Answer:
The expression written by Emma to represent the amount of the money that Juan earned is WRONG.
Step-by-step explanation:
Here, the amount earned by Juan by walking dogs = w
The amount earned byJuan lowing lawns = (2w - 6)
Now, the total amount earned by Juan
= Amount earned from (Mowing Lawn + Walking dogs)
= w + (2 w - 6)
On further simplification, we get:
w + (2 w - 6) = w + 2w - 6 = 3w-6 = 3(w-2)
So, the total amount earned by Juan = 3(w-2)
Now, the expression written by Emma = 2(3w-6)
and 2(3w-6) ≠ 3(w-2)
Hence, the expression written by Emma to represent the amount of the money that Juan earned is WRONG.
A transformation named T maps triangle XYZ to triangle X'Y'Z'. The transformation is a(n) _____.
The transformation T could be translation, rotation, reflection, or dilation, altering the position, orientation, or size of the triangle.
The transformation T mapping triangle XYZ to triangle X'Y'Z' could be any of the basic rigid transformations in geometry: translation, rotation, reflection, or dilation.
- A translation involves shifting the entire triangle by a certain distance in a certain direction without changing its orientation or shape.
- A rotation rotates the triangle around a fixed point (e.g., the origin or a specific vertex), altering its orientation but maintaining its shape and size.
- A reflection flips the triangle across a line (e.g., the x-axis, y-axis, or an arbitrary line), producing a mirror image with reversed orientation.
- A dilation scales the triangle uniformly, either enlarging or shrinking it while maintaining its shape and proportions.
To determine which transformation T specifically represents, additional information is needed, such as the coordinates of the vertices X, Y, Z and X', Y', Z', or any constraints or properties of the transformation provided. Each type of transformation has distinct characteristics that can be identified through geometric properties and transformations of coordinates.
HELP PLZZZ ASAPP!!! (Plz answer em right!)
Answer:
1) B, C, and D
2) plane BFD
Step-by-step explanation:
1) Collinear points are points on the same line. There are two lines shown. One of them shows points A, C, and E on it. The other line shows points B, D, and D on it. The latter set of points is listed among the answer choices.
__
2) A plane can be named by three points in the plane that are not on the same line. The points shown as being in the plane are B, C, D, F, with points B, C, and D being on the same line (see question 1). So, the plane can be named with point F and any two of B, C, and D. Plane BFD is an appropriate name.
2918+49
Which numbers should be used?
The estimated quotient is?
Answer: Use all numbers to add.
2918+49=2967
Estimated.... 2918+49 ≈ 3050
Hope this helps you out.
Answer:use all the numbers
Step-by-step explanation:
2918+49 ≈ 3050
what is the difference between Compound Interest and simple interest???
Answer:
Simple interest is calculated using initial principle while compound interest is calculated considering the interest also .
Step-by-step explanation:
Interest is the cost of borrowing money, where the borrower pays a fee to the lender for using his money. The interest, typically expressed as a percentage, can either be compounded or simple .
Simple interest is based on the principal amount , while compound interest is based on the principal amount and the interest that adds onto it in every period and the final principle is used for calculating the interest.
Simple interest is calculated on the principal amount of a loan and it's easier to find out than compound interest.
Let f(x) be a polynomial such that f(cos θ) = cos(4) θ for all θ. Find f(x). (This is essentially the same as finding cos(4) θ in terms of cos θ we structure the problem this way so that you can answer as a polynomial. Be sure to write your polynomial with the terms in order of decreasing degree.)
Answer:
f(x) = 8x⁴-8x²+1
Step-by-step explanation:
I will assume that f(cos θ) = cos(4θ). Otherwise, f would not be a polynomial. lets divide cos(4θ) in an expression depending on cos(θ). We use this properties
cos(2a) = cos²(a) - sin²(b)sin(2a) = 2sin(a)cos(a)sin²(a) = 1-cos²(a)cos(4θ) = cos(2 * (2θ) ) = cos²(2θ) - sin²(2θ) = [ cos²(θ)-sin²(θ) ]² - [2cos(θ)sin(θ)]² = [cos²(θ) - ( 1 - cos²(θ) ) ]² - 4cos²(θ)sin²(θ) = [2cos²(θ)-1]² - 4cos²(θ) (1 - cos²(θ) ) = 4 cos⁴(θ) - 4 cos²(θ) + 1 - 4 cos²(θ) + 4 cos⁴(θ) = 8cos⁴(θ) - 8 cos²(θ) + 1
Thus f(cos(θ)) = 8 cos⁴(θ) - 8 cos²(θ) + 1, and, as a result
f(x) = 8x⁴-8x²+1.
