a)
The robot made the turn at the point (-20, 15).
b)
The equation of the second line on which the robot traveled is:
y = (4/3)x + 35.
c)
The distance between the starting position and the final position is 20.51 feet.
d)
The equation of the line the robot needs to travel along in order to return and recharge is y = (-3/4)x + 5.
What is an equation of a line?The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
Example:
The slope of the line y = 2x + 3 is 2.
The slope of a line that passes through (1, 2) and (2, 3) is 1.
We have,
a.
To find the coordinates of the point at which the robot made the turn, we need to determine its position after traveling for 60 seconds along the initial northeast path.
The distance the robot covers in 60 seconds is:
distance
= speed x time
= (1/2) x 60 = 30 feet
The initial path has a slope of -3/4 (since it passes through (4, -3) and
(0, 5)), so the robot's new path after turning right will have a slope of the negative reciprocal of -3/4, which is 4/3.
We know that the robot traveled 30 feet along the initial path, so we can use the slope and distance traveled to find the endpoint of the initial path:
y = mx + b, where m is the slope and b is the y-intercept
-3 = (-3/4)(4) + b
b = 0
So the equation of the initial path is y = (-3/4)x.
The endpoint of the initial path is (-20, 15) (since the initial path passes through (0, 5) and (-20, 15)).
The robot made the turn at the point (-20, 15).
b.
Since the robot turned right, its new path will be perpendicular to the initial path.
We know the robot's position after turning (i.e., the endpoint of the initial path), so we can use this point and the slope of the new path to find the equation of the new path.
The slope of the new path is 4/3, so the equation of the new path is:
y - 15 = (4/3)(x + 20)
Simplifying this equation, we get:
y = (4/3)x + 35
The equation of the second line on which the robot traveled is:
y = (4/3)x + 35.
c.
From the last equation in part (b), we have:
9(y - 5)² = 29 - 16x² - 12x
Taking the square root of both sides and solving for y, we get:
y = 5 ± √((29 - 16x^2 - 12x)/9)
Since the robot is traveling along the new path for 80 seconds, we want to find y when t = 80.
Substituting t = 80 into the equation for y and simplifying, we get:
y = 5 + √(17/9)
So the robot's final position is approximately (20.50, 6.63).
The distance between the starting position and the final position is:
distance
= √((20.50 - 0)² + (6.63 - 5)²)
= √(421.17)
= 20.51 feet.
d.
The robot needs to travel along a path that is perpendicular to the second line on which it traveled in order to return to its starting position.
The slope of the second line is 4/3, so the slope of the path back to the starting position is -3/4 (the negative reciprocal of 4/3).
We know the robot's final position and the slope of the desired path, so we can use the point-slope form to find the equation of the path:
y - 5 = (-3/4)(x - 0)
Simplifying this equation, we get:
y = (-3/4)x + 5
So the equation of the line the robot needs to travel along in order to return and recharge is y = (-3/4)x + 5.
Setting the two equations equal to each other and solving for x, we get:
(4/3)x + 35 = (-3/4)x + 5
Multiplying both sides by 12 to eliminate fractions, we get:
16x + 420 = -9x + 60
Solving for x, we get:
x = -16.8
Substituting this value of x into either equation, we get:
y = (4/3)(-16.8) + 35
y = 13.60
So the intersection point is approximately (-16.8, 13.60).
To find how long it will take the robot to get there, we can use the distance formula between the robot's final position and the intersection point:
distance = √((-16.8 - 20.50)² + (13.60 - 6.63)²) ≈ 40.49 feet.
Since the robot travels at a constant speed of 1/2 foot per second, it will take approximately 80.98 seconds (rounded to two decimal places) for the robot to return and recharge.
Thus,
The robot made the turn at the point (-20, 15).
The equation of the second line on which the robot traveled is:
y = (4/3)x + 35.
The distance between the starting position and the final position is 20.51 feet.
The equation of the line the robot needs to travel along in order to return and recharge is y = (-3/4)x + 5.
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Aaron bought a large basket of 59 apples for $22.95. He also bought a small basket of 18 apples for $8.95. Which am amount is the closest estimate of the cost per apple?
Calculate the cost per apple for large and small baskets, $0.39 per apple is the closest estimate.
To estimate the cost per apple, one must calculate the price of each apple in the large and small baskets, then average these amounts, resulting in approximately 44 cents per apple.
To find the cost per apple:
Calculate the cost per apple in the large basket: $22.95 / 59 apples = $0.39 per apple.
Calculate the cost per apple in the small basket: $8.95 / 18 apples = $0.50 per apple.
Compare the two costs: $0.39 is the closest estimate of the cost per apple.
