The tangential component of the acceleration vector at point t = 1 is aT(1) = 233/3 and The normal component of the acceleration vector at point t = 1 is aN(1) = (1/3)√10459
How do we calculate the tangential component?The acceleration vector can be found from the following formula:
[tex]a(t) = r''(t) = (-3,-10t,-8t3).[/tex]
To find the tangential component of the acceleration vector, we first need the velocity vector v(t).
[tex]v(t) = r'(t) = (-3,-10t,-8t3) .[/tex]
Next, we need to normalize the velocity vector using the following formula:
[tex]T(t) = v(t) / ||v(t)||,[/tex]
Where ||v(t)|| is the magnitude of the velocity vector.
[tex](1) = (-3,-10,-8) / \sqrt{(3^2 + 10^2 + 8^2)} = (-3/3, -10/3, -8/3) = (-1 , -10/3, -8/3) .[/tex]
Then, the tangential component of a(1) is:
[tex]aT(1) = a(1) T(1) = (-3, -10, -8) (-1, -10/3, -8/3) = 3 + 100/3 + 64/3 = 233/3.[/tex]
How do we calculate the normal component?To find the normal component of a(1), we simply need to find the magnitude of the tangential component and subtract it from the magnitude of the acceleration vector.
[tex]aN(1) = \sqrt{ (a^2 - aT(1)^2)} = \sqrt{(3^2 + (10)^2 + (8)^2 - (233/3)^ 2)} = \sqrt{(9 + 100 + 64 - 54289/9)} = \sqrt{(10459/9)} = (1/3)\sqrt{10459}[/tex]
Therefore, the tangential and normal components of the acceleration vector at the point t = 1 are:
[tex]aT(1) = 233/3[/tex] and [tex]aN(1) = (1/3)\sqrt{10459}[/tex]
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19. Assertion(A): The graph of the linear equation 7x - 2y = 6 cuts the Y-axis at the point (0, -3). Reason(R): The coordinates of any point on the Y-axis is (a, 0), where a is any real number. pls help
get the answer
Answer:
Pretty sure its C
Step-by-step explanation:
To cut through y axis, x axis is always 0, So it would be (0, a) where a is any real number, not (a,0) as is given in reason.
A city's population was 84,000 at the beginning of 2020. If the city's population increases by 4% per year, how many years will it take for city's population to reach 132,000 people? a. The answer to this question is the solution to what equation? [Let a represent the number of years since the beginning of 2020.] Preview b. Solve the equation in part (a)
a. The answer to this question is the solution to the equation:
84000(1 + 0.04)^a = 132000
Where "a" represents the number of years since the beginning of 2020, and 0.04 is the decimal equivalent of 4%.
We use the formula for compound interest, which is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we assume that the population growth rate is compounded annually, so n = 1.
b. Solving the equation in part (a), we get:
(1 + 0.04)^a = 132000/84000
1.04^a = 1.5714
a = log(1.5714)/log(1.04)
a ≈ 9.9 years
Therefore, it will take approximately 9.9 years for the city's population to reach 132,000 people, assuming a 4% annual growth rate.
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Will give brainlest to first correct answer!!!
Evelyn has a bag that contains 3 red marbles and 2 blue marbles.
Evelyn randomly pulls a marble from the bag and then puts it back in the bag. She repeats this 20 times. How many times should she expect to draw a red marble from the bag?
Answer:
She will draw 120 times for a red marble
Step-by-step explanation:
Can someone help me find the elevation of the sun I need the answers that are highlighted in yellow please help image below
By using Pythagοrean theοrem, the angle οf elevatiοn tο the sun, vertical height, hοrizοntal distance, slant height, tangent and angle οf elevatiοn is 51°
What is Pythagοrean theοrem ?The Pythagοrean theοrem is a fundamental result in Euclidean geοmetry that describes the relatiοnship between the three sides οf a right-angled triangle.
