The measure of all the angles is given below :
angle 1 = angle 2 = angle 3 = angle 4 = angle 5 = 60°
Given, ΔWXY is an equilateral triangle and ΔWZY is an isosceles triangle.
We have to find the measure of all the angles,
As, ΔWXY is an equilateral triangle
so, ∠WXY = ∠XYW = ∠YWX = 60°
Now, let ∠WZY = ∠WYZ = x and ∠ZWY = y
so, on applying angle sum property, we get
2x + y =180°
so, y = 60°
and 2x = 120°
x = 60°
Hence, angle 1 = angle 2 = angle 3 = angle 4 = angle 5 = 60°
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for a fixed location, the number of sunlight hours in a day fluctuates throughout the year. suppose that the number of daily sunlight hours in a particular location can be modeled by the following. l(t)
On October 24, 218 days after March 20, there will be 10 hours of sunlight.
The number of daily sunlight hours of a particular location is modelled by the trigonometric equation,
l(t) = 12 + 3.5sin (2[tex]\pi[/tex]/365t)
Here , L(t)= Number of sunlight hours in a day.
t = Number of days after 20th March .
Now we are required to determime the day during first 365 on which there are 10 sunlight hours ,that is, we need the value of "t" for which L(t) =10 hours. So in the given equation by putting L(t)= 10 and solving for t , follows as,
l(t) = 12 + 3.5sin (2[tex]\pi[/tex]/365t)
10 = 12 + 3.5sin (2[tex]\pi[/tex]/365t)
10-12 = 3.5sin (2[tex]\pi[/tex]/365t)
-2 = 3.5sin (2[tex]\pi[/tex]/365t)
-2/3.5 = sin (2[tex]\pi[/tex]/365t)
-0.574 = sin (2[tex]\pi[/tex]/365t)
For the inverse function of y = sin x
sin ⁻¹(y) =x + 2n[tex]\pi[/tex]
Now as we are required to calculate the day having 10 sunlight hours in first 365 days so putting n=0 we get,
t = -35.3521 or t =217.9422
Now t can not be negative as we want to find the day after march 20 having 10 sunlight hours. So neglecting t=-35.3521 . Now we get t=217.9422 .
Rounding off we get ,t=218 which is the required answer.
Hence, 218 days after march 20 i.e on October 24 it will have 10 sunlights.
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which of the following transducers can steer the sound beam electronically creating a parallelogram-shaped image?
Answer:
Step-by-step explanation:
u
Line u has an equation of y = -x + 7. Line v is perpendicular to line u and passes through
(-8, -1). What is the equation of line v?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified
proper fractions, improper fractions, or integers.
The equation of line v is y - x = 7 and The equation in slope-intercept form is y = x + 7.
Given:
y = -x + 7
Compare it to Slope-intercept form y = mx + c (∵ m = slope or gradient)
The gradient (m₁) is -1.
The gradient of the perpendicular line v is given by
m₂= -1/m₁
m₂=-1/-1
m₂=1
To find the duration of the perpendicular line v use the formula
y- y₁ = m(x-x₁) (i.e., one slope, one point formula)
y₁=-1, x₁ =-8 , m₂ =1
y-(-1) = 1[x-(-8)]
y + 1 = x + 8
y - x = 8 - 1
y - x = 7
So, The equation of line v is y - x = 7 and The equation in slope-intercept form is y = x + 7.
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The following list shows the number of goals scored by a soccer team in each of 9 games.
0, 0, 1, 1, 1, 3, 3, 4, 5
How does the median number of goals scored compare with the mean number of goals scored?
answer choices:
The median is equal to the mean
The median is less than the mean by 1.
The median is greater than the mean by 1.
The median is less than the mean by 2.
The median for a particular set is one less than the mean.
Given;
List of order for goals scored by a soccer team in 9 matches
= 0, 0, 1, 1, 1, 3, 3, 4, 5
To find the relation between mean and median;
The obtained mean = Sum of observations/number of observations
= 0 + 0 + 1 + 1 + 1 + 3 + 3 + 4 + 5
= 18/9
= 2
Mean = 2
For median;
let us write the given observations in ascending order;
⇒ 0, 0, 1, 1, 1, 3, 3, 4, 5
Here no. of observations is r = 9 which is an odd number.
Therefore observation at 5th is a middle position which has 1 as a value.
Median = 1
Therefore, the median is less than the mean by 1.