Find the volume of a right circular cone that has a height of 17.5 ft and a base with a diameter of 6.1 ft. Round your answer to the nearest tenth of a cubic foot.
Please help asap! 45 points
Theo budgets $154 for karate classes. He buys a karate uniform, called a dogi, for $12. If it costs $8 to attend each karate class, which inequality below represents the number of classes. c, that Theo can take?
Final answer:
The inequality representing the number of karate classes Theo can take is c ≤ 17, considering his budget after buying a dogi and the cost per class.
Explanation:
The question asks us to find the inequality that represents the number of karate classes, c, Theo can take given his budget scenario. Theo budgets $154 for his karate classes and spends $12 on a dogi. This leaves Theo with $154 - $12 = $142 for classes. Since each class costs $8, we divide the remaining budget by the cost per class to find the maximum number of classes he can take:
$142 ÷ $8 = 17.75.
Since Theo cannot attend a fraction of a class, we round down to 17. The inequality representing the number of classes Theo can take would be c ≤ 17, where c is the number of classes Theo can attend.
The inequality that represents the number of classes, [tex]\( c \)[/tex], that Theo can take is:
[tex]\[ 8c + 12 \leq 154 \][/tex]
To derive this inequality, we start by considering the total amount Theo has budgeted for karate classes, which is $154. From this budget, Theo has already spent $12 on purchasing a karate uniform. Therefore, the remaining amount available for attending the classes is [tex]\( 154 - 12 \)[/tex].
Now, each karate class costs $8, so the total cost for attending [tex]\( c \)[/tex] classes is [tex]\( 8c \)[/tex]. To ensure that Theo does not exceed his budget, the total cost including the uniform should be less than or equal to $154.
Hence, the inequality that represents this situation is:
[tex]\[ 8c + 12 \leq 154 \][/tex]
This inequality states that the cost of the uniform plus the cost of the classes must be less than or equal to Theo's budget. Solving for [tex]\( c \)[/tex] will give us the maximum number of classes Theo can attend without exceeding his budget.
Ogether, teammates pedro and ricky got 2675 base hits last season. pedro had 283 more hits than ricky. how many hits did each player have?
Please help meeeee.....
What is the smallest positive integer n such that 2n is a perfect square and 3n is a perfect cube?
a club has 15 members. How many ways can the club choose a president, vice president, and treasurer? (club rules forbid one person from holding more than one office.) show all work
There are 2730 different ways for a club with 15 members to choose a president, vice president, and treasurer, with each office being held by a different person.
Explanation:To determine the number of ways the club can choose a president, vice president, and treasurer, we can use the concept of permutations since the order of selection matters here, and no person can hold more than one office.
The first office to be filled is the president's.
There are 15 potential members to fill this role.
Once the president has been chosen, there are 14 remaining members who could be the vice president.
Finally, for the treasurer, there are 13 members left to choose from.
To find the total number of ways to choose these officers, we multiply the number of choices for each office together.
Number of ways = 15 (for president) × 14 (for vice president) × 13 (for treasurer) = 2730 ways.
So, there are 2730 different ways in which the club can select a president, vice president, and treasurer.
Pls explain 4 brainliest!
May the 4th be with you
George has a lawn care business. He charges $70.00 per yard that he cuts. It costs George $3.49 per yard for lawn mower maintenance and $24.04 per yard for gasoline. Approximately how much profit will George make if he cuts 6 yards? A. $402.00 B. $582.00 C. $276.00 D. $258.00
When finding the area of a circle do you times your raduis by 2?
After a busy day of orders, a florist has 6 roses, 7 lilies, 3 carnations, 35 chrysanthemums, 27 sunflowers, 18 violets, and 9 tulips left in her inventory. part
a.write a ratio that compares the number of roses to the number of violets. part
b.describe what three other comparisons have the same ratio. part
c.provide reasoning why more than one comparison can be made with the same ratio.
A glass cylinder is completely filled with 24x^5y^8 cubic inches of salt solution. The jug has a base area of 4x^5 square inches. The hight of the jug, in inches, is represented by the expression 24x^5y^8/ 4x^5. Which of the following simplified expressions represents the height of the jug in inches?
A: 6y^3
B: 6y^8
C: 6xy^8 (I think it's this one....)
D: 6x^10y^8
A weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time t = 0 seconds. When the weight it released, it oscillates and returns to its original position at t = 3 seconds. Which of the following equations models the distance, d, of the weight from its equilibrium after t seconds?
a. d=-9cos(pi/3)t
b. d=-9cos(2pi/3)t
c. d=-3cos(pi/9)t
d. d=-3cos(2pi/9)t
For a better understanding of the explanation provided here kindly go through the file attached.
