a woman at a point a on the shore of a circular lake with radius 1.2 mi wants to arrive at the point B diametrically opposite A on the other side of the lake in the shortest possible time (see the figure). she can walk at the rate of 5 mi/h from C to B and row a boat at 3.5 mi/h from A to C . setting T (x) function to the time by interlaced AC, x is interlacted AC. how to move for shortest.
The angle θ to the diameter that she should row is; θ = 56.44°
How to find the angle of an arc?It should be noted that;
a = 90° - θ. Thus;
b = 2θ
For any circle, the angle subtended (in radians) is equal to the arc length (d_w ) divided by the radius (1.2). Thus; d_w = 4θ .
The time required to walk this distance is given by;
t_w = 4θ/5 = 0.8θ
The distance rowed is given by: cos θ = d_r/2.4
d_r = 2.4cos θ while the time needed for it is;
t_r = 2.4cos θ/3.5
t_r = 0.6857θ
Thus, total time is;
T(θ) = t_r + t_w
T(θ) = 0.6857θ + 0.8θ
T(θ) = 1.4857θ
Now we need the critical numbers:
T'(θ) = 1 - 1.2 sin θ
At T'(θ) = 0, we have;
1 - 1.2 sin θ = 0
1/1.2 = sin θ
θ = sin⁻¹(1/1.2)
θ = 56.44°
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Find three consecutive integers such that the
sum of the first and the third is 32.
The three consecutive integers are 15 16 and 17.
Step 1:
Let X be the first integer. Since they are consecutive, it means that the second number will be X + 1 and the third number will be X + 2 and sum of the first and the third should add up to 32. Therefore, you can write the equation as follows:
(X) + (X + 1) + (X + 2) be the three consecutive integers.
Step 2:
The sum of the first and the third should add up to 32.
X + ( X + 2 )= 32
2X + 2= 32
2X = 32 - 2
2X = 30
X = 30/2
X=15
Hence the first number is 15, the second number is 15 + 1, and the third number is 15 + 2.
Therefore, three consecutive integers in which the sum of the first and the third is 32 are 15 16, and 17.
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(LOOK AT PHOTO) What is the quotient of the rational expression below?
x²-49 x²-14x+49
x+2
3x+6
The quotient [tex]\frac{x^2 - 49}{x + 2} \div \frac{x^2 - 14 +49}{3x + 6}[/tex] is [tex]\frac{3(x +7)}{(x -7)}[/tex]
How to determine the quotient?The expression is given as:
[tex]\frac{x^2 - 49}{x + 2} \div \frac{x^2 - 14 +49}{3x + 6}[/tex]
Express x^2 - 49 as difference of two squares
[tex]\frac{(x + 7)(x- 7)}{x + 2} \div \frac{x^2 - 14 +49}{3x + 6}[/tex]
Factorize other expressions
[tex]\frac{(x + 7)(x- 7)}{x + 2} \div \frac{(x -7)(x-7)}{3(x + 2)}[/tex]
Express as product
[tex]\frac{(x + 7)(x- 7)}{x + 2} \times\frac{3(x + 2)}{(x -7)(x-7)}[/tex]
Cancel the common factors
[tex]\frac{(x + 7)}{1} \times\frac{3}{(x -7)}[/tex]
Evaluate the product
[tex]\frac{3(x +7)}{(x -7)}[/tex]
Hence, the quotient [tex]\frac{x^2 - 49}{x + 2} \div \frac{x^2 - 14 +49}{3x + 6}[/tex] is [tex]\frac{3(x +7)}{(x -7)}[/tex]
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Christmas bulbs made of different colours are set to light after 8seconds,10 seconds and 14 seconds. How many times will they light simultaneously in one hour if they start together
Answer:
36 seconds will be the answer... by looking down here↓:
Step-by-step explanation:
6 secs x 6 cycles = 36 seconds
9 secs x 4 cycles = 36 secs
12 ses x 3 cycles = 36 secs
What are the zeros of the quadratic function f(x) equals 6X squared +12 X -7
The zeroes of the function will be [tex]\rm -1 \pm \dfrac{\sqrt{78}}{6}}[/tex]
What is a Quadratic Function ?A quadratic function is given by ax² +bx+c = f(x)
The given quadratic function is
f(x) = 6x² +12x - 7
The zeroes of the function will be given by
f(0) = 0
6x² +12x - 7 = 0
The zeroes are given by
[tex]\rm \dfrac {-b \pm \sqrt{b^2 -4ac}}{2a}[/tex]
here
a = 6
b = 12
c = -7
Therefore the zeroes of the function will be
[tex]\rm \dfrac {-12 \pm \sqrt{12^2 -4*6*(-7)}}{2 *6}[/tex]
[tex]\rm \dfrac {-12 \pm \sqrt{312}}{12}[/tex]
[tex]\rm -1 \pm \dfrac{\sqrt{78}}{6}}[/tex]
Therefore these are the two zeroes of the function.