In equatiοn fοrm, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
Where c is the length οf the hypοtenuse, and a and b are the lengths οf the οther twο sides (alsο knοwn as the legs) οf the right-angled triangle.
a) The Name οf the angle οf elevatiοn is: angle οf elevatiοn tο the sun
b) The Name οf the Hypοtenuse side is: vertical height (bοdy height)
c) The Name οf the Oppοsite side is: hοrizοntal distance (shadοw length)
d) The Name οf the Adjacent side is: slant height (hypοtenuse οf the triangle fοrmed by the bοdy, the tip οf the shadοw, and the sun)
e) The Trig Ratiο I will be using is: tangent (οppοsite/hypοtenuse) because we are given the οppοsite side and the hypοtenuse.
f) Shοw the wοrk tο find the angle οf elevatiοn. Rοund tο the nearest degree:
tan(angle οf elevatiοn) = οppοsite/hypοtenuse
tan(angle οf elevatiοn) = 45/34
angle οf elevatiοn [tex]= tan^{-1(45/34)[/tex]
angle οf elevatiοn ≈ 51°
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Barbara is going to buy a shirt for 15.00$ and a pair of pants for 25.00$. She has to also pay 7% sales tax. If Barbara pays with a 50$ bill, how much change will she get back?
Barbara will get $7.20 back in change.
What exactly is a sales tax?A sales tax that is typically added to the selling price by the seller.
Without tax, the total cost of the shirt and pants is:
$15.00 + $25.00 = $40.00
The sales tax is 7% of $40.00, which is as follows:
0.07 x $40.00 = $2.80
So, with tax, the total cost of the shirt and pants is:
$40.00 + $2.80 = $42.80
If Barbara pays with a $50 bill, she will receive the following change:
$50.00 - $42.80 = $7.20
As a result, Barbara will receive $7.20 in change.
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7) 99 was divided by some number, then added to 15. Next, this sum was
multiplied by 8, which gave a product of 48. Find this number.
Answer:
-11
Step-by-step explanation:
48 ÷8=6-15=-9
99÷-9=-11
Bernie helped design a magician costume for a school play. He used 5.75 feet of ribbon for the magician's wand. He used 11.75 feet of the same ribbon to decorate the magician's robe. If Bernie used $10.50 of ribbon in all, how much did the ribbon cost per foot?
The best solution gets brainlist
The ribbon cost $0.60 per foot.
Solution:[tex]\text{Total number of ribbon used} = (5.75 + 11.75) = 17.50 \ \text{feet}[/tex]
[tex]\text{17.50 feet of ribbon cost} = \$10.50[/tex]
[tex]\text{1 foot of ribbon cost = x}[/tex]
Cross Multiply[tex]17.50 \times x = 1 \ foot \times \$10.50[/tex]
[tex]x = 1 \ foot \times 10.50\div17.50[/tex]
[tex]x = \$0.60[/tex]
Therefore, The ribbon cost per foot $0.60 per foot.
Idil drove 12 miles in 1/5 of an hour. On average how fast did she drive in miles per hour
If Idil drove 12 miles in 1/5an hour on average, then she drove at an average speed of 60 miles per hour.
To calculate the average speed of Idil's car in miles per hour, we need to divide the distance she drove (12 miles) by the time it took her to drive that distance (1/5 hour):
Average speed = distance ÷ time
Average speed = 12 miles ÷ (1/5) hour
To divide by a fraction, we can multiply by its reciprocal, so:
Average speed = 12 miles × 5/1 hour
Average speed = 60 miles per hour
Therefore, Idil drove at an average speed of 60 miles per hour.
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A principal gathered data about the distance in miles that his teachers and bus drivers live from the school
A principal gathered data about the distance in miles that his teachers and bus drivers live from the school. then, the interquartile range of the distances for the bus drivers in 5 miles less than the interquartile range of the distance for the teacher.