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/. Which equation represents the line that passes through the points (-8, 6) and (-11, -6)?
[A] (v-8)=-4(x+6)
[B] (v+6)= 4(x-8)
[C] (v-6)=-4(x+8)
[D] (v-6) = 4(x+8)
The equation (D) y - 6 = 4(x+8) represents the line passing through the points (-8, 6) and (-11, -6).
What is the point-slope form of a line?The point-slope form can be used to get the equation of a straight line that traverses a specified point and is inclined at a particular angle to the x-axis. A line exists if and only if each point on it fulfills the equation for the line. This suggests that a linear equation in two variables can represent a line.
The given points are
(x₁, y₁) = (-8, 6)
(x₂, y₂) = (-11, -6)
So, the slope is given by
m = (y₂- y₁)/ (x₂- x₁)
= (-6 -6)/ (-11 -(-8)
= -12/ -3
= 4
Put (x₁, y₁) = (-8, 6) in the equation (y - y₁) = m (x- x₁)
y - 6 = 4(x-(-8)
y - 6 = 4(x+8)
So, the correct option is (D).
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An office manager booked 72 airline tickets. He booked 7 more tickets on Airlines A than on Airline B. On Airlines C, he booked 5 more than thrice as many tickets as on Airline B. How many tickets did he book on each airline?
He booked ___ tickets on Airlines A, ___ tickets on Airlines B, and ___ on Airlines C.
Answer:
Hence, he booked 19 tickets for Airlines A, 12 tickets for Airline B, and 41 tickets for Airlines C.
Step-by-step explanation:
Suppose the office manager booked a+b+c = 72 tickets.
a = b + 7
c = 3b + 5.
So, we have (b + 7) + b +(3b + 5) = 72, or
5b + 12 = 72, or
5b = 60, or
b = 12.
Can someone please help
The angle 1 is 88 degree, 2 is 42 degree and the 3 is 113 degree.
We know that the sum of the angles of the triangle is 180, so in one triangle where the 3 angle is missing we can write as:
42 + 25 + ∠3 = 180
∠3 + 67 = 180 ,
∠3 = 113°
now in the triangle where 1 and 2 angles are missing, we can say that ∠2 = 42° because of transversely equal angles:
so, ∠ 2 = 42°
now again for a triangle,
∠1 + 42 + 50 = 180
∠1 + 92 = 180
∠1 = 88°
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Kaden drinks 1 3/4 glasses of milk each day. At this rate, how many days will it
take him to drink a total of 14 glasses?
Answer:
It will take 8 days
Step-by-step explanation:
Mixed numbers in equations
1 3/4 * d = 14
Change the mixed number to an improper fraction
1 3/4 = ( 4*1 + 3) /4 = 7/4
7/4 * d = 14
Multiply each side by 4/7
4/7 * 7/4 *d = 14 * 4/7
d = 8
It will take 8 days.
A family has two cars. The first car has a fuel efficiency of 25 miles per gallon of gas and the second has a fuel efficiency of 30 miles per gallon of gas. During one particular week, the two cars went a combined total of 1525 miles, for a total gas consumption of 55 gallons. How many gallons were consumed by each of the two cars that week?
The volume of gas consumed by the first and second cars is 25 gallons and 30 gallons, respectively.
A family has two cars. The first car has a fuel efficiency of 25 miles per gallon of gas, and the second has a fuel efficiency of 30 miles per gallon of gas. During one particular week, the two cars travelled a combined total of 1525 miles, for a total gas consumption of 55 gallons.
Let the volumes of gas consumed by the first car and the second car be denoted by the variables "x" and "y". We can form the given two equations using the given data.
x + y = 55
25x + 30y = 1525
We will substitute the value of "x" from the first equation into the second equation.
x = 55 - y
25x + 30y = 1525
25(55 - y) + 30y = 1525
1375 - 25y + 30y = 1525
5y = 150
y = 30
x = 55 - y = 55 - 30 = 25
Hence, the gas consumed by the first and second cars is 25 gallons and 30 gallons, respectively.
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What is the answer pls?
The initial value of f(x) is -36.
The initial value of g(x) is -44
The range of f(x) is {y | -36 ≤ y ≤ -20}
The range of g(x) is {y | -44 ≤ y ≤ -20}
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
{x | -6 ≤ x ≤ -2}
g(x) = 6x - 8
From the table, we get that,
The initial value of f(x) is -36.
The initial value of g(x).
g(x) = 6 x (-6) - 8 = -36 - 8 = -44
The range of f(x).