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A right cylinder has a radius of 5 and a height of 9. What is its surface area?
A. 45 units²
B. 140 units²
C. 90 units²
D. 70 units²
Answer:
the answer is B. 140 units²
The midpoint of FG is point H at (–5, 2). One endpoint is G(–9, –6). What is the y-coordinate of the other endpoint? The y-coordinate of the other endpoint is
The y-coordinate of the other endpoint is 10 units.
The given coordinates are H (-5, 2) and G (–9, –6).
We need to find the y-coordinate of the other endpoint.
What is the midpoint formula?The mid-point formula is [tex](x, y)=(\frac{x_{1}+x_{2} }{2} , \frac{y_{1}+y_{2} }{2})[/tex]
Now, [tex](-5, 2)=(\frac{x_{1} +(-9)}{2} , \frac{y_{1} +(-6)}{2})[/tex]
⇒[tex]2=\frac{y_{1}-6 }{2}[/tex]
⇒[tex]4=y_{1}-6[/tex]
⇒[tex]y_{1}=10[/tex]
Therefore, the y-coordinate of the other endpoint is 10 units.
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There is a mound of g pounds of gravel in a quarry. Throughout the day, 200 pounds of gravel is added to the mound. Two orders of 700 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.
Answer:
[tex]\fbox {2,400 pounds}[/tex]
Step-by-step explanation:
Information we have :
200 pounds is added2 orders of 700 pounds is removed1,200 pounds remainWhat we need to find :
Original amountSolving :
g + 200 - 2(700) = 1200g + 200 = 1200 + 1400g + 200 = 2600g = 2,400 pounds of gravelIf the length of rectangle is thrice of its breadth and it's perimeter is 32 cm then finds its area.
Answer:
48 cm²
Explanation:
Let the breadth be b, then the length is 3b
P = 2(Length + Breadth)
32 = 2(3b + b)
32 = 2(4b)
8b = 32
b = 4
Breadth is 4 cm
Length: 3b = 3(4) = 12 cm
Area of rectangle:
Length × Breadth
12 × 4
48 cm²
Given :-
Length of rectangle is thrice of its breadth.Its perimeter is 32 cm.To Find :-
Area of rectangle?Solution :-
Let breadth of rectangle be x cm As it is stated in question that length of rectangle is thrice its breadth so length of rectangle will be 3x cmUsing formula;
Perimeter of rectangle = 2(L + B)Where;
L denotes length of rectangleB denotes a breadth of rectangleWe have;
Perimeter of rectangle = 32 cmLength of rectangle (L) = 3xBreadth of rectangle (B) = xBy putting all values in formula we get;
→ 2(3x + x) = 32
→ 2(4x) = 32
→ 2 × 4x = 32
→ 8x = 32
→ x = 32/8
After dividing 32 with 8, we get;
→ x = 4
Hence;
Length (L) = 3x = 3 × 4 = 12 cmBreadth (B) = x = 4 cmNow, using formula;
Area of rectangle = L × BWhere;
L denotes length of rectangleB denotes a breadth of rectangleWe have;
Length of rectangle (L) = 12 cmBreadth of rectangle (B) = 4 cmArea of rectangle = ?By putting all values in formula we get;
→ Area of rectangle = 12 × 4
By multiplying 12 with 4, we get;
→ Area of rectangle = 48 cm²
Hence, area of rectangle is 48 cm².The planet Earth is about 193,000,000 miles away from the sun, and the speed of light is approximately 1.86 × 10 5 miles per second. Which expression could you use to figure out how many seconds it takes the light from the sun to reach Earth?
