Final answer:
To find the position vector r(t) given its derivative and an initial condition, we integrate each component of the derivative and apply the initial condition to solve for constants of integration. The final position function is r(t) = t³i + t⁴j + (0.5t² - 0.5)k.
Explanation:
To find r(t) given that r'(t) = 3t²i + 4t³j + tk and r(1) = i + j, we integrate each component of r'(t) with respect to t. The integral of a derivative returns the original function plus a constant of integration, which we can solve using the initial condition provided.
For the i component: ∑ 3t² dt = t³+ C1
For the j component: ∑ 4t³ dt = t⁴ + C2
For the k component: ∑ t dt = 0.5t² + C3
Applying the initial condition r(1) = i + j, we substitute t = 1 into r(t) to solve for the constants of integration:
(1) + C1 = 1, so C1 = 0
(1) + C2 = 1, so C2 = 0
0.5(1) + C3 = 0, so C3 = -0.5
Therefore, the position function r(t) is given by t³i + t⁴j + (0.5t² - 0.5)k.
r(t) = t^3i + t^4j + ((1/2)t^2 - 1/2)k.
To find r(t) given r'(t) = 3t^2i + 4t^3j + tk and r(1) = i + j, we need to integrate r'(t) with respect to t.
Step 1: Integrate each component of r'(t) separately:
∫3t^2 dt = t^3 + C1 (integration with respect to t)
∫4t^3 dt = t^4 + C2 (integration with respect to t)
∫tk dt = (1/2)t^2k + C3 (integration with respect to t)
Step 2: Combine the results to get r(t):
r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2k + C3)
Step 3: Use the given initial condition r(1) = i + j to find the values of the constants C1, C2, and C3:
r(1) = (1^3 + C1)i + (1^4 + C2)j + ((1/2)(1)^2k + C3)
i + j = i + j + (1/2)k + C3
Comparing the coefficients of k, we get:
(1/2) + C3 = 0
C3 = -1/2
Therefore, r(t) = (t^3 + C1)i + (t^4 + C2)j + ((1/2)t^2 - 1/2)k
The formula V = πd2h 8 is used to find the volume of a parabolic cone. In the formula "d" represents the diameter of the cone and "h" represents the height. What is the volume of the parabolic cone that is 6 cm in diameter and 12 cm in height? A) 36π cm3 B) 48π cm3 C) 54π cm3 D) 60π cm3
Answer:
V = 54π cm³ ⇒ answer (C)
Step-by-step explanation:
∵ V = (πd²h)/8
∵ d = 6 cm
∵ h = 12 cm
∴ V = (π × 6² × 12) ÷ 8 = 54π cm³
Final answer:
To find the volume of a parabolic cone with a diameter of 6 cm and height of 12 cm using the formula V = πd²h/8, calculate the radius (3 cm), square it, multiply by the height, divide by 8, and then multiply by π, resulting in 54π cm³(Option C).
Explanation:
The student's question pertains to calculating the volume of a parabolic cone using a specific formula, which is V = πd²h/8, where d is the diameter and h is the height of the cone.
Given the cone's diameter (d) of 6 cm and height (h) of 12 cm, we can plug these values into the formula to find the volume:
First, calculate the radius (r) of the cone by dividing the diameter by 2: r = d / 2 = 6 cm / 2 = 3 cm.Next, plug the radius and height into the formula and calculate the volume:Therefore, the volume of a parabolic cone with a diameter of 6 cm and a height of 12 cm is 54π cm³.
If a ploynominal has four terms 3x^3+5x+6x^2+10 which factoring method can be considered
Answer:
(c) Factor by Grouping
a cylinder shaped drum is used as a garbage container. The drum has a height of 4 ft and a radius of 1.25 ft how many cubic feet of garbage does the drum hold ? enter your answer as a decimal rounded to the nearest hundreth
Answer:
19.63
Step-by-step explanation:
The price of a shirt was $38. It was reduced by 20% and then again by 10%. What would the price of be if it were reduced by 30% from the original
Original Price of the shirt is $38.