= {y | -36 ≤ y ≤ -20}
g(x) = 6x - 8
g(x) = 6 x (-6) - 8 = -36 - 8 = -44
g(x) = 6 x (-2) - 8 = -12 - 8 = -20
The range of g(x)
= {y | -44 ≤ y ≤ -20}
Thus,
The initial value of f(x) is -36.
The initial value of g(x) is -44
The range of f(x) is {y | -36 ≤ y ≤ -20}
The range of g(x) is {y | -44 ≤ y ≤ -20}
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Use the distance formula to find the distance, to the nearest tenth, from R(7, -2) to
W(-2, 3).
The distance between (7,-2) and (-2,3) is 11.2 units using the distance formula.
What is distance formula?Distance is a measurement of how far apart two objects or points are, either numerically or occasionally qualitatively. Distance can refer to a physical length in physics or to an estimate based on other factors in common usage. a measurement of the distance between two points along a line or line segment. It's formula is "distance formula=√(x2-x1)²+(y2-y1)²".
Here,
distance formula=√(x2-x1)²+(y2-y1)²
=√(-2-7)²+(3--2)²
=√(-9)²+5²
=√81+45
=√126
=11.225 units
rounding to nearest tenth,
=11.2 units
Using the distance formula, the distance between (7,-2) and (-2,3) is 11.2 units.
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What is the vertex and the equation of the axis of symmetry of the graph of Y=x^2-6x-7
help asap
For the given quadratic equation:
y = x² - 6x - 7
The vertex is (3, -16) and the axis of symmetry is at x = 3.
How to get the vertex and axis of symmetry?For a quadratic equation of the form:
y = a*x² + b*x + c
The x-value of the vertex (h, k) is:
h = -b/2a
And the axis of symmetry is the vertical line x = h.
In this case, the quadratic equation is:
y = x² - 6x - 7
The x-value of the vertex is at:
h = -(-6)/2*1 = 3
And the y-value of the vertex is at:
y = (3)² - 6*3 - 7
y = 9 - 18 - 7
y = -16
Then the vertex is at (3, -16)
And the axis of symmetry is x = 3
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*
10 The points A, B and C have coordinates A(2, 8), B(9, 7) and C(k, k-2).
a Given that AB = BC, show that a possible value of k is 4 and find the other possible value of k.
4, find the equation of the line that bisects angle ABC.
b For the case where k = 4， find the equation of the line that bisects angle ABC
By using distance formula the following results are obtained
a) Other value of k = 14
b) For k = 4, the equation of bisector of [tex]\angle ABC[/tex] is
[tex]y = \frac{1}{3}x + 4[/tex]
What is distance formula?
Distance formula is used to find the distance between two coordinates.
If the coordinates of one point be ([tex]x_1, y_1[/tex]) and coordinates of other point is ([tex]x_2, y_2[/tex]), then distance between the two points is
[tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2[/tex]
The points A, B and C have coordinates A(2, 8), B(9, 7) and C(k, k-2).
Length of AB =
[tex]\sqrt{(9 - 2)^2 + (7 - 8)^2}\\\\\sqrt{49 +1}\\\\\sqrt{50}[/tex]
Length of BC =
[tex]\sqrt{(k - 9)^2 + (k - 2 - 7)^2}\\\\\sqrt{(k - 9)^2 + (k - 9)^2}\\\\\sqrt{2(k-9)^2}\\\\(k-9) \sqrt{2}[/tex]
By the problem,
[tex](k-9)\sqrt{2} = \sqrt{50}\\k - 9 = \sqrt\frac{50}{2}}\\k - 9 = \sqrt{25}\\k - 9 = \pm 5\\k - 9 = 5 \ or \ k - 9 = -5\\k = 5 + 9 \ or \ k = 9 -5\\k = 14 \ or \ k = 4[/tex]
The other value of k is 14
b) When k = 4
Coordinate of C = (4, 2)
Since AB =AC , the bisector of [tex]\angle ABC[/tex] passes through the midpoint of AC
Midpoint of AC =
[tex](\frac{2 + 4}{2} , \frac{8+2}{2})\\(3, 5)[/tex]
The line also passes through (9, 7)
Slope of the bisector of [tex]\angle ABC[/tex] = [tex]\frac{7 -5}{9 - 3}[/tex]
= [tex]\frac{1}{3}[/tex]
Equation of the bisector of [tex]\angle ABC[/tex]
[tex]y - 5 = \frac{1}{3}(x - 3)\\\\y - 5 = \frac{1}{3}x -1\\\\y = \frac{1}{3} x - 1 + 5\\\\y = \frac{1}{3} x + 4[/tex]
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i really need help with this
Rita has $20,000 to invest for five years and she found an interest rate of
4%. How much money will she have at the end of five years if the interest
rate compounds monthly?