Interquartile Range:
In descriptive statistics, the interquartile range (IQR) is a measure of the statistical distribution, the distribution of data. IQR can also be referred to as bad reading, 50% mean, fourth distribution, or H-distribution. It is defined as the difference between the 75th and 25th percentile of the data. To calculate the IQR, the data set is divided into quartiles, or four parts of equal rank by linear interpolation. These quartiles are represented by Q1 (also known as the lower quartile), Q2 (the median) and Q3 (also known as the upper quartile).
The lower quartile is the 25th percentile and the upper quartile is the 75th percentile, so IQR = Q3 − Q1
Basically, the interquartile range represents the width or “spread” of the collection. The interquartile range is determined by the difference between the upper quartile (highest 25%) and lower quartile (lowest 25%) points of a data set.
From the figure, we can find:
The interquartile range of the bus driver is: 20 -10 = 10
The interquartile range of the teacher is: 30 -15 = 15
Therefore,
the interval interquartile distance is bus The driver's distance was 5 miles less than the interquartile range of the teacher's distance.
Complete Question:
A principal gathered data about the distance in miles that his teachers and bus drivers live from the school .The box plots below show these data.
(1) The inter Quartile range of the distances for the bus driver is twice the interquartile range of the distances for the teachers.
(2) The range of the distances for the teachers is twice the range of the distances for the bus drivers.
(3) the interquartile range of the distances for the bus drivers in 5 miles less than the interquartile range of the distance for the teacher.
(4) The range of the distances for the teachers is 5 miles less than the range of the distances for the bus driver.
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Answer: C. The interquartile range of the distances for the bus drivers is 5 miles less than the interquartile range of the distances for the teachers.
if the shaded cross sections of the solids have the same area, which of the following corresponds to the value of a: the side length of the base of the square prism?
Therefore, the value of 'a' when the shaded cross-sections of the solids have the same area is
:a = √(4A/π).
To find out the value of a (side length of the base of the square prism) when the shaded cross-sections of the solids have the same area, we need to use the formula for the area of a square.
The area of a square is given by the formula: A = a², where 'a' is the side length of the square.
Now, let's look at the two solids given in the question.
The first solid is a square prism with base 'a' and height 'a'.The second solid is a right circular cylinder with radius 'a' and height '2a'.The cross-section of the square prism is a square with side length 'a'.The cross-section of the right circular cylinder is a circle with radius 'a'.
Given that the shaded cross-sections of the solids have the same area, we can equate the area of the square with the area of the circle.
A = πr², where 'r' is the radius of the circle Substituting the values of 'r' and 'a', we get:A = πa²/4 = a²/4Multiplying both sides by 4, we get:4A = πa²
Now we can solve for 'a' by taking the square root of both sides a = √(4A/π).
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3) Jiya walks 6 km due east and then 8 km due north. How far is she from her
starting place?
6 km
8 km
Answer:
10km
Step-by-step explanation:
Pythagoras
Formula=[tex]a^2+b^2=c^2[/tex]
[tex]6^{2} +8^{2} =c^2\\c^2=100\\c=\sqrt{100} \\c=10km[/tex]
I need help I need to show my work please help
Answer: 19
Step-by-step explanation:
This is an isosceles trapezoid. Note that because this is an isosceles trapezoid, QN and MP are equal. Use the lengths given and solve:
3x + 1 + 6 = 6x - 5
3x = 12
x = 4
Plug x = 4 in 3x + 1 + 6, and we get 3(4) + 1 + 6 = 12 + 7 = 19
Hope this helps!
FOR BRAINLIEST!!
Directions: Solve for x. The figure is a parallelogram
Answer:
Answer is in the picture
Step-by-step explanation:
Hope you understand :)
Answer:
x = 10-----------------------------
According to one of the properties of a parallelogram, any two consecutive interior angles are supplementary.
We know supplementary angles add up to 180°.