Since, the weight attached is already at the lowest point at time, t=0, therefore, the equation will have a -9 as it's "amplitude" and it will be a Cosine function. This is because in cosine function, the function has the value of the amplitude at t=0.
Now, we know that the total angle in radians covered by a cosine in a given period is [tex] 2\pi [/tex] and the period given in the question is t=3 seconds. Therefore, the angular velocity, [tex] \omega [/tex] of the mentioned system will be:
[tex] \omega=\frac{2\pi}{3} [/tex]
Combining all the above information, we see that the equation which models the distance, d, of the weight from its equilibrium after t seconds will be:
[tex] d=-9cos(\frac{2\pi}{3})t [/tex]
Thus, Option B is the correct option. The attached diagram is the graph of the option B and we can see clearly that at t=3, the weight indeed returns to it's original position.
Answer: B.
Step-by-step explanation:
answer on edge
What is the ratio of the length of the radius and the length of the diameter for all circles?
What is the approximate surface area of the sphere 15yd?
A: 225 yd^2
B: 707 yd^2
C: 1,767 yd^2
D: 5,301 yd^2
The approximate surface area of the sphere will be 707 yd^2
What is sphere?Sphere is a round solid figure, or its surface, with every point on its surface equidistant from its centre.A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. How to solve this problem?The steps are as follow:
Given the diameter of sphere is 15 ydSo the radius will be half of diameter which will be 7.5 ydThe surface area of sphere will be as follow:r = 7.5 yd
Surface area of sphere = 4πr^2
Surface area of sphere = 4*π*(7.5)^2
Surface area of sphere = 706.85 yd^2
Surface area of sphere = 707 yd^2
So the approximate surface area of the sphere will be 707 yd^2
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a cars fuel tank can hold 20 gallons of gas total it is only half full you stop and buy 5 gallons of gas how full is he fuel tank after you buy gas
An instructor makes $15 more per hour than an assistant. If the combined hourly wage of the instructor and assistant is $65, what is the hourly wage of the assistant? A. $15 B. $20 C. $25 D. $30
A recipe that makes 36 oatmeal cookies requires 1.5 cups of oatmeal. How many cups of oatmeal would be needed to make 60 oatmeal cookies?
To solve this problem, we are going to use a proportion, letting the variable x represent the unknown number of cups of oatmeal that we need.
1.5 cups/36 cookies = x cups/60 cookies
To simplify a proportion like this, we can use cross-multiplication. This means multiplying the numerator of one fraction by the denominator of the other fraction and setting it equal to the product of the other numerator and denominator. This is modeled below:
(36)(x) = (1.5)(60)
To simplify, we begin by multiplying out each side.
36x = 90
Finally, we divide both sides of the equation by 36.
x = 2.5
This means that we need 2.5 cups of oatmeal to make the cookies.
Hope this helps!
an interval has the notation (-9,-7). find the distance from the midpoint of the interval to either end point
An equation that is true for all allowed values of the variable
Which algebraic rule describes the 180° counterclockwise rotation about the origin
Answer:
[tex](x,y)\rightarrow (-x,-y)[/tex]
Step-by-step explanation:
We are asked to find the the algebraic rule describes the 180° counterclockwise rotation about the origin .
We know that when we rotate a point 180° counterclockwise rotation about the origin , the x and y-coordinates change their sign.
[tex](x,y)\rightarrow (-x,-y)[/tex]
Therefore, our required rule is [tex](x,y)\rightarrow (-x,-y)[/tex].
Graph the solution of 2 ≥ 4 - v
Answer:
Ok so ppl it just A
Step-by-step explanation:
To solve this problem, we must first subtract 4 from both sides of the inequality. This gives us -2 ≥ -v. Next divide both sides by -1. Remember, when you divide by a negative in an inequality, it becomes necessary to flip the direction of the inequality symbol. Therefore, the answer to the inequality is 2 ≤ v, which is equivalent to v ≥ 2.
A student is using the completing the square method to transform the equation x2-10x+ 36 = 0 to the form (x-h) 2+k = 0. which step would give him the correct answer?
A dog is attached to a 49 foot rope fastened to the outside corner of a fenced-in garden that measures 42 feet by 51 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander.
The area that a dog can roam, attached to a 49-foot rope at the corner of a 42x51 foot garden, was initially calculated as a quarter-circle's area with mistakes acknowledged in the approach. The correct calculation involves more intricate geometrical considerations, considering the question's unique constraints.
Explanation:To determine the area that a dog can wander, given it's attached to a 49-foot rope fastened to the outside corner of a fenced-in garden measuring 42 feet by 51 feet, we need to consider the shape of the area the dog can cover. The dog can cover a quarter-circle around each of two adjacent sides of the garden and a semi-circle along the side opposite the corner where the rope is fastened, provided the rope's length exceeds the garden's dimensions on those sides.