Double discount :
Given that it was reduced by 20%, means
[tex] \\ Cost \; of \; Shirt\; after\; 1^{st}\; discount (20\%)\\ \\= \; (100\; - \; 20) \%\; of \; Original Price=\; 80\%\; of\; \$38\\ \\ = \frac{80}{100} \times 38\; =\; \frac{3040}{100}\; =\; \$30.40\\ \\ Cost \; of \; Shirt \; after\; 2^{nd}\; discount(10\%)\\ \\ (100 - 10)\%\; of\; $30.40=90\%\; of\; \$30.40\\ [/tex]
[tex] =\frac{90}{100} \times 30.40 = \frac{2736}{100}= \$27.36 \\ \\ After \; double \; \; discount,(20\%\; of\; Original\; Price, again\; 10\%\; reduced\; )\\\\ Double \; Discounted \; Price = \$ 27.36\\ \\ But \; After\; single \; discount\; of\; 30\%\; of\; Original \; Price, \; we\; get\\ \\ (100-30)\%\; of \; \$38=70\% \times \$38=\; \frac{70}{100} \times 38 =\frac{2660}{100}=\$26.6 [/tex]
Please help me questions 9 and 10
If the diameter of a circle changes from 18 cm to 6 cm, how will the circumference change?
A) multiplies by 1/3
B) multiplies by 3
C) increases by 10
D) decreases by 10
What is the product of 3 and (5/4n+1.8)?
Evaluate the function at the indicated value of x. Round your result to three decimal places
F(x)=500e(0.05)^x with a value x=2
The given function is represented below:
F(x)=500e^(0.05)x with a value x=2
or, F(x) = 500 * e^(0.05*2)
( putting the value of x as 2 )
or, F(x) = 500 * e^(0.1)
or, F(x) = 500 * e^0.1
or, F(x) = 500 * e^(0.1)
or, F(x) = 500 * 1.10517092
or, F(x) = 552.585459
The value of the function at x = 2, rounded off to 3 decimal places is given by 552.585
Hope this helps..!!
Thank you :)
Integral of y/((y^2)-1)
Final answer:
The integral of y/((y^2)-1) is (1/2) ln|y^2 - 1| + C.
Explanation:
To evaluate the integral of y/((y^2)-1), we can perform a substitution by letting u = y^2 - 1. Then, du = 2y dy, and the integral becomes:
∫ y/((y^2)-1) dy = (1/2) ∫ 1/u du = (1/2) ln|u| + C
Substituting back u = y^2 - 1, the final answer is (1/2) ln|y^2 - 1| + C.
If the polynomial P(x) has roots −5, −1, 2, and 2, which of the following represents the factored form of function P(x)?
The graph of y = x^2 has been translated 7 units to the left. The equation of the resulting parabola is _____.
Answer:
The equation of the resulting parabola is:
y=(x+7)^2
Step-by-step explanation:
We, know that the transformation of the type:
f(x+a) is a translation of the parent function f(x) to the left or right depending on the sign of the constant 'a'.
if:
a<0 then the translation is to the right.
and if a>0 then the translation is to the left.
It is given that the graph of the function,
let f(x)=y=x^2 is translated 7 units to the left.
This means that the equation of the resulting function will be:
y=(x+7)^2
Birthweights at a local hospital have a normal distribution with a mean of 110 oz. and a standard deviation of 15 oz. what z-value corresponds to a birthweight of 138 oz.? (round your answer to the nearest hundredth.)
help and explain............
The mean finish time for a yearly amateur auto race was 185.19185.19 minutes with a standard deviation of 0.3410.341 minute. the winning car, driven by rogerroger, finished in 184.14184.14 minutes. the previous year's race had a mean finishing time of 110.4110.4 with a standard deviation of 0.1370.137 minute. the winning car that year, driven by sallysally, finished in 110.05110.05 minutes. find their respective z-scores. who had the more convincing victory? rogerroger had a finish time with a z-score of nothing. sallysally had a finish time with a z-score of nothing. (round to two decimal places as needed.)