Apply this property to the given parallelogram and set up the following equation:
35 + 14x + 5 = 18014x + 40 = 18014x = 140x = 10the combined score on this test ranges from 400 to 1600. if you were to randomly draw five numbers from a 400-1600 number set, what is the probability that the medium score of the actual 2022 sat results is contained in between the highest and lowest value of these five random numbers?
The combined score on the 2022 SAT test ranges from 400 to 1600. If you were to randomly draw five numbers from a 400-1600 number set, the probability that the medium score of the actual 2022 SAT results is contained in between the highest and lowest value of these five random numbers is approximately 0.004%
How do we find the probability?To find the probability that the medium score of the actual 2022 SAT results is contained in between the highest and lowest value of these five random numbers, we need to find the probability of the following event: “the three other random numbers drawn lie between the highest and lowest values.
The probability of choosing one of the five numbers that falls within the range is (1600 – 400)/1201 = 1/2.25.
The first number can be any number within the 400-1600 range, so the probability is 1.The second number must lie within the range created by the highest and lowest values of the first number, which has a width of 1201. Thus, the probability is 1201/3201.
The third number must lie within the range created by the highest and lowest values of the first two numbers, which has a width of 801. Thus, the probability is 801/2401.The fourth and fifth numbers must lie within the range created by the highest and lowest values of the first three numbers, which has a width of 401.
Thus, the probability is 401/1601.Therefore, the probability of the medium score of the actual 2022 SAT results being between the highest and lowest values of these five random numbers is (1/2.25) * (1201/3201) * (801/2401) * (401/1601) * 1 = 0.000038 or approximately 0.004%.
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Assume that women have heights that are normally distributed with mean of 63.6 inches and a standard deviation of 2.5 inches. Find the value of the quartile Q3. A) 66 inches B) 65.3 inches C) 67.8 inches D) 64.3 inches 3
The correct option is B. The value of the quartile Q3 is 65.3 inches
Quartile is a type of statistical value that represents one-fourth of a dataset. It divides a dataset into four equal parts. Q1, Q2, and Q3 are the first, second, and third quartiles, respectively.
The formula for the third quartile is as follows:
Q3 = μ + (0.67 × σ), where μ is the mean and σ is the standard deviation.
Given:
Mean = μ = 63.6 inches
Standard deviation = σ = 2.5 inches
The value of the quartile Q3 can be found by substituting the given values in the formula.
Q3 = μ + (0.67 × σ)
Q3 = 63.6 + (0.67 × 2.5)
Q3 = 63.6 + 1.675
Q3 = 65.275 ≈ 65.3
Therefore, the value of the quartile Q3 is 65.3 inches. Therefore, option B is the correct answer.
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To refresh your memory of 9th grade math read Chapter 12. Complete the following table of angle and side measurements. Right Angles Angle A B C a b c
1 40 __ 90 __ __ 55 2 __ 55 __ __ 77 __
3 35 __ __ __ 68 83
4 __ __ __ 50 70 __
To refresh your memory of 9th-grade math, read Chapter 12. Complete the following table of angle and side measurements.
Right AnglesAngleA B C a b c
1 40 50 90 50 40 552 35 55 90 55 35 77
3 35 55 90 55 35 684 30 60 90 60 30 855 60 30 90 60 30 906 45 45 90 45 45 63
The above given table represents the values of angles and sides of the right-angled triangles. In the right-angled triangle, one angle measures 90°, and this angle is known as the right angle.
Further, the sum of other two angles measures 90°.In the above table, a, b and c are sides, and angles A, B, and C are opposite angles of a, b, and c, respectively.
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What is the slope of the line passing through the points (4, -6) and (2, 3)?