The amount of money Rita will get after 5 year with interest rate compounds monthly is $4,420.
Explain the term compound interest?The interest on either a deposit that is calculated using both the initial principle and the accrued interest from prior periods is known as compound interest. In other words, compounding is interest that is earned on interest. Different frequency patterns, such as daily, monthly, or yearly, are provided for compounding interest.The formula for the calculation of the compound interest is-
A = P(1 + r/100n)∧nt
In which,
A = amount after compounding
P = principal amount ( $20,000 )
n = number of time principal amount compounded in a year. ( 12 )
t = total time in years. (5 )
r = rate of interest (4%)
Put the value in eq.
A = 20,000(1 + 4/1200)∧5×12
A = 20,000×1.22
A = 24420
CI = A - P
CI = 24420 - 20000
CI = 4,420.
Thus, the amount of money Rita will get after 5 year with interest
rate compounds monthly is $4,420.
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PLEASE ALOT OF POINTS I need an answer quick
Graph the line through (0, -4) with m = -3
Answer:
Step-by-step explanation:
hope it helps
A data set comparing a woman's shoe size to her height is represented by the table.
Shoe Size Height (inches)
7.5 63
9 72.5
11 70
7 62
9 69.5
10 72
12 72.5
13 73
13 70
What is the equation for the line of best fit for a woman's height, y, based on her shoe size, x?
Answer:
10 72
Step-by-step explanation:
Laura received $10 from her aunt on her 8th birthday. On
her 9th birthday, her aunt gave her $20. For her 10th
birthday, her aunt gave her $40. How much do you think
Laura's aunt will give her on her 11th birthday? Justify your
answer (prove your answer is right or reasonable).
Vanya gathered some mushrooms. He dried 2/5 of them. Then he pickled 5/9 of the remaining mushrooms and fried 28 that were left. How many mushrooms did he gather.
Need this stat this is due tomorrow
Answer:
105
Step-by-step explanation:
1. x = mushroom he gather
2. he dry 2/5 from he gather, 2/5 from x
3. he pickled 5/9 of remaining mushroom
find the remaining mushroom value first, x-2/5x = 3/5x
then multiply with 5/9, 5/9(3/5x) = 15/45x
4. total mushroom - dry - pickled, he still have 28 to fry
5. write the equation like this
x - 2/5x - 15/45x = 28
45/45x - 18/45x -15/45x = 28
12/45x = 28
x = 28(45) / 12
x = 105
Can someone pls give me the answer to that?
The values of x and y which are the missing terms of the given triangle have their values as; x = ⁵/₂ and y = ⁹/₄
How to Interpret Division of line segments?
From the given triangle, we can see that the transverse line intersects both side lengths at their midpoints. Thus, we can say that;
3x - 4 = x + 1 ------(1)
⁴/₃y + 2 = 4y - 4 ------(2)
Let us solve equation 1 by rearranging to get;
3x - x = 1 + 4
2x = 5
x = ⁵/₂
Similarly, we can say that for equation 2;
4y - ⁴/₃y = 2 + 4
4y - ⁴/₃y = 6
Multiply through by 3 to get;
12y - 4y = 18
8y = 18
y = 18/8
y = ⁹/₄
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the graph of a linear function is shown . which word describes the slope of the line
Answer:
positive
Step-by-step explanation:
In isosceles triangle dac, ad is congruent to ac. Kiran knows that the base angles of an isosceles triangle are congruent. What additional information does kiran need to know in order to show that ab is a perpendicular bisector of segment cd?.
If DB and BC are congruent, then AB must be a perpendicular bisector. We conclude that AB is congruent to AB by the reflexive characteristic because DB and BC are congruent.
ADB is congruent to ACB according to the Side-Side-Side, or SSS, congruency rule because the sides AD, AB, and DB in triangle ADB are congruent to the equivalent sides AC, BC, and AB in triangle ACB.
As a result, CPCTC ABD = ABC according to the notion of congruency, where Congruent Parts of Congruent Triangles are Congruent.