Answer:
Let X the random variable that represent the mean finish time for a yearly amateur auto race a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(185.19,0.341)[/tex]
Where [tex]\mu=185.19[/tex] and [tex]\sigma=0.341[/tex]
The z score is given by this formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
And for a time of 184.14 we have the following z score:
[tex] z = \frac{184.14-185.19}{0.341}= -3.08[/tex]
Let Y the random variable that represent the mean finish time for the previous year auto race a population, and for this case we know the distribution for X is given by:
[tex]Y \sim N(110.4,0.137)[/tex]
Where [tex]\mu=110.4[/tex] and [tex]\sigma=0.137[/tex]
The z score is given by this formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
And for a time of 110.05 we have the following z score:
[tex] z = \frac{110.05-110.4}{0.137}=-2.557[/tex]
As we can see we have a higher z score for the case of the previous year so then we have a more convincing victory on this case since represent a higher quantile in the normal standard distribution.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the mean finish time for a yearly amateur auto race a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(185.19,0.341)[/tex]
Where [tex]\mu=185.19[/tex] and [tex]\sigma=0.341[/tex]
The z score is given by this formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
And for a time of 184.14 we have the following z score:
[tex] z = \frac{184.14-185.19}{0.341}= -3.08[/tex]
Let Y the random variable that represent the mean finish time for the previous year auto race a population, and for this case we know the distribution for X is given by:
[tex]Y \sim N(110.4,0.137)[/tex]
Where [tex]\mu=110.4[/tex] and [tex]\sigma=0.137[/tex]
The z score is given by this formula:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
And for a time of 110.05 we have the following z score:
[tex] z = \frac{110.05-110.4}{0.137}=-2.557[/tex]
As we can see we have a higher z score for the case of the previous year so then we have a more convincing victory on this case since represent a higher quantile in the normal standard distribution.
A convex polygon has 6 sides what is the sum of its interior angles
Ted take home pay is 1900 a month. He spend 17%of his take home pay on groceries. How much do groceries cost tedeach month
Megan buys 3 bracelets and 3 necklaces. Each bracelet costs 5. Megan pays on 40 and gets 4 change. what is the cost of one necklace?
How many years does it take for an annuity of $ 1,000 to grow to $ 20,000, assuming k = 7%?
a. 12.94
b. 13.02
c. 14.18
d. 15.67
e. none of the above?
A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top. if water is poured into the cup at a rate of 2cm3/s, how fast is the water level rising when the water is 5 cm deep?
The water level in a paper cup shaped like a cone with radius 3 cm and height 10 cm, filled at a rate of 2 cm³/s, rises at approximately 0.025 cm per second when the water is 5 cm deep.
Explanation:This question involves related rates, a concept in Calculus. We know that the volume, V, of a cone with a radius r and height h is given by the formula V = (1/3)πr²h. Given that the shape of the cup is conical, the radius and the height of the water in the cup are proportional, so we can express r as r=3h/10.
Thus, we can rewrite the volume formula in terms of h: V = 1/3 * π * (3h/10)² * h = πh³/100. Differentiating both sides with respect to time t, we get dV/dt = πh² dh/dt. We want to find dh/dt (the rate at which the water level rises) when h=5 cm and given that dV/dt (the rate at which water is poured into the cup) is 2 cm³/s.
Plugging these values into the differentiated formula, we get: 2 = π(5)² * dh/dt. Solving for dh/dt, we find that dh/dt = 2/(25π) or about 0.025 cm/s. So, the water level is rising at a rate of approximately 0.025 cm per second when the water is 5 cm deep.
Learn more about Related Rates here:https://brainly.com/question/29898746
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whats the lcd of 6/7, 3/5 and 1/4
the surface area of a pyramid is 533 square meters. what is the slant height? (base=13m) (width=13m)
what is b(-10) from the given
Answer:
6
Step-by-step explanation:
Replace x with -10
[tex]b(-10) = |(-10) +4| = |-6| = 6[/tex]
A women has twice as many dimes as quarters in her purse.If the dimes were quarters and the quarters were dimes,she would have $1.20 more than she now has.How many of each does she have?
Answer:
16 dimes and 18 quarters!!