Answer:
m = -9/2
Step-by-step explanation:
Slope = rise/run or (y2 - y1) / (x2 - x1)
Points (4, -6) (2, 3)
We see the y increase by 9, and the x decrease by 2, so the slope is
m = -9/2
Who ever helps me, Get 100 points
Step-by-step explanation:
a) Area=144m²
side²= 144
side=12m
b) perimeter=32m
4×side=32
side=32/4
side=8m
Which number has a 3 with a value 100 times greater than the three in 62,091.384
Answer:
You didn't give any options, but I'm guessing one of them has a 3 in the tens digits, which should be the right answer
Step-by-step explanation:
300?
Can someone please help me with this?
Answer:
Yes they are congruent
Option B is correct
Step-by-step explanation:
Opisite sides of parallelogram are congurent so,
BA is congruent to CD (side)
And
BE I congruent to ED (Side)
AE is congruent to EC (Side)
So according to side-side-side congurence theorem triangle ABE is congruent to triangle CDE
Hope you understand :)
Amina was given 25% trade discount on an article listed 0.75p she sold it with a cash discount of 5% find Amina's percentage profit
Amina's percentage profit is 25%, as the percentage profit must be rounded down to the nearest whole number.
What is profit?Profit is an important concept in economics and finance, referring to the difference between the cost of production and the revenue generated from the sale of goods and services. It is a measure of success and financial performance and is used to analyze the profitability of a business. Profit is generally expressed as a percentage of revenue or as a dollar amount.
To calculate this, first find the cost of the article after the trade discount:
Article Cost = 0.75p - (0.75p * 0.25) = 0.75p * 0.75 = 0.5625p
Next, find the cost of the article after the cash discount:
Article Cost = 0.5625p - (0.5625p * 0.05) = 0.5625p * 0.95 = 0.5343p
Finally, calculate the percentage profit:
Percentage Profit = (0.75p - 0.5343p) / 0.75p * 100 = 0.2167 / 0.75 * 100 = 28.89%
Amina's percentage profit is 25%, as the percentage profit must be rounded down to the nearest whole number.
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If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is A(x) p(x) * (a) Find A'(x). Why does the company want to hire more workers if A'(x) > 0? (b) Show that A'(x) > 0 if p'(x) is greater than the average productivity.
The average productivity of the workforce, A(x), is given by
A(x) = p(x)/x,
where p(x) is the total value of the production when there are x workers in the plant.
A'(x) is the derivative of A(x) with respect to x.
A'(x) = (p(x)/x)' = (p'(x)x - p(x))/x^2.
The company wants to hire more workers if A'(x) > 0 because it means that the average productivity of the workforce is increasing with additional workers.
A'(x) > 0 if p'(x) is greater than the average productivity, which is p(x)/x.
Therefore, A'(x) > 0 if p'(x) > p(x)/x.
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if other factors are held constant, which of the following sets of data would produce the largest t-statistic for an independent-samples t-test?
The set of data that would produce the largest t-statistic for an independent-samples t-test is the one with the largest difference between the sample means and the smallest variability within each sample.When other factors are held constant, the t-statistic for an independent-samples t-test is given by the following formula:
t=(x1 − x2)/ SEwhere x1 and x2 are the means of the two samples, and SE is the standard error of the difference between the means. The standard error of the difference between the means is given by:SE = sqrt [s1^2/n1 + s2^2/n2]where s1 and s2 are the standard deviations of the two samples, and n1 and n2 are the sample sizes. Therefore, the t-statistic can be maximized by maximizing the difference between the sample means and minimizing the variability within each sample.Therefore, the set of data that would produce the largest t-statistic for an independent-samples t-test is the one with the largest difference between the sample means and the smallest variability within each sample.
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(5 points) For each of the following vector spaces, write down the zero vector. No justification required.a. (1 point) The vector space R 4 with the standard vector addition/scalar multiplication.b. (1 point) The vector space P3 with standard polynomial addition/scalar multiplication.c. (1 points) The vector space M22 with standard matrix addition/scalar multiplication.d. (2 points) The vector space M23 with standard matrix addition/scalar multiplication.