Due to the fact that ABD and ABC are linear pair angles,
ABD + ABC = 180°,
ABD + ABD = 2, and ABD = 180°
by virtue of the substitution feature,
ABD = 180°/2 = 90°.
As AB is perpendicular to CD and AB is a perpendicular bisector of CD, ABD = 90° = ABC.
Therefore, the option; DB and BC are congruent is what Kiran needs to know in order to demonstrate that AB is a perpendicular bisector of segment CD.
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g(n)=1+5(n−1)
Complete the recursive formula of g(n)=g(n−1)+ ?
The recursive formula which correctly represents the given formula; g ( n ) = g ( n − 1 ) + 5.
What is the recursive formula for the given arithmetic progression?It follows from the task content that the expression which represents the recursive formula.
By observation, the common difference, d of the arithmetic progression is; 5.
On this note, the recursive formula can be written as follows;
g ( n ) = g ( n − 1 ) + 5.
Consequently, the required recursive formula for the given arithmetic progression is; g ( n ) = g ( n − 1 ) + 5.
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A data set comparing a woman's shoe size to her height is represented by the table.
Shoe Size Height (inches)
7.5 63
9 72.5
11 70
7 62
9 69.5
10 72
12 72.5
13 73
13 70
What is the equation for the line of best fit for a woman's height, y, based on her shoe size, x?
y = 1.24x + 70.1
y = −1.24x − 70.1
y = 1.36x + 55.6
y = −1.36x − 55.6
The equation for the line of best fit for a woman's height, y, based on her shoe size, x is y = 1.36x + 55.6.
From the given table:
To find the equation we need to insert the points (x,y) in calculator.
Points are:
Sum of X = 91
Sum of Y = 624.5
Mean X = 10.1111
Mean Y = 69.3889
Sum of squares (SSX) = 42.8889
Sum of products (SP) = 57.6111
Regression Equation = ŷ = bX + a
b = SP/SSX = 57.61/42.89 = 1.34326
a = MY - bMX = 69.39 - (1.34*10.11) = 55.80699
y = 1.34326X + 55.80699
aproximate:
y = 1.36 + 55.6.
Therefore The equation for the line of best fit for a woman's height, y, based on her shoe size, x is y = 1.36x + 55.6.
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I need help please!!
the slope of the given point is m=−7/2
look at points c and d on the graph what is the distance in units between points c and d round to nearest hundredth
Answer:
7.07
Step-by-step explanation:
Draw a horizonal line and a vertical line to creat a right triangle. The horizontal measurement would be 5 and so would the horizontal measurement.
Use the Pythagorean theorem
[tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex] The c side is the length that you are looking for
[tex]5^{2}[/tex] + [tex]5^{2}[/tex] = [tex]c^{2}[/tex]
25 + 25 = [tex]c^{2}[/tex]
50 = [tex]c^{2}[/tex]
[tex]\sqrt{50}[/tex] = [tex]\sqrt{c^{2} }[/tex]
7.07 = c Rounded to the nearest hundredths
Help my with this problem please 20 points
The difference of the polynomials [tex]2x^2[/tex] - 9x + 8 and [tex]6x^2[/tex] - 5x - 12 is -[tex]4x^2[/tex] - 4x 20
The first polynomial is given that
[tex]2x^2[/tex] - 9x + 8
The second polynomial is
[tex]6x^2[/tex] - 5x - 12
Here we have to find the difference, to find that we need to subtract the second polynomial from the second polynomial
Then,
[tex]2x^2[/tex] - 9x + 8 - ( [tex]6x^2[/tex] - 5x - 12 )
Open the bracket using the distributive property
[tex]2x^2[/tex] - 9x + 8 - [tex]6x^2[/tex] + 5x +12
Combine the like terms
[tex]2x^2[/tex] - [tex]6x^2[/tex] -9x +5x + 8 +12
Apply the arithmetic operations on like terms
-[tex]4x^2[/tex] - 4x 20
Hence, the difference of the polynomials [tex]2x^2[/tex] - 9x + 8 and [tex]6x^2[/tex] - 5x - 12 is -[tex]4x^2[/tex] - 4x 20
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On a plane there were 6 airline
workers and 240 passengers.
If each worker looked after the same
number of passengers, how many
passengers did each worker look after?