One pound of tomatoes costs $1.59.One pound of bananas costs $1.87. How much more do 5 pounds of tomatoes costs than 2 pounds of bananas? Show your work
when you multiply $1.59 by 5 you get $7.95
when you multiply $1.87 by 2 you get $4.74
so when you subtract them you get $3.21
Mark wants to use a grid like the ones in exercises 1 and 2 to model the percent equivalent of the fraction 2/3.How many grid squares should he shade? What percent would his model show?
Answer:
67 squares or 66.66 squares.
2/3 turned into a decimal is 66.66 or rounded, 67 squares.
Hope this helped! (plz mark me brainliest!)
Suppose you can work at most a total of 25 hours per week. Baby-sitting, x, pays $6 per hour and working at the grocery store, y, pays $9 per hour. You need to earn $171 a week to pay for your expenses.
Answer:
y=7 number of hours at grocery store
x=18 number of hours at baby- sitting
Step-by-step explanation:
According to the information provided.
x is number of hours at baby- sitting
y is number of hours at grocery store
total number of hours worked
1) x+y =25
total earn in a week
2) x*$6 + y* $9 = $171
from equation 1
x+y=25
x= 25-y
we place the above derived equation in equation 2
x*$6 + y* $9 = $171
(25-y)*$6 + y* $9 = $171
(25*6) -6y +9y =171
150+3y=171
3y=171-150
3y=21
y=7 number of hours at grocery store
x= 25-y
x= 25-7
x=18 number of hours at baby- sitting
In the rectangular coordinate system, points (4,0) and (–4,0) both lie on circle C. What is the maximum possible value of the radius of C ?
Answer:
There is no finite maximum value of the radius of C.
Step-by-step explanation:
We have been given that in the rectangular coordinate system, points (4,0) and (–4,0) both lie on circle C. We are asked to find the maximum possible value of the radius of C.
If the center of the circle C would have been at origin (0,0), then the radius of the circle C would be 4 units.
Since there is no information given about center of circle, so radius can be infinitely long. Therefore, there is no finite maximum value of the radius of C.
All 231 students in the Math Club went on a field trip. Some students rode in vans which hold 7 students and some students rode in buses which hold 25 students each. How many of each type of vehicle did they use if there were 15 vehicles in total?
answer : 8 vans and 7 buses
A(n) is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."
Answer:
Prototype
Step-by-step explanation:
Before a product is made, an original model of the product is first formulated. This is called a prototype. It is not simply a part of the design, but a very important part that the process of creating the final product solely depends on it. It is largely described as a highly representative example of a concept or product.
A prototype is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."
What is a prototype?In Engineering, a prototype can be defined as a rudimentary (early) model, release, or sample of a particular product, so as to avail the developers and manufacturers an opportunity and ability to test a concept, functionality, or process within it.
For instance, the main purpose of the prototype of a robot is to determine and show whether or not the robot and all of its features are in tandem with the design specifications and market requirements.
Read more on prototype here: brainly.com/question/7509258
#SPJ3
What is the first speed at which the ratio of stopping distance to speed is greater than 3 to 1
50 mph
40 mph
20 mph
Answer:
40 mph
Step-by-step explanation:
3 times the speed in mph will be ...
3 × 20 = 60 . . . . . more than 43
3 × 40 = 120 . . . . less than 126
The stopping distance in feet is more than 3 times the speed in mph for a speed of 40 mph.
_____
The answer will depend on the units of the ratio. Here, we are apparently to use units of feet per (mile per hour), that is, (ft·h/mi). The answer would be different for distance in meters and speed in km/h, for example.
Melissa wants to purchase a digital camera.The listed price of the camera is $195.99.The camera is no sale for 10% off and Melissa has a coupon for 5% off the sales tax is 7%. How much money will the 10% off sale save Melissa
Answer:
The 10% off sale will save Melissa $10.49.
Step-by-step explanation:
Given:
The listed price of the camera is $195.99.
The camera is no sale for 10% off and Melissa has a coupon for 5% off the sales tax is 7%.
Now, to find the money Melissa will save of the 10% off sale.
So, to get the price of camera of the 5% off sale:
[tex](195.99-5\%\ of\ 195.99)[/tex]
[tex]=(195.99-\frac{5}{100}\times 195.99)[/tex]
[tex]=(195.99-9.80)[/tex]
[tex]=186.19[/tex]
Now, adding the sales tax:
[tex]186.19+7\%\ of\ 186.19[/tex]
[tex]=186.19+\frac{7}{100} \times 186.19[/tex]
[tex]=186.19+13.03[/tex]
[tex]=199.22[/tex]
Thus, the price is $199.22.