Answer:
Speed of light 186000 miles/second.
it takes 500 seconds (approx 8 minutes) for light from the sun to reach Earth.
Step-by-step explanation:
Earth travels around the sun in a slightly oval-shaped orbit, known as an ellipse. Therefore, the planet's distance from the sun changes throughout the year.
However, the average distance from Earth to the sun is about 93 million miles (150 million kilometres). Scientists also call this distance one astronomical unit (AU).
A universe is a big place, and sometimes researchers use astronomical units to communicate how far celestial objects are separated from one another. For example, Jupiter orbits about 5 AU from the sun.
The distance between the earth and the sun d= velocity of light × time taken to reach light from the sun to earth
Distance from Earth to Sun=93000000 miles.
Speed of light = 186000 miles/second.
=> [tex]\frac{93000,000}{186000}[/tex]
=> 500 seconds.
=> 8.333 minutes
it takes 500 seconds (approx 8 minutes) for light from the sun to reach Earth.
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Calcular cuántos números enteros diferentes de tres dígitos se pueden formar con los dígitos 2, 3 ,4 ,5 ,6 ,7 , 8 si los dígitos no pueden repetirse
There are 210 different three-digit whole numbers that can be formed with the digits 2, 3, 4, 5, 6, 7, 8 if the digits cannot be repeated.
To calculate how many different three-digit whole numbers can be formed with the digits 2, 3, 4, 5, 6, 7, 8 if the digits cannot be repeated, we need to use the permutation formula.
The number of permutations of n objects taken r at a time, where order matters and objects cannot be repeated, is given by:
P(n,r) = n! / (n-r)!
In this case, we have 7 digits to choose from, and we want to form a three-digit number, so r = 3. Therefore, the number of different three-digit whole numbers that can be formed is:
P(7,3) = 7! / (7-3)! = 7! / 4! = 7 x 6 x 5 = 210
Each of these numbers will be unique, as we are not allowed to repeat any of the digits.
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Evaluate 21 + (-3c) when c=-2
27
Solution:
Substitute c with -221 + ( -3 ) ( -2 )
Simplify21 + 6
27
Therefore, The answer is 27.
3. When using set builder notation, what does it mean to 'Define a set'?
Answer:
A set is a well-defined collection of distinct mathematical objects.
3. What's the difference between 126 1/4 and 78 2/3?
O A. 57 7/12
OB. 47 7/12
O C. 58 5/12
O D. 48 1/3
Answer:
B. 47 7/12
Step-by-step explanation:
=126 1/4 - 78 2/3
Change the two mix fraction to improper fractions.
= 505/4 - 236/3
= 1515-944/12
= 571/12
= 47 7/12
IN THE GIVEN FIG, FIND THE VALUE OF x.
The answer is simple
First find the angle measures of first triangle:
statement:
we know that the sum of 45 and 30 sums up to the opposite angle and that is equal to 75, then to find the next remaining angle:
180-(75+20)=85
The you find the supplement of 85:
180-85=95
so x = 95
Hope it helps
Each point has been reflected over the x-axis or y-axis.
Select the axis over which each point has been reflected.
The reflection is
x-axisx-axisy-axisWhat is reflection over axis?A reflection of a point, a line, or a figure in the X axis involved reflecting the image over the x axis to create a mirror image. In this case, the x axis would be called the axis of reflection.
For reflecting over the X axis is to negate the value of the y-coordinate of each point, but leave the x-value the same.
and, for reflecting over the Y axis is to negate the value of the x-coordinate of each point, but leave the y value the same.
So, by considering the above value rules the reflection of the given points as follows over respective axis.
E(7, 1) ⇒ (7, -1)
Here, the reflection is over x-axis because the y value is changing
F(-3, 5) ⇒ (-3, -5)
Here, the reflection is over x-axis because the y value is changing
G(6, -2) ⇒ (-6, -2)
Here, the reflection is over y-axis because the x value is changing.
