So, the equation of line is : x= -5
A tourist in ireland wants to visit seven different cities. if the route is randomly selected, what is the probability that the tourist will visit the cities in alphabetical order? round your answer to five decimal places.
Answers
n----------> number of cities
the probability that the tourist will visit the cities in alphabetical order is (1/n!)
P=(1/n!)
n!-------------> (7*6*5*4*3*2*1)=5040
therefore
1/5040=0.000198-----------> (five decimal places)--------> 0.00020 (0.020%)
the answer is 0.00020 (0.020%)
The probability that the tourist will visit the cities in alphabetical order is 0.00020.
There are a total of 7! ways to visit 7 different cities, but there is only one way to visit them in alphabetical order. So, the probability that the tourist will visit the cities in alphabetical order is:
1/7! = 1/5040 = 0.0002
To five decimal places, the probability is 0.00020.
Here is a step-by-step explanation of the calculation:
We can think of the route as a permutation of 7 letters, with no of the letters being identical.
The number of permutations of 7 letters is 7!.
So, the number of ways to visit the cities in alphabetical order is 1.
The probability that the tourist will visit the cities in alphabetical order is 1/7!.
Here are some additional things to consider:
The probability that the tourist will visit the cities in alphabetical order is very small, because there are many other possible routes.
The probability would be 1 if there were only one city to visit.
The probability would be 0 if there were 7 cities and the tourist had to visit them all in alphabetical order.
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Consider the expression 625(5xy)^-3/ (5x)^2 y^7
Answers
625
= -------------------- / 25x^2y^7
125 x^3y^3
= 5/x^3y^3 / 25x^2y^7
= 5/x^3y^3 * (1/ 25x^2y^7)
= 1 / 5x^5y^10
answer
1
-----------------
5x^5y^10
Answer:
1
-----------------
5x^5y^10
Step-by-step explanation:
It takes Makhaya 75 minutes and 3 paper sheets to complete a writing assignment, and it takes him 15 minutes and 1 paper sheet to complete a math assignment. Makhaya is required to spend more than 300 minutes to complete assignments, and he can use at most 20 paper sheets.
Let W denote the number of writing assignments he completes and M the number of math assignments he completes.
Write an inequality that represents the condition based on the number of minutes.
Write an inequality that represents the condition based on the number of paper sheets.
Answers
Answer:
Part a) [tex]75W+15M > 300\ minutes[/tex]
Part b) [tex]3W+M \leq 20\ paper\ sheets[/tex]
Step-by-step explanation:
Let
W------> the number of writing assignments that Makhaya completes
M-----> the number of math assignments that Makhaya completes
so
case A) Based on the number of minutes
[tex]75W+15M > 300[/tex] -----> inequality that represent the situation based on the number of minutes
The solution is the shaded area between the positive values of W and M
see the attached figure N 1
case B) Based on the number of paper sheets
[tex]3W+M \leq 20[/tex]-----> inequality that represent the situation based on the number of paper sheets
The solution is the triangular shaded area between the positive values of W and M
see the attached figure N 2
Answer:
First box: 75W+15M>300
Second box: 3W+M≤ 20
Step-by-step explanation:
Did it on Khan
Which algebraic expression represents the phrase “the quotient of negative eight and the sum of a number and three”?
Answers
Don't need 20 characters
We have:
the quotient:
() / ()
negative eight:
(-8) / ()
The sum of a number and three:
Let:
x: the number
(-8) / (x + 3)
Answer:
An algebraic expression that represents the phrase is
(-8) / (x + 3)
"The quotient of negative eight and the sum of a number and three"
Jim's family went on vacation and rented a car. The rental car agency charged $64.75 plus an additional $0.03 for each mile the car was driven. If Jim's family paid a total of $71.14 for the car rental, how many miles did the family drive the car? Explain how you set up an equation to solve this word problem
Answers
Each mile = $0.03
$6.39 ÷ $0.03 = 213 miles.