Now, to get the price of 10% off sale:
[tex]195.99-10\%\ of\ 195.99[/tex]
[tex]=195.99-\frac{10}{100} \times 195.99[/tex]
[tex]=176.39[/tex]
So, adding sales tax:
[tex]176.39+7\%\ of\ 176.39[/tex]
[tex]=176.39+\frac{7}{100} \times 176.39[/tex]
[tex]=188.73[/tex]
Hence, the price is $188.73.
Now, to get the money 10% off sale will save Melissa:
[tex]199.22-188.73[/tex]
[tex]=10.49[/tex]
Therefore, the 10% off sale will save Melissa $10.49.
Final Answer:
The 10% off sale will save Melissa approximately $19.60.
Explantion:
To calculate the savings from the 10% off sale on the listed price of the digital camera, follow these steps:
1. Determine the listed price of the camera. In this case, the listed price is $195.99.
2. Calculate the discount amount by multiplying the listed price by the discount percentage (expressed as a decimal). For a 10% discount, you would convert that percentage to a decimal by dividing by 100:
10% = 10/100 = 0.10.
3. Multiply the listed price by the decimal form of the discount percentage to find the savings:
Savings = Listed Price × Discount Percentage
Savings = $195.99 × 0.10
4. Calculate the actual savings:
Savings = $195.99 × 0.10 = $19.599
5. Since savings are typically represented in dollar format (rounded to two decimal places), we round the savings to the nearest cent:
Savings ≈ $19.60
So, the 10% off sale will save Melissa approximately $19.60.
In the triangle below, what ratio is sin θ?
Answer:
[tex]\displaystyle \frac{12}{13} = sin\:θ\:[0,3743340836π ≈ θ][/tex]
Step-by-step explanation:
Remember this?!
Extended Information on Trigonometric Ratios
[tex]\displaystyle \frac{OPPOSITE}{HYPOTENUSE} = sin\:θ \\ \frac{ADJACENT}{HYPOTENUSE} = cos\:θ \\ \frac{OPPOSITE}{ADJACENT} = tan\:θ \\ \frac{HYPOTENUSE}{ADJACENT} = sec\:θ \\ \frac{HYPOTENUSE}{OPPOSITE} = csc\:θ \\ \frac{ADJACENT}{OPPOSITE} = cot\:θ[/tex]
I am joyous to assist you anytime.
Chris has 7/8 yards of purple ribbon and 1 1/6 yards of green ribbon. Chris has a total of 3 2/3 yards of ribbon. How many yards of Chris's ribbon are neither purple nor green.
Answer:The number of yards of Chris's ribbon are neither purple nor green is 1.625 yards
Step-by-step explanation:
Chris has a total of 3 2/3 yards of ribbon. Converting 3 2/3 yards to improper fraction, it becomes 11/3 yards. Chris has 7/8 yards of purple ribbon. He also has 1 1/6 yards of green ribbon. Converting to improper fraction, it becomes 7/6 yards of green ribbon.
The total length of purple and green ribbon that Chris has would be 7/8 + 7/6 = (21+28)/24 = 49/24
The number of yards of Chris's ribbon are neither purple nor green is 11/3 - 49/24 = (88 - 49)/24
= 39/24 = 1.625 yards
Mister Rogers is fencing another new rectangular garden in his neighborhood. One side of the garden faces the road and needs to be pretty. The other three sides just need to be functional. The pretty fencing costs $35 per linear foot and the functional fencing costs $18 per linear foot. Mr. Rogers has $ 3000 to build his fence. What dimensions of the garden give him the maximum area?
Answer:
length of the pretty side and length of the side oppositte to the pretty side = 37.91 ft
length of the other two sides = 27.52 ft
Step-by-step explanation:
The mathematical problem is:
Max A = b1*h
subject to: 35*b1 + 18*(2*h + b2) <= 3000
Where
A: area of the garden
b1: length of the pretty side
b2: length of the side oppositte to the pretty side
h: length of the other two sides
Replacing with b1 = b2 and taking only the equality sign in the restriction (in the maximum all the money will be spent), we get:
35*b1 + 18*(2*h + b1) = 3000
35*b1 + 36*h + 18*b1 = 3000
53*b1 + 36*h = 3000
b1 = 3000/53 - (36/53)*h
Substituing in Area's formula
A = (3000/53 - (36/53)*h)*h
A = (3000/53)*h - (36/53)*h^2
In the maximum, the derivative of A is equal to zero
dA/dh = 3000/53 - 2*(36/53)*h =
3000/53 - 72/35*h = 0
h = (3000/53)*(35/72)
h = 27.52 ft
then,
b1 = 3000/53 - (36/53)*27.52
b1 = 37.91 ft =b2
The dimensions of the garden that give Mr. Rogers the maximum area are approximately: Length (L) = 28.3 feet and Width (W) = 41.7 feet.
