3. A vacuum robot is in a room and charging at position (0, 5). Once charged, it begins moving on a northeast path at a constant speed of 1/2 foot per second along the line 4 − 3 = −15. After 60 seconds, it turns right 90° and travels in the new direction.
a. What are the coordinates of the point at which the robot made the turn?
b. Find an equation for the second line on which the robot traveled.
c. If after turning, the robot travels 80 seconds along this line, how far has it traveled from its starting position?
d. What is the equation of the line the robot needs to travel along in order to return and recharge? How long will it take the robot to get there?
Show your work
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.
Learn more about equation of a line here:
https://brainly.com/question/23087740
#SPJ3
Two consecutive positive odd integers have a product of 63. what is the smaller number
Which is the correct simplified form of the expression ?(4m-2n8/9m-6n-8)1/2
Answer:
[tex]\frac{2m^2n^8}{3}[/tex]
Step-by-step explanation:
The given expression is:
[tex](\frac{4m^{-2}n^8}{9m^{-6}n^{-8} })^{\frac{1}{2}}[/tex]
Upon simplifying the above equation, we get
Move [tex]m^{-6}[/tex] to teh numerator and Using the negative exponent rule, we get
=[tex](\frac{4m^{-2}n^8m^6}{9n^{-8} })^{\frac{1}{2}}[/tex]
Move [tex]n^{-8}[/tex] to teh numerator and Using the negative exponent rule, we get
=[tex](\frac{4m^{-2}n^8m^6n^8}{9})^{\frac{1}{2}}[/tex]
=[tex](\frac{4m^{4}n^{16}}{9})^{\frac{1}{2}}[/tex]
=[tex](\frac{(2)^2(m^2)^2(n^8)^2}{(3)^2})^{\frac{1}{2}}[/tex]
=[tex]\frac{2m^2n^8}{3}[/tex]
which is the correct simplified form of the given expression.
The volume of an oblique pyramid with a square base is V units3 and the height is h units.
Which expression represents the area of the base of the pyramid?
A)3v/h units
B)(3V – h) units
C)(V – 3h) units
D)V/3H units
Answer: A. 3V/ h units^2
Step-by-step explanation:
Look at the cylinder below. which of these could not be a cross section of the cylinder? a circle b square c triangle d rectangle
Final answer:
A square cannot be a cross section of a traditional circular cylinder when sliced perpendicular to its height. However, circles, rectangles, and triangles can be cross sections depending on the shape of the cylinder and the slicing.
Explanation:
When considering the cross sections of a cylinder, it is important to remember that the cross-sectional shape remains consistent along the height of the cylinder. A cross section made perpendicular to the height of a traditional circular cylinder is always a circle. Therefore, the correct answer to which shape could not be a cross section of a cylinder is a square, option b. Circles, rectangles, and triangles can all be cross sections of a cylinder depending on the shapes of the top and bottom faces of the cylinder and how the cross section is sliced.
If the cylinder is a right circular cylinder, the only possible perpendicular cross-sectional shape is a circle. A square cross section can occur in a cylinder if the top and bottom faces are square and the cylinder is not circular. Similarly, a rectangular cross section can occur if the cylinder has a rectangular type of cross-section to start with. However, for a circular cylinder, a triangle cannot be a cross section because slicing it will always yield a circular shape due to the round nature of its base.
This week, sandy was out sick on monday and tuesday. Last week, Jared was out sick on Thursday and Friday. The week before, Elisa was out sick on Wednesday and Thursday. What generalization can you make about these three students absents? Can you make a second generalization?
A generalization is a broad statement or an idea that is applied to a group of people or things. Most often, generalizations are not entirely true, because usually, the pattern they see in the relationship of a certain group of individuals or situations does not apply. In our example, where Elisa, Sandy, and Jared got sick and were absent from school for two days straight, within the 3 week period, we can make the following generalizations patterns:
1) There is an outbreak of sickness in the school for 3 consecutive weeks.
2) The said sickness lasts for two days.
A quadrilateral has angles that measure 90°, 46°, and 120°. What is the measure of the fourth angle?
find the height of the triangle
If SAT scores are normally distributed with a mean of 1518 and a standard deviation of 325, find the probability that a randomly selected SAT score is between 1550 and 1575.
A.
0.5714
B.
0.9684
C.
0.0316
D.
0.5398
Answer:
C.
0.0316
Step-by-step explanation:
Under the normal curve, approximately what percent of scores fall between and -1 to +1 standard deviations around the mean?
Which function has an inverse that is also a function?