Here are the solutions:
a) R4 = {(a, b, c, d) | a, b, c, d ∈ R}
b) P3 = {a0 + a1x + a2x^2 + a3x^3 | a0, a1, a2, a3 ∈ R}
c) M22 = {(aij) | i, j = 1, 2}
d) M23 = {(aij) | i = 1, 2, j = 1, 2, 3}
For the given vector space R4 with the standard vector addition/scalar multiplication, the zero vector is the following: 0 = [0, 0, 0, 0].b) For the given vector space P3 with standard polynomial addition/scalar multiplication, the zero vector is the following: 0(x) = 0.c) For the given vector space M22 with standard matrix addition/scalar multiplication, the zero vector is the following:[0 0] [0 0]d) For the given vector space M23 with standard matrix addition/scalar multiplication, the zero vector is the following:[0 0 0][0 0 0].
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Can you guys please help me with this question I think it’s A. but I’m not sure if it’s correct?!PLEASEEE
Answer:
A is correct
Step-by-step explanation:
Whatever value you put in for "y" will keep both of the equations equal
A triangle is shown on the coordinate plane below.
1 unit
H
y
(-1,9) of
-(-1, 2)
7
6
5
-3
3 d
(7,2)
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
-11
-2
What is the area of the triangle?
Il unit
X
You answer would be mate it would be x=66
Let A denote the balance at the end of Month n for each month where the balance is positive. To find a formula for An, we can do the following. For Month 1, note that A1 = 4000(1.015) - 100 For Month 2, note that A2 =[4000(1.015 ) – 100] (1.015) – 100
= 4000(1.015)^2 – 100 - 100(1.015) For Month 3, note that A3 = [4000(1.015)^2 - 100 - 100(1.015)] (1.015) - 100 = [4000(1.015)^3 - 100 - 100(1.015)] - 100(1.015)^2
B. (6 pts) (Formulas for An) i. Give a recursive formula for An. Make sure to show that the formula is consistent with the results for n= 1,2,3 on the previous page. ii. Give an explicit formula for An in summation notation that captures the pattern exhibited at the bottom of the previous page. Make sure to show that the formula is consistent with the results for n = 1,2,3 on the previous page.
(a) The recursive formula for An is: An = (1.015)An-1 - 100.
(b) The explicit formula for An in summation notation is: An = 4000(1.015)^n - 100[1 + (1.015) + (1.015)^2 + ... + (1.015)^(n-1)].
(a) This formula says that to find the balance for month n, we take the balance for month n-1, multiply it by 1.015 (to account for the interest rate), and subtract 100 (to account for the withdrawal). This formula is consistent with the results for n=1,2,3 on the previous page.
(b) The explicit formula for An in summation notation is: An = 4000(1.015)^n - 100[1 + (1.015) + (1.015)^2 + ... + (1.015)^(n-1)].
This formula says that to find the balance for month n, we take the starting balance of 4000 multiplied by the interest rate raised to the power of n, and then subtract the sum of 100 multiplied by the geometric series (1 + r + r^2 + ... + r^(n-1)), where r = 1.015. This formula is consistent with the results for n=1,2,3 on the previous page.
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Which two angles in the triangles below are complementary?
I NEED HELPPPPPPPP
Answer:
∠ BAC and ∠ CAD
Step-by-step explanation:
• Complementary angles sum to 90°
∠ BCA and ∠ ACD are a linear pair and sum to 180°
∠ BCA + 110° = 180° ( subtract 110° from both sides )
∠ BCA = 70°
the sum of the 3 angles in a triangle = 180°
in Δ ABC
∠ BAC + 55° + 70° = 180°
∠ BAC + 125° = 180° ( subtract 125° from both sides )
∠ BAC = 55°
in Δ ACD
∠ CAD + 110° + 35° = 180°
∠ CAD + 145° = 180° ( subtract 145° from both sides )
∠ CAD = 35°
then
∠ BAC + ∠ CAD = 55° + 35° = 90°
∠ BAC and ∠ CAD are complementary
Answer:
(a) ∠BAC and ∠CAD
Step-by-step explanation:
You want to know which angles in the figure are complementary.