Calculating the area involves adding the areas of these three parts:
The quarter-circle around the 42-foot side: Area = 1/4 * π * (49)^2The quarter-circle around the 51-foot side: This area is already included in the first quarter-circle since the radius (49 feet) covers both sides.The semi-circle along the side opposite to the corner: Since the rope extends beyond the garden, this area is not applicable.Therefore, the total area that the dog can wander is effectively the area of a quarter-circle with a radius of 49 feet. This can be calculated as:
Area = 1/4 * π * (49)^2 square feet
Calculating this gives an area of 1884.96 square feet (rounded to two decimal places).
However, upon reevaluation, it's clear that a mistake was made in the initial explanation; the correct approach is to consider not just a quarter-circle but also the additional areas the rope covers beyond the garden's dimensions.
The exact area that the dog can wander is [tex]{2401\pi} \)[/tex] square feet.
Identify the radius: The radius of the circle that the dog can wander is the length of the rope, which is 49 feet.
Calculate the area of the circle: Use the formula for the area of a circle, [tex]\( A = \pi r^2 \)[/tex], where [tex]\( r = 49 \)[/tex] feet.
[tex]\[ A = \pi \times 49^2 \][/tex]
Compute [tex]\( 49^2 \)[/tex]:
[tex]\[ 49^2 = 2401 \][/tex]
Multiply to find the area:
[tex]\[ A = \pi \times 2401 \][/tex]
The exact area that the dog can wander is [tex]{2401\pi} \)[/tex] square feet.
This calculation confirms that the dog can wander an area of [tex]\( 2401\pi \)[/tex] square feet within the fenced-in garden while attached to the 49-foot rope.
(20 PTS!!)
The dot plot below shows the amount of time two random groups of students took to brush their teeth:
Based on visual inspection of the dot plots, which of the following groups, if any, shows a greater average time required to brush their teeth?
Group R
Group S
Both groups show about the same average time.
No conclusion about average time can be made from the data.
Answer:
The correct option is C.
Step-by-step explanation:
In a dot plot, the number of dots above a number represents the frequency of that number.
From the given dot plot it is clear that number of dots for group R is 21 and the time taken by the students of groups R are
45, 50, 50, 55, 60, 60, 70, 75, 80, 85, 90, 90, 95, 100, 100, 105, 110, 110, 115, 120, 120
The average of the data is
[tex]Average=\frac{\sum x}{n}[/tex]
[tex]Average=\frac{45+50+50+55+60+60+70+75+80+85+90+90+95+100+100+105+110+110+115+120+120}{21}[/tex]
[tex]Average=\frac{1785}{21}[/tex]
[tex]Average=85[/tex]
From the given dot plot it is clear that number of dots for group S is 17 and the time taken by the students of groups S are
50, 55, 60, 60, 65, 70, 70, 80, 90, 90, 95, 100, 105, 110, 110, 115, 120
[tex]Average=\frac{50+55+60+60+65+70+70+80+90+90+95+100+105+110+110+115+120}{17}[/tex]
[tex]Average=\frac{1445}{17}[/tex]
[tex]Average=85[/tex]
The average time of both groups are same.
Since both groups show about the same average time, therefore the correct option is C.
Lance rolls a die 5 times. What is the probability that he rolls an even number all five times?
Lance rolls a die 5 times
In a six sided dice, the even numbers are 2, 4 and 6. So 3 even numbers
the probability that he rolls an even number = [tex]\frac{3}{6} = \frac{1}{2}[/tex]
the probability that he rolls an even number all five times
= [tex]\frac{1}{2} *\frac{1}{2} * \frac{1}{2} * \frac{1}{2} * \frac{1}{2}= \frac{1}{32}[/tex]
So answer is [tex]\frac{1}{32}[/tex]
50 POINTS!!!!
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The Dust Bowl hit middle America a few years after the Great Depression started, and the Dust Bowl was considered one of the causes that continued the Great Depression during the 1930’s. Explain in three sentences how this natural disaster affected America’s economy in the 1930’s?
I need help with this question!
what is the product of 2.31 ans 0.21
What is the product of 2.31 and 0.21?
Solution:
We need to find 2.31*0.21
Let us first find 231*21, then we would consider decimals
231*1=231
231*2=462
So, 231*21=231+4620=4851
So, we get 231*21=4851
But, we need to find, 2.31*0.21
There are two numbers after decimal in 2.31 (3 and 1). and, there are two numbers after decimal in 0.21(2 and 1)
So, In result of, 2.31*0.21 there must be four digits after decimal.
As, 231*21=4851
So, 2.31*0.21=0.4851 (four digits after decimal)
Answer:0.4851