To find the dimensions of Mr. Rogers's garden that will give him the maximum area while staying within his budget, we need to set up and solve a problem involving optimization with constraints.
First, let's define the variables:
Let the length of the garden (which faces the road and requires pretty fencing) be denoted as [tex]L[/tex].Let the width of the garden be denoted as [tex]W[/tex].The cost of fencing:
Pretty fencing costs $35 per linear foot, and it is only needed for one side of the garden (the length L).Functional fencing costs $18 per linear foot, and it is needed for the other three sides (two widths W and the remaining length L).The total cost of fencing can be expressed as follows:
[tex]35L + 18(2W + L) = 3000[/tex]
Simplifying this equation:
[tex]35L + 36W + 18L = 3000[/tex]
[tex]53L + 36W = 3000[/tex]
To find the dimensions that maximize the area, we need to express the area in terms of one variable and use calculus to find the maximum. Let's solve for one variable in terms of the other. We'll solve for [tex]W[/tex]:
[tex]53L + 36W = 3000[/tex]
[tex]36W = 3000 - 53L[/tex]
[tex]W = \frac{3000 - 53L}{36}[/tex]
Now, express the area [tex]A[/tex] as a function of [tex]L[/tex]:
[tex]A = L \cdot W[/tex]
[tex]A = L \left(\frac{3000 - 53L}{36}\right)[/tex]
[tex]A = \frac{3000L - 53L^2}{36}[/tex]
To find the maximum area, we take the derivative of [tex]A[/tex] with respect to [tex]L[/tex] and set it to zero:
[tex]\frac{dA}{dL} = \frac{3000 - 106L}{36}[/tex]
Set the derivative to zero and solve for [tex]L[/tex]:
[tex]\frac{3000 - 106L}{36} = 0[/tex]
[tex]3000 - 106L = 0[/tex]
[tex]106L = 3000[/tex]
[tex]L = \frac{3000}{106}[/tex]
[tex]L \approx 28.3[/tex]
Now we use this value of [tex]L[/tex] to find the corresponding value of [tex]W[/tex]:
[tex]W = \frac{3000 - 53 \times 28.3}{36}[/tex]
[tex]W \approx \frac{3000 - 1499.9}{36}[/tex]
[tex]W \approx \frac{1500.1}{36}[/tex]
[tex]W \approx 41.7[/tex]
If there were 400 foreclosures this year and 500 last year what is the percent of change?
Answer:
-20% ?
Step-by-step explanation:
I'm not sure but 2% of 5 is 1 so 20% of 500 is 100. so minus 20% would give you 400.
To find the percent of change in foreclosures, subtract the new number from the old number, divide by the old number, and multiply by 100. The foreclosures went from 500 to 400, resulting in a 20 percent decrease.
To calculate the percent of change in the number of foreclosures from one year to the next, you take the difference in the numbers (new number - old number), divide by the old number (the number from the earlier year), and then multiply the result by 100 to obtain a percentage. In this case, there were 500 foreclosures last year and 400 this year. Therefore, the calculation is as follows:
(Old Number - New Number) / Old Number × 100 = Percent Change
(500 - 400) / 500 × 100 = 100 / 500 × 100
0.2 × 100 = 20%
There has been a 20 percent decrease in foreclosures from last year to this year.
Worldwide quarterly sales of a brand of cell phones were approximately q = −p + 126 million phones when the wholesale price was $p. (a) If the cellphone company was prepared to supply q = 9p − 354 million phones per quarter at a wholesale price of $p, what would have been the equilibrium price? $ 48 Correct: Your answer is correct. (b) The actual wholesale price was $43 in the fourth quarter of 2004. Estimate the projected shortage or surplus at that price. HINT [See Example 4.] There is an estimated Correct: Your answer is correct. of Incorrect: Your answer is incorrect. million phones.
Answer:
a) equilibrium price = $48
b) shortage of 50 million phones
Step-by-step explanation:
Quarterly sales of a brand/quantity demanded (q) =
-p+126 million phones
a) if supply(q) = 9p - 354 million phones, at equilibrium quantity demanded = quantity supplied
-p + 126 = 9p - 354
-p - 9p = -354 - 126
-10p = -480
p = -480/-10
p = $48
The price at equilibrium = $48
b) actual wholesale price in the fourth quarter of 2004= $43
Quantity demanded = -p + 126
= -43 + 126
= 83 million phones
Quantity supplied = 9p - 354
= 9(43) - 354
= 387 - 354
= 33 million phones
Since quantity supplied is less than quantity demanded, there will be a shortage.