Complementary anglesTwo angles are complementary when the total of their measures is 90°. In the given figure, triangle ABD is a right triangle with the right angle at vertex A.
The two acute angles at vertices B and D will be complementary:
∠ABC and ∠CDA . . . . . not on the list of choices
Ray AC divides the right angle into two complementary parts:
∠BAC and ∠CAD . . . . . first choice on the list
__
Additional comment
The two smaller triangles are isosceles. Angles CDA and CAD are both 35°, so have a sum of 70°. Likewise, angles ABC and BAC are both 55°, with a sum of 110°.
Angles BCA and ACD are a linear pair, so have a sum of 180°. They are supplementary, not complementary.
the centroid of a triangle is located 12 units from one of the vertices of a triangle. find the length of the median of the triangle drawn from that same vertex
a. 16
b. 18
c. 24
d. 36
e. 48
Length of the median of the triangle is b. 18
How to find the length of the median of a triangle?
Given that the centroid of a triangle is located 12 units from one of the vertices of a triangle. We need to find the length of the median of the triangle drawn from that same vertex.
Let ABC be a triangle.
Let AD be median of the triangle.
Let E be centroid of ΔABC
Length of the median of triangle = AD
AD = Length of AE + Length of ED
But,
AE = 12
ED = x
So,
AD = 12 + x
Since, centroid divides median in 2:1 ratio.
Then,
[tex]12 : x = 2 : 1\\12/x = 2/1\\\\12= 2x\\[/tex]
Divide by 2 each side
[tex]x = 6[/tex]
So, AD = 12 + x = 12 + 6 = 18
Thus, Length of the median of triangle is 18.
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A 17-ft ladder is leaning against the side of a house. The top of the ladder is sliding down the house at rate of 5 ft/sec. a) Determine how fast the bottom of the ladder is sliding away when the top of the ladder is 8 feet from the ground. b) At what rate is the angle o between the ladder and the ground is changing then.
a) d/dt [√(289 - 8²)] = d/dt [√(255)] = d/dt [√(5²*17)] = 5/2√17 m/s.
b) The required value is the rate of change of x when the top of the ladder is 8 feet from the ground, then y = 15 feet.
a) Determine how fast the bottom of the ladder is sliding away when the top of the ladder is 8 feet from the ground. As the ladder leans against the side of the house, it makes a right angle with the ground. Using Pythagorean Theorem, the length of the ladder is given by.
Ladder length = √(hypotenuse)² - (height)²= √(17² - height²) meters. Taking the derivative of both sides of this equation, we get: dL/dt = [d/dt √(289 - h²)] meters/sec. Let h = 8 feet, dL/dt = -5 feet/sec.
Thus, the rate of change of the bottom of the ladder is d/dt [√(289 - h²)] meters/sec when the top of the ladder is 8 feet from the ground, and the bottom of the ladder is sliding away at a rate of 5 feet/sec.
b) At what rate is the angle o between the ladder and the ground changing then. Let y be the height of the ladder and the angle it makes with the ground be x. From the given values, y = 17 cos x.
Taking the derivative of both sides with respect to time: dy/dt = -17 sin x (dx/dt)In the given problem, dx/dt = -5/17sin x. Then, dy/dt = 5 sin x cos x= (5/2) sin 2x.
Solving this value of sin x by Pythagorean Theorem, sin x = 8/17. Substituting the value in the equation for the derivative, we get: dy/dt = (5/2) * sin 2x= (5/2) * sin 2(arc sin(8/17))= (5/2) * (2*8/17 * 15/17)= (600/289) ft/sec.
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