Each worker looked 40 passengers
From the question, we have
On a plane there were 6 airline workers and 240 passengers.
each worker looked after the same number of passengers
number of passengers = 240/6
=40
Each worker looked 40 passengers
Divide:
Repetitive subtraction is the process of division. It is the multiplication operation's opposite. It is described as the process of creating equitable groups. When dividing numbers, we divide a larger number down into smaller ones such that the larger number obtained will be equal to the multiplication of the smaller numbers. One of the four fundamental mathematical operations, along with addition, subtraction, and multiplication, is division. Division is the process of dividing a larger group into smaller groups so that each group contains an equal number of things. It is a mathematical operation used for equal distribution and equal grouping.
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4. Each year a certain amount of money is deposited in an account which pays an
annual interest rate of r so that at the end of each year the balance in the account is
multiplied by a growth factor of x = 1+r. $1,000 is deposited at the start of the first
year, an additional $300 is deposited at the start of the next year, and $500 at the
start of the following year.
a. Write an expression for the value of the account at the end of three years in
terms of the growth factor x.
b. Determine (to the nearest cent) the amount in the account at the end of three
years if the interest rate is 4%.
a)An expression for the value of the account at the end of three years will be:
$1000[tex]x^3[/tex] +300[tex]x^2[/tex] + 500x
b) The amount in the account at the end of three years if the interest rate is 4% will be:
= 1000 × [tex]1.04^3[/tex] + 300 × [tex]1.04^2[/tex] + 500 × 1.04
= $ 1969.344
Given,
In the question:
A growth factor of x = 1+r. $1,000 is deposited at the start of the first year,
An additional $300 is deposited at the start of the next year, and $500 at the start of the following year.
To find the:
a) Write an expression for the value of the account at the end of three years in terms of the growth factor x.
b) Determine (to the nearest cent) the amount in the account at the end of three years if the interest rate is 4%.
Now, According to the question:
a) Growth factor x = 1 + r
Based on the given conditions:
An expression for the value of the account at the end of three years will be:
$1000[tex]x^3[/tex] +300[tex]x^2[/tex] + 500x
b) The amount in the account at the end of three years if the interest rate is 4% will be:
1 + 4% = 1 + 0.04 = 1.04
= 1000 × [tex]1.04^3[/tex] + 300 × [tex]1.04^2[/tex] + 500 × 1.04
= $ 1969.344
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a boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore (see figure). a woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. if she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? if she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)?
(a) she land to minimize the total travel time is 2.4min
(b) the minimum speed at which she must row so that the quickest way to the restaurant is to row directly is 6
(a)
T(x) = time rowing + time walking
= [tex]\frac{\sqrt{x^{2} +16} }{2}[/tex] +[tex]\frac{6 - x}{3}[/tex]
T(x) = 1/2[tex]\sqrt{(x^{2} + 16)}[/tex] + 2 - [tex]\frac{x}{3}[/tex]
first derivation
T'(x) = 1/4[tex]\sqrt[-1/2]{(x^{2} + 16)}[/tex] .2x - 1/3
= [tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex] - 1/3
T'(x) = 3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] / 6[tex]\sqrt{(x^{2} + 16)}[/tex]
T'(x) = 0
3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] / 6[tex]\sqrt{(x^{2} + 16)}[/tex] = 0
3x - 2 [tex]\sqrt{(x^{2} + 16)}[/tex] =0
3x/2 = [tex]\sqrt{(x^{2} + 16)}[/tex]
9[tex]x^{2}[/tex]/4 = [tex]x^{2}[/tex] + 16
9[tex]x^{2}[/tex] = 4 [tex]x^{2}[/tex] + 64
5 [tex]x^{2}[/tex] = 64
[tex]x^{2}[/tex] =64/5
x = [tex]\sqrt{64/5}[/tex]
x = 8/[tex]\sqrt{5}[/tex]
x = 8[tex]\sqrt{5}[/tex]/5
total travel time = 6 - 8[tex]\sqrt{5}[/tex]/5 = 2.4 min
she land to minimize the total travel time is 2.4min
(b) T(x) = time rowing
= [tex]\frac{\sqrt{x^{2} +16} }{2}[/tex]
first derivation
T'(x) = 1/4[tex]\sqrt[-1/2]{(x^{2} + 16)}[/tex] .2x
= [tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex]
T'(x) = 0
[tex]\frac{x}{2\sqrt{(x^{2} + 16)} }[/tex] =0
x = 0
minimum speed = 6-x = 6-0=6
the minimum speed at which she must row so that the quickest way to the restaurant is to row directly is 6
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