Shortage = 83 -33
= 50 million phones
The equilibrium price is determined to be $48 by equating the demand and supply. At an actual price of $43, there is a shortage of 50 million phones. This is found by comparing the quantities demanded and supplied at that price.
Equilibrium Price and Surplus/Shortage Estimation
We are given the demand and supply functions for cell phones, where the demand function is q = −p + 126 and the supply function is q = 9p − 354. To find the equilibrium price, we set the demand function equal to the supply function:
−p + 126 = 9p − 354Solving for p:
126 + 354 = 9p + p480 = 10pp = 48Hence, the equilibrium price is $48.
For part (b), we need to estimate the shortage or surplus at an actual wholesale price of $43. We substitute p = 43 into both demand and supply functions:
Demand: qd = −43 + 126 = 83 million phonesSupply: qs = 9(43) − 354 = 387 − 354 = 33 million phonesTherefore, there is a shortage of 83 - 33 = 50 million phones at the price of $43.
Maggie graphed the image of a 90° counterclockwise rotation about vertex A of △ABC. Coordinates B and C of △ABC are (2, 6) and (4, 3) and coordinates B’ and C’ of it’s image are (–2, 2) and (1, 4). What is the coordinate of vertex A?
Answer:
A(2, 2)
Step-by-step explanation:
I find it useful to graph the given points. The center of rotation is at the place where the perpendicular bisectors of BB' and CC' meet. That point is ...
A = (2, 2).
__
The graph shows you the slope of BB' is 1, so its perpendicular bisector will have a slope of -1. The midpoint of BB' is (2-2, 6+2)/2 = (0, 4). This is the y-intercept of the line, so the perpendicular bisector of BB' has equation ...
y = -x +4
The slope of CC' is -1/3, so its perpendicular will have a slope of -1/(-1/3) = 3. The midpoint of CC' is (4+1, 3+4)/2 = (5/2, 7/2). In point-slope form the equation of the perpendicular bisector of CC' is ...
y -7/2 = 3(x -5/2)
2y -7 = 3(2x -5) . . . . multiply by 2
2(-x+4) -7 = 3(2x -5) . . . substitute for y
-2x +1 = 6x -15 . . . . .eliminate parentheses
16 = 8x . . . . . . . . . . add 2x+15
x = 2 . . . . . . divide by 8
y = -2+4 = 2 . . . . from the equation for y
The intersection of the perpendicular bisectors of BB' and CC' is the center of rotation:
A = (2, 2)
Final answer:
The coordinate of vertex A is (2, 2), found by equating B' to (y, x) and using the given B' coordinates.
Explanation:
To find the coordinates of vertex A, we can use the properties of a rotation. A 90° counterclockwise rotation about a point involves switching the coordinates and negating the y-coordinate. Let's denote the coordinates of A as (x, y). After the rotation, A becomes B', so we have:
B' = (y, x)
From the given coordinates of B' (–2, 2), we can equate:
y = 2
-2 = -x
Solving these equations, we find that x = 2. Therefore, the coordinates of vertex A are (2, 2). So, Maggie's image graph depicts a 90° counterclockwise rotation about vertex A (2, 2) of triangle ABC.
Two chess players, A and B, are going to play 7 games. Each game has three possible
outcomes: a win for A (which is a loss for B), a draw (tie), and a loss for A (which is
a win for B). A win is worth 1 point, a draw is worth 0.5 points, and a loss is worth 0
points.
(A) How many possible outcomes for the individual games are there, such that overall
player A ends up with 3 wins, 2 draws, and 2 losses?
(B) How many possible outcomes for the individual games are there, such that A ends
up with 4 points and B ends up with 3 points?
(C) Now assume that they are playing a best-of-7 match, where the match will end as
soon as either player has 4 points. For example, if after 6 games the score is 4 to 2 in
favor of A, then A wins the match and they don’t play a 7th game. How many possible
outcomes for the individual games are there, such that the match lasts for 7 games and
A wins by a score of 4 to 3?
Answer:
A) 210
B) 357
C) 267
Step-by-step explanation:
A) Among 7 games, we can first choose 3 wins, and then among remaining 4 games, we can choose 2.
To calculate the possibility, we will use Combination.