{ ( -1 , 3 ) , ( 0,4 ), ( 1 , 14 ) , ( 5, 6 ) , ( 7, 2 )}
Further explanationFunction is a relation which each member of the domain is mapped onto exactly one member of the codomain.
There are many types of functions in mathematics such as :
Linear Function → f(x) = ax + bQuadratic Function → f(x) = ax² + bx + cTrigonometric Function → f(x) = sin x or f(x) = cos x or f(x) = tan xLogarithmic function → f(x) = ln xPolynomial function → f(x) = axⁿ + bxⁿ⁻¹ + ...If function f : x → y , then inverse function f⁻¹ : y → x
Let us now tackle the problem!
According to the definition above, it can be concluded that a function cannot have the same x value.
Of the four tables available in choices, table option C has an inverse that is also a function. This is because x values and y values are all different.
[tex]\{(-1,3) , ( 0,4} ) , ( 1,14 ) , ( 5, 6) , ( 7, 2) \}[/tex]
Option A doesn't have inverse because there is the same value of y i.e 4
[tex]\{(-1,-2) , ( 0,\boxed{4} ) , ( 1,3 ) , ( 5, 14) , ( 7, \boxed {4}) \}[/tex]
Option B doesn't have inverse because there is the same value of y i.e 4
[tex]\{(-1,-2) , ( 0,\boxed{4} ) , ( 1,5 ) , ( 5, \boxed {4}) , ( 7, 2) \}[/tex]
Option D doesn't have inverse because there is the same value of y i.e 4
[tex]\{(-1,\boxed {4}) , ( 0,\boxed{4} ) , ( 1,2 ) , ( 5, 3) , ( 7, 1) \}[/tex]
Learn moreInverse of Function : https://brainly.com/question/9289171Rate of Change : https://brainly.com/question/11919986Graph of Function : https://brainly.com/question/7829758Answer detailsGrade: High School
Subject: Mathematics
Chapter: Function
Keywords: Function , Trigonometric , Linear , Quadratic
Solve this inequality: 3q + 11 + 8q > 99. A. q > 8 B. q > 1/8 C. q > 88 D. q > 11
Factor this expression.
4x – 20
A. 4(x – 16)
B. 4(x – 6)
C. 4(x – 5)
D. 4(x – 10)
Write the equation of the given circle.
center (-2, -2)
radius of 6
Given: KLMN is a parallelogram m∠K : m∠KLM=1:3 LF ⊥ KN , LD ⊥ NM Find: m∠FLD
The measure of the angle m∠FLD is 45 degrees.
What is parallelogram?A parallelogram is a special type of quadrilateral that has equal and parallel opposite sides. The given figure shows a parallelogram ABCD which as AB parallel to CD and AD parallel to BC.
In a parallelogram, opposite angles are equal and since it is a quadrilateral, the angles add up to 360°.
[tex]\rm \angle NKL = \angle LMN = X\\\\\angle KLM = \angle KNM = 3X\\\\ 2(x+3x) = 360\\\\ 80x = 360\\\\ x=\dfrac{360}{80}\\\\ x = 45[/tex]
The figure LFND is another quadrilateral where:
[tex]\rm \angle LFN = \angle NDL =90\\\\ \angle FND = 3x[/tex]
The measure of the, m∠FLD is;
[tex]\rm (90\times 2)+3x+ (angle\ FLD) = 360\\\\m \angle\ FLD = 360- 180- (3\times 45)\\\\ m \angle\ FLD = 360 - 315\\\\ m \angle\ FLD = 45[/tex]
Hence, the measure of the angle m∠FLD is 45 degrees.
Learn more about parallelogram here;
https://brainly.com/question/13039289
#SPJ3
Professor Scott has 84 students in his college mathematics lecture class. The scores on the midterm exam are normally distributed with a mean of 72.3 and a standard deviation of 8.9. How many students in the class can be expected to receive a score between 63.4 and 81.2? Express the answer to the nearest student.
A suitable calculator can show you the number of students having a score in that range is expected to be about 57.
Kristin decides to spend at most $50 for a birthday dinner at a restaurant, including a 15% tip.,
Write an inequality in one variable to represent this situation.
What is the most that her meal can cost before a tip?
The representation of this situation as an inequality is 1.15x ≤ 50. To find the maximum possible cost of Kristin's meal before the tip, the inequality is solved for x, which results in the value of $43.48.
Explanation:In this scenario, the cost of Kristin's meal before the tip can be represented by the variable x. The tip is 15% of the cost of the meal, which can be represented as 0.15x. The total cost of the meal, including the tip, is therefore x + 0.15x, which is 1.15x. Given that the total cost cannot exceed $50, we can write the following inequality to represent this situation: 1.15x ≤ 50.