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2/5x - 1/2 = 1/3 + 5/6x
Answer:
[tex]x = \frac{25}{13} [/tex]
Step-by-step explanation:
Given:
[tex] \frac{2}{5} x - \frac{1}{2} = \frac{1}{3} + \frac{5}{6} x[/tex]
Combine like terms:
[tex] \frac{2}{5} x - \frac{5}{6} x = \frac{1}{3} + \frac{1}{2} [/tex]
Find GCF :
[tex] \frac{12}{30} x - \frac{25}{30} = \frac{2}{6} + \frac{3}{6} [/tex]
Subtraction and Addition :
[tex] \frac{13}{30} x = \frac{5}{6} [/tex]
Divide both sides by 13/30 :
[tex] \frac{ \frac{13}{30} }{ \frac{13}{30} } x = \frac{ \frac{5}{6} }{ \frac{13}{30} } [/tex]
Simplify :
Multiplication (Reciprocal) :
[tex]x = \frac{5}{6} \times \frac{30}{13} [/tex]
[tex]x = \frac{150}{78} [/tex]
[tex]x = \frac{25}{13} [/tex]
Answer :
[tex]x = \frac{25}{13} [/tex]
Hope it helps
Solve the following system of equations and show all work.
y = 2x2
y =
3x -1 (10 points)
Answer: [tex]\left(\frac{1}{2}, 1 \right), (1, 2)[/tex]
Step-by-step explanation:
We have [tex]y=2x^{2}[/tex] and [tex]y=3x-1[/tex].
Since both of the equations are set equal to y, we can conclude that:
[tex]2x^2 = 3x-1\\\\2x^{2}-3x+1=0\\\\(2x-1)(x-1)=0\\\\x=\frac{1}{2}, 1[/tex]
If [tex]x=\frac{1}{2}[/tex], then [tex]y=3\left(\frac{1}{2} \right)-1=1[/tex]
If [tex]x=1[/tex], then [tex]y=3-1=2[/tex]
Therefore, the solutions are [tex]\left(\frac{1}{2}, 1 \right), (1, 2)[/tex]
how many units are -7 and its opposite from zero?
Answer:
-7 is -7 units away from zero. Its opposite is 7 from zero.
Step-by-step explanation: Count from 0 how many digits away it is
Anu's age exceeds Sumbo's age by 15 The sum of the square of their ages is 725. What are their ages?
Answer:
Anita= 25 years old
Sumbo= 10 years old
Step-by-step explanation:
Start by forming 2 equations that represent the given information.
Let Anu's and Sumbo's ages be A and S respectively.
A= S +15 -----(1)
A² +S²= 725 -----(2)
Now, solve for A and S by substitution.
Substitute (1) into (2):
(S +15)² +S²= 725
Expand:
S² +2(S)(15) +15² +S²= 725
2S² +30S +225= 725
-725 on both sides:
2S² +30S -500= 0
Divide both sides by 2:
S² +15S -250= 0
Factorise:
(S +25)(S -10)= 0
S +25= 0 or S -10= 0
S= -25 (reject) or S= 10
Sumbo's age cannot be a negative value hence -25 is rejected.
Substitute S= 10 into (1):
A= S +15
A= 10 +15
A= 25
A menu has five choices of appetizers
The number of the meals possible is 350.
The complete question is given below:-
A menu has 5 choices of appetizers, ten main courses, and seven desserts. How many meals are possible?
What is the combination?The arrangement of the different things or numbers in a number of ways is called the combination.
Given that:-
A menu has 5 choices of appetizers, ten main courses, and seven desserts.The number of the combination of the meals will be calculated as:-
N = 5 x 10 x 7
N = 350
Therefore the number of the meals possible is 350.
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Step 1: Choose the lowest common denominator.
02 03 04 05 06 08 010 012
3/4
50/80
O
Find the sum: -and
The expression written in equivalent form with a
common denominator is
The sum is
The expression written in equivalent form with a common denominator is -1/6
Adding fractionsFractions are written as ratio of two integers. For instance a/b is a fraction.
Given the sum of the fractions shown;
-3/4 and 5/8
Sum = -3/4 + 5/8
Sum = 5/8 - 3/4
Sum = 5-6/8
Sum = -1/8
Hence the sum of the given fraction -1/3 and 5/8 is -1/6
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What is the explicit formula for this sequence?
60, 30, 15, 7.5, ...
A. an = 60.
(9)
OB. an= = (1.60
OC. an = 60.
• (+)-
OD. an=3.2(n-1)
60(n-1)
(n-1)
i beilive this is unknown
Which number is irrational?
OA. 0.45
OB. 0.636363...