[tex]C(7,3)*C(4,2) = 35 *6 = 210[/tex]
B) Player A can get 4 points with the following cases:
4 wins and 3 loses
3 wins, 2 draws and 2 loses
2 wins, 4 draws and 1 lose
1 win and 6 draws
Indeed, these cases matches for Player B too to get 3 points.
So again, we will use Combination to calculate the possibility.
[tex]C(7,4) + C(7,3)*C(4,2) + C(7,2)*C(5,4) + C(7,1) = 357[/tex]
C) Here, we need to find two possibilities after 6 games and add them, while Player A has 3 points and wins the 7th game, and Player A has 3.5 points and draws the 7th game.
[tex][C(6,3)+C(6,2)C(4,2)+C(6,1)C(5,4)+C(6,6)] + [C(6,3)C(3,1)+C(6,2)C(4,3)+C(6,1)] =[20+90+30+1]+[60+60+6]=141+126=267[/tex]
Need help please!!! Just the numbers 4-7!
What is the equation of the line which
includes the points (2, 4) and (14, –2)?
A. y=5x−12
B. y=−12x−5
C. y=−12x+5
D. y=−2x+5
Answer:
The equation of the line is y = (-1/2)x + 5
Step-by-step explanation:
[tex]m = \frac{y2 - y1}{x2 - x1 } [/tex]
First of all, have to find gradient using the formula above :
(2,4) & (14,-2)
m = (-2-4) / (14-2)
= -6 / 12
= -1/2
Second, using y = mx + b as b is a constant and is a y-intercept. Using any of these 2 coordinates to find the value of b with given gradient :
y = mx + b
Let y=4 & x=2
4 = (-1/2)(2) + b
b = 4 + 1
= 5
Lastly, put the value of gradient and y-intercept into the equation :
y = mx + b
Let m=-1/2 & b=5
y = (-1/2)x + 5
To estimate the length of the lake, caleb starts at one end of the lake and walk 95m. He then turns a 60° angle and walks on a new path and walks 8m more then arrives at the other end of the lake. Approximately how long is the lake?
Answer:
Length of the lake is 97.30 m
Step-by-step explanation:
We have given Caleb starts at one end of the lake and walk 95 m
So [tex]d_1=95m[/tex]
And then he turns at an angle of 60°
So [tex]\Theta =60^{\circ}[/tex] and then again walk 8 m
So [tex]d_2=8m[/tex]
We have to fond the total length of the lake , that is d
Total length of the lake is given by [tex]d=\sqrt{d_1^2+d_2^2+2d_1d_2cos\Theta }=\sqrt{95^2+8^2+2\times 95\times 8\times cos60^{\circ}}=97.30m[/tex]
So length of the lake is 97.30 m
Final answer:
Using the Law of Cosines with Caleb's path measurements of 95 meters and 103 meters at a 60-degree angle, the length of the lake is found to be approximately 104 meters.
Explanation:
To estimate the length of the lake, we can represent Caleb's path as a triangle, where the length of the lake forms one side of the triangle. Caleb starts by walking 95 meters along one side, then makes a 60° angle and walks 8 meters more than the length of the first path, forming the second side of the triangle. The length of the lake, which is the final side, can be calculated using the Law of Cosines.
The Law of Cosines is given by c² = a² + b² - 2ab×cos(γ), where γ is the enclosed angle and a, b, and c are the lengths of the sides of the triangle.
In this case, a = 95m, b = 95m + 8m = 103m, and γ = 60°. Plugging these values into the Law of Cosines, we will find the length of the lake (c).
c² = 95² + 103² - 2×95×103×cos(60°)
c² = 9025 + 10609 - 19570×0.5
c² = 9025 + 10609 - 9785
c² = 10849
c ≈ √10849
c ≈ 104.16 meters
Therefore, the length of the lake is approximately 104 meters long.
The second hand on a clock is 8 \text{ cm}8 cm8, start text, space, c, m, end text long. What is the distance the tip of the second hand travels in 101010 minutes?
The second hand of a clock is 8 cm long what is the distance of the tip of the second hand travel in 10 minutes
Answer:
The distance travelled by tip of second hand in 10 minutes is 502.4 cm
Solution:
Length of second hand = 8 cm long
In one revolution it travels a circumference of a circle of which radius is 8 cm
The circumference of circle = [tex]2 \pi r[/tex]
[tex]c = 2 \times \pi \times 8 = 16\pi[/tex]
We are aksed to find the distance of the tip of the second hand travel in 10 minutes
Second hand complete 1 revolution in 1 minute.