To find the most that her meal can cost before a tip, we need to solve this inequality for x. So, we divide both sides by 1.15: x ≤ 50/1.15. After calculation, we find that x (the cost of the meal before the tip) must be ≤ about $43.48.
Learn more about Inequality here:https://brainly.com/question/30231017
#SPJ3
A tree is 100 feet tall casts a shadow that is 150 feet long . Determine the angle at which the rays of of tge sun hit the ground , to tye nearest degree
During boot camp, the drill sergeant measured the weight of the men in his unit. he found the average weight of the men to be 142 pounds and the standard deviation 14 pounds. the data is normally distributed. find the interval in which 68% of the data lies. what is the probability that a man picked at random from the unit will weigh more than 170 pounds? that he will weigh less than 128 pounds?
To find the interval of 68% of the data, use the empirical rule. The probability a man weighs more than 170 lbs or less than 128 lbs can be found using the normal distribution curve.
Explanation:To find the interval in which 68% of the data lies, we can use the concept of the empirical rule. According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean.
Since the standard deviation is 14 pounds and the mean is 142 pounds, one standard deviation above the mean is 142 + 14 = 156 pounds, and one standard deviation below the mean is 142 - 14 = 128 pounds.
Therefore, the interval in which 68% of the data lies is from 128 pounds to 156 pounds.
The probability that a man picked at random from the unit will weigh more than 170 pounds can be found by calculating the area under the normal distribution curve to the right of 170 pounds.
The probability that a man picked at random from the unit will weigh less than 128 pounds can be found by calculating the area under the normal distribution curve to the left of 128 pounds.
Solve d = c π for π. A) π = cd B) π = c d C) π = d c D) π = c − d
Answer:
The answer is B
Step-by-step explanation:
Twenty different statistics students are randomly selected. for each of them, their body temperature ( degrees °c) is measured and their head circumference (cm) is measured. if it is found that r=0, does that indicate that there is no association between these two variables?
what is the best approximation of the volume, in cubic units, of a cone of diameter 20 and height 13
The best approximation for the volume of the cone in cubic units is 1361.357 units³.
What is Volume?Volume of a three dimensional shape is the space occupied by the shape.
Given that,
Diameter of the cone = 20
Radius of the cone, r = Half of diameter
= 20/2 = 10
Height of the cone, h = 13
Volume of the cone = 1/3 π r² h
= 1/3 π (10)² (13)
= 1361.357 units³
Hence the volume is 1361.357 units³.
Learn more about Volume here :
https://brainly.com/question/29767724
#SPJ2
Please help me find the value of x in the triangle in the link!
On a 10 item test, three students in prof. miller's advanced chemistry seminar received scores of two, five, and eight, respectively. for this distribution of test scores, standard deviation is equal to the square root of
The standard deviation for the given test scores can be calculated by finding the mean, computing the squared differences from the mean, averaging those squared differences to get the variance, and then taking the square root of that variance. The standard deviation for the scores is approximately 2.449.
The student is likely seeking assistance with the concept of standard deviation, which is a measure of dispersion or distribution around the mean in a set of data. The question at hand requires calculating the standard deviation for a small dataset consisting of three scores from a chemistry seminar. To find the standard deviation, one would first need to calculate the mean of the test scores, then find the variance by averaging the squared differences between each score and the mean, and finally, take the square root of the variance.
Step-by-step Calculation:
Calculate the mean (average) of the scores: (2 + 5 + 8) / 3 = 15 / 3 = 5.Find the squared differences from the mean: (2 - 5)² + (5 - 5)² + (8 - 5)² = 3² + 0² + 3² = 9 + 0 + 9.Calculate the variance: (9 + 0 + 9) / 3 = 18 / 3 = 6.The standard deviation is the square root of the variance: √(6) = 2.449The standard deviation of the test scores in Prof. Miller's advanced chemistry seminar is approximately 2.449.
Anja collected data about the number of dogs 9th-grade students own. She created this histogram to represent the data and determined that it is skewed right. Which statement is true about Anja’s claim?
What is a polynomial function in standard form with zeroes 0, 1, 4, and –1?
(Please help and thanks in advance!)
A regular octagonal pyramid has a base edge of 3m and a lateral area of 60m^2. Find its slant height.
The slant height is 5 m
What is the end behavior of the polynomial function? PLEASE HELP!!!
Please help asap!!!!!!!!!!!!
Can someone please help me with this!!
Will give brainliest!!