O C. √25
OD. √6
Answer:
D. Sqroot6
Step-by-step explanation:
6 is not a perfect square. So it is a non-repeating decimal that never ends
sqroot6 ~=
2.4494897428...
this is an irrational number.
The rest on the numbers can be written as a ratio:
0.45=45/100=9/20
0.63636363
= 63/99 = 7/11
sqrt25 = 5 = 5/1
that means they are rational.
Identify the value of the variable in the equivalent expressions.
What is the value of n in the simplified expression?
(4k7)3= 4n ·(k7) 3= 64k21
n =
What is the value of m in the simplified expression?
(Negative 3 r Superscript negative 4 Baseline) Superscript negative 4 Baseline = (negative 3) Superscript negative 4 Baseline times (r Superscript negative 4 Baseline) Superscript negative 4 Baseline = StartFraction 1 Over 81 EndFraction r Superscript m
m =
The value of expression is n=3 and r= 16
What is expression?An expression is a set of terms combined using the operations +, – , x or , /.
Given:
[tex](4k^{7})^{3} = 4^{n}* (k^{7})^{3} = 64 k^{21}[/tex]
[tex](64 k^{21}) = 4^{n}* k^{21} = 64 k^{21}[/tex]
[tex]4 ^{3} * k^{21}= 4^{n}* k^{21}[/tex]
On comparing
n=3
Now,
[tex](-3r^{-4})^{-4} = (-3)^{-4} * (r^{-4})^{-4} = 1/81* r^{m}[/tex]
[tex](-3)^{-4} * (r^{-4})^{-4} = 1/81* r^{m}\\1/ (-3)^{4} * 1/(r^{-4})^{4} = 1/81* r^{m}\\1/ (-3)^{4} * (r^{16}) = 1/81* r^{m}[/tex]
1/81 * r^16 = 1/81 * r^{m}
On compairing
r = 16.
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Answer:
answer in picture
Step-by-step explanation:
y = ( x + z) ( x+ 2x)
Answer:
3x^2 + 3xz
Step-by-step explanation:
Under the assumption that I need to distribute,
(x + z)(x + 2x)
(x + 2x) is (3x)
3x(x + z) becomes 3x*x and 3x*z
3x*x = 3x^2
3x*z = 3xz
combine, 3x^2 + 3xz
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Theresa uses a unique box in the shape of a trapezoidal prism for her specialty candles. The area of the bas
of the box for the smaller candle is 125 cm² and the box is 12 cm tall. The box used for the larger candle has
a base whose area is 160% that of the smaller box, and it is 6 centimeters taller.
What percent of the volume of the smaller box is the volume of the larger box?
Pls explain ur Answer<3
Answer:
eyyyyyyyyyy wassupppppppp
Answer:
240%
Step-by-step explanation:
Volume of a prism is Bh (area of the base * height)
Let's start with the small prism.
B=125
h=12
Volume = 125*12 = 1500 cubic cm
Bigger prism base are is 160% of 125. 1.6 * 125 = 200
Height is 6 cm more = 12+6 = 18
V = 200 * 18 = 3600
To find the percentage, divide 3600 by 1500. You get 2.4, which is 240%
Evaluate the function f(r) = √r + 1 - 1 at the given values of the independent variable and simplify.
a. f(-1) b. f(24) c. f(x-1)
Step-by-step explanation:
I'm going to assume you meant to write [tex]\sqrt{r+1}[/tex] in the equation as +1 - 1 wouldn't make much sense since they would just cancel out.
a. [tex]f(-1) = \sqrt{-1 + 1} - 1\\ f(-1) = \sqrt{0} - 1\\f(-1) = 0-1\\f(-1) = -1[/tex]
b. [tex]f(24) = \sqrt{24 + 1} - 1\\ f(24) = \sqrt{25} - 1\\f(24) = 5 - 1\\f(24) = 4[/tex]
c. [tex]f(x-1) = \sqrt{(x-1)+1} -1\\f(x-1) = \sqrt{x} - 1[/tex]
Plss HELPP I neeed help on this question
Answer: 180 degrees
Step-by-step explanation:
The measure of the arc of a semicircle is 180 degrees.
Eduardo and his friends want to start a band, so he decides to take guitar lessons. He purchases a 6-lesson package at Sharp Notes music store, which comes out to $22.50 per lesson.