Therefore, in 10 minutes it revolves 10 revolution,
In 1 revolution tip of second covers = 16π
Hence distance travelled is given as:
[tex]\rightarrow 16 \pi \times10 = 160 \pi[/tex]
We know that π is a constant equals to 3.14
[tex]\rightarrow 160 \pi = 160 \times 3.14 = 502.4[/tex]
Thus the distance travelled by tip of second hand in 10 minutes is 502.4 cm
Answer:
502.4
Step-by-step explanation:
The derivative of the function B is given by B′(t)=8e0.2cost, and B(2.2)=4.5. If the linear approximation to B(t) at t=2.2 is used to estimate B(t), at what value of t does the linear approximation estimate that B(t)=9 ?
Answer:
t=2.83
Step-by-step explanation:
Linear Approximation Of Functions
The equation of a line is given by
[tex]y=y_o+m(x-x_o)[/tex]
Where m is the slope of the line and [tex](x_o,y_o)[/tex] are the coordinates of a point through which the line goes.
Given a function B(t), we can build an approximate line to model the function near one point. The value of m is the derivative of B in a specific point [tex](t_o, B_o)[/tex]. The equation becomes
[tex]B(t)=B_o+B'(t_o)(t-t_o)[/tex]
Let's collect our data.
[tex]B'(t)=8e^{0.2cost},\ B_o=B(2.2)=4.5[/tex]
Let's find the required values to build the approximate function near [tex]t_0=2.2[/tex]. We evaluate the derivative in 2.2
[tex]B'(2.2)=8e^{0.2cos2.2}=7.11[/tex]
The function can be approximated by
[tex]B(t)=4.5+7.11(t-2.2)[/tex]
Once we have B(t), we are required to find the value of t, such that
[tex]B(t)=9[/tex]
Or equivalently:
[tex]4.5+7.11(t-2.2)=9[/tex]
Rearranging
[tex]\displaystyle t-2.2=\frac{9-4.5}{7.11}[/tex]
Solving for t
[tex]\displaystyle t=\frac{9-4.5}{7.11}+2.2[/tex]
[tex]\boxed{t=2.83}[/tex]
We find the equation of the linear approximation at t=2.2 using B(2.2) and B'(2.2). We then solve this equation for t when the linear approximation equals 9. This gives us the time at which the linear approximation estimates B(t) = 9.
Explanation:To solve for the time when the linear approximation gives an output of 9, we need to find the equation of the tangent line (i.e., the linear approximation) at t = 2.2. The linear approximation to a function at a particular point is given by the formula L(t) = f(a) + f'(a)*(t-a), where f is the function, a is the point, and f' denotes the derivative of the function.
Given that B(2.2) = 4.5 and B'(t) = 8e0.2cos(t), we can substitute these into the linear approximation formula to get L(t) = 4.5 + 8e0.2cos(2.2)*(t-2.2).
Next, we solve this equation for t when L(t) = 9: 9 = 4.5 + 8e0.2cos(2.2)*(t-2.2). Solving the equation for t gives the time at which the linear approximation estimates B(t) = 9.
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If m arc CD = 143°, and m arc AB = 39°, what is m∠ DEC ?
Answer:
[tex]m\angle DEC=91^o[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The measure of the interior angle is the semi-sum of the arcs comprising it and its opposite
so
[tex]m\angle DEC=\frac{1}{2}[arc\ AB+arc\ DC][/tex]
substitute the given values
[tex]m\angle DEC=\frac{1}{2}[39^o+143^o][/tex]
[tex]m\angle DEC=\frac{1}{2}[182^o][/tex]
[tex]m\angle DEC=91^o[/tex]
PLEASE HELP ASAP!!!!
There are 30 students in Bobby's class. 40% of the students have blue eyes. How many students in Bobby's class have blue eyes?
A. 12
B. 18
C. 28
D. 42
E. 75
Answer:
12
Step-by-step explanation:
Number of students in Bobby's class = 30
40% of the students have blue eyes.
It means 40 out of 100 students will have blue eyes.
Then we have to find how many out of 30 students will have blue eyes.
Let us take the number of blue-eyed students out of 30 to be x.
Then:
[tex]\frac{40}{100}=\frac{x}{30}\\x=\frac{40}{100}\times30\\x=12[/tex].
Hence 12 out of 30 students will have blue eyes.
Brayden has math and reading homework tonight. Brayden can solve each math problem in 4 minutes and he can read each page in 2.5 minutes. The number of pages Brayden read is twice the number of math problems he solved. And it took him 45 minutes to complete all of his homework. Determine the number of math problems Brayden solved and the number of pages he read.
Answer:
Brayden Solved 5 math problem and read 10 pages.
Step-by-step explanation:
Let the number of Math Problem solved be x.
Also Let the Number of Pages he read be y.
Given:
The number of pages Brayden read is twice the number of math problems he solved.
Hence equation can be framed as;
[tex]y =2x[/tex]
Also Given:
Ayden can solve each math problem in 3 minutes and he can read each page in 1 minute and it took him 50 minutes to complete all of his homework.
Hence the equation can be framed as;
[tex]4x+2.5y =45[/tex]
Now Substituting the value of y in above equation we get;
[tex]4x+2.5(2x) =45\\\\4x+5x=45\\\\9x=45\\\\x=\frac{45}{9}=5[/tex]
Now Substituting the value of x to find the value of y we get;
[tex]y =2x =2\times5 =10[/tex]
Hence Brayden Solved 5 math problem and read 10 pages.
Quadrilateral ABCD is a parallelogram if both pairs of opposite sides are congruent. Show that quadrilateral ABCD is a parallelogram by finding the lengths of the opposite side pairs. What is the length of BC?
In parallelogram ABCD, opposite sides AB and DC are both 8 units long, and AD and BC are both 5 units long.
To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent. We can do this by finding the lengths of the opposite side pairs and showing that they are equal.
Let's denote the points as follows:
- A, B, C, D are the vertices of the quadrilateral ABCD.
- AB and DC are opposite sides, and AD and BC are opposite sides.
Given that AB = 8 units and AD = 5 units, we need to find the lengths of DC and BC.
Since ABCD is a parallelogram, opposite sides are congruent. Therefore, DC = AB = 8 units.
To find BC, we know that BC = AD = 5 units.
Thus, we have:
- AB = DC = 8 units
- AD = BC = 5 units
Since both pairs of opposite sides are congruent, quadrilateral ABCD is a parallelogram.
The length of BC is 5 units.
In the given case, The length of BC is 8 units.
To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent.
Let's denote the lengths of the sides as follows: AB, BC, CD, and DA.
Given that ABCD is a parallelogram, we know that opposite sides are equal in length.
Therefore, we can equate the lengths of AB to CD and BC to DA.
Let's assume that the length of side AB (and thus CD, since they are opposite and equal) is given as 8 units.
We are asked to find the length of side BC (which will be equal to the length of side DA).
Since we do not have any additional information such as angles or diagonals, we cannot calculate the length of BC directly.
However, if we are given that ABCD is a parallelogram, then by definition, the lengths of opposite sides are equal.
Therefore, without loss of generality, we can state that the length of BC is also 8 units, which is equal to the length of AB.
Thus, we have shown that both pairs of opposite sides are congruent:
AB = CD = 8 units
BC = DA
Since AB = 8 units and AB = CD, it follows that CD = 8 units.
Similarly, since AB = CD and AB = 8 units, by the properties of a parallelogram, BC must also be equal to 8 units.
Therefore, the length of BC is 8 units, confirming that quadrilateral ABCD is indeed a parallelogram.
The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 4 inches.
How far should The light bulb should be placed from the vertex?
Answer:
1 in
Step-by-step explanation:
We are given that
Diameter of casting=8 in
Radius of casting=[tex]\frac{Diameter}{2}=4 in[/tex]
Depth of casting=4 in
We have to find the distance of light bulb should be from the vertex
It means the reflector passing through the point (4,4).
Equation of parabola along y-axis is given by
[tex]x^2=4ay[/tex]
Using the equation substitute x=4 and y=4
[tex]16=16a[/tex]
[tex]a=\frac{16}{16}=1[/tex]
The focus of parabola is at (0,a).
Therefore, the focus of reflector=(0,1)
Hence, the light bulb should be placed 1 in far from the vertex.
The distance of light bulb can be calculated using equation of parabola. The parabola is a plane curve which is U-shaped.
The distance of light bulb is [tex]1\:\rm in[/tex].
Given:
The diameter is [tex]8\:\rm in[/tex].
The radius is [tex]=\frac{d}{2}=\frac{8}{2}=4 \:\rm in[/tex].
The depth is [tex]4\:\rm in[/tex].
Since the depth is [tex]4\:\rm in[/tex] and radius is [tex]4\:\rm in[/tex] so reflector passes through the point [tex](4,4)[/tex].
Write the equation of parabola for y-axis.
[tex]x^2=4ay[/tex]
Putting [tex]4[/tex] for [tex]x[/tex] and [tex]4[/tex] for [tex]y[/tex].
[tex]4^2=4\times a\times 4\\a=\frac{16}{16}\\a=1[/tex]
The focus of a parabola is [tex](0,1)[/tex].
The distance of light bulb is [tex]1\:\rm in[/tex].
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