0.5x + 0.1x = 0.25x + 0.2x + 1.20
0.6x = 0.45x + 1.20
0.6x - 0.45x = 1.20
0.15x = 1.20
x = 1.20/0.15
x = 8 quarters
2x = 16 dimes
hope this helped :D
How to get from 92 to 280 in 4 jumps. This is supposed to be an exponential equation. You have to multiply the same number everytime.
jennifer made $74,900 last year she pay 6% state income tax, 15% federal income tax 6.2% for social security and 1.45% for medicare what is her monthly income? what was her net monthly income
To calculate Jennifer's monthly income, divide her annual income by 12. Then, subtract the deductions and taxes to find her net monthly income.
Explanation:To calculate Jennifer's monthly income, we need to divide her annual income by 12, since there are 12 months in a year. So, Jennifer's monthly income would be $74,900 / 12 = $6,241.67.
To calculate Jennifer's net monthly income, we need to subtract the deductions and taxes from her monthly income. First, let's calculate the deductions:
State income tax: 6% of $6,241.67 = $374.50Federal income tax: 15% of $6,241.67 = $936.25Social Security: 6.2% of $6,241.67 = $386.67Medicare: 1.45% of $6,241.67 = $90.39Now, we can subtract these deductions from Jennifer's monthly income:
Net monthly income = $6,241.67 - $374.50 - $936.25 - $386.67 - $90.39 = $4,453.86
Therefore, Jennifer's net monthly income is $4,453.86.
A sample of 26 elements from a normally distributed population is selected. the sample mean is 10 with a standard deviation of 4. the 95% confidence interval for μ is
The 95% confidence interval for the population mean ( μ) is approximately (8.46,11.54), calculated from a sample of 26 elements with a mean of 10 and a standard deviation of 4.
To calculate the 95% confidence interval for the population mean (μ), we can use the formula:
Confidence interval=Sample mean±(Critical value× Sample size/Standard deviation )
Given:
Sample mean (xˉ ) = 10
Standard deviation (σ) = 4
Sample size (n) = 26
Confidence level = 95%
Step 1: Find the critical value from the Z-table for a 95% confidence level.
Since it's a two-tailed test, we'll find the Z-value corresponding to a cumulative probability of 0.975.
Z α/2 =1.96
Step 2: Plug the values into the formula:
Confidence interval=10±(1.96× 26/4 )
Step 3: Calculate the margin of error:
Margin of error≈1.96×45.099
Margin of error≈1.96× 5.0994
Margin of error≈1.96×0.785
Margin of error≈1.5376
Step 4: Calculate the confidence interval:
Lower limit=10−1.5376
Lower limit≈8.4624
Upper limit=10+1.5376
Upper limit≈11.5376
So, the 95% confidence interval for
μ is approximately (8.4624,11.5376).
Read the proof. Given: m∠H = 30°, m∠J = 50°, m∠P = 50°, m∠N = 100° Prove: △HKJ ~ △LNP Statement Reason 1. m∠H = 30°, m∠J = 50°, m∠P = 50°, m∠N = 100° 1. given 2. m∠H + m∠J + m∠K = 180° 2. ? 3. 30° + 50° + m∠K = 180° 3. substitution property 4. 80° + m∠K = 180° 4. addition 5. m∠K = 100° 5. subtraction property of equality 6. m∠J = m∠P; m∠K = m∠N 6. substitution 7. ∠J ≅ ∠P; ∠K ≅ ∠N 7. if angles are equal then they are congruent 8. △HKJ ~ △LNP 8. AA similarity theorem Which reason is missing in step 2? CPCTC definition of supplementary angles triangle parts relationship theorem triangle angle sum theorem
Answer:
Step-by-step explanation:
Given: m∠H = 30°, m∠J = 50°, m∠P = 50°, m∠N = 100°
To prove: △HKJ ~ △LNP
Proof:
Step 1. m∠H = 30°, m∠J = 50°, m∠P = 50°, m∠N = 100° (Given)
Step 2. m∠H + m∠J + m∠K = 180° (Triangle angle sum theorem)
Step 3. 30° + 50° + m∠K = 180°(substitution property)
Step 4. 80° + m∠K = 180°(addition Property)
Step 5. m∠K = 100°(subtraction property of equality)
Step 6. m∠J = m∠P; m∠K = m∠N(substitution)
Step 7. ∠J ≅ ∠P; ∠K ≅ ∠N (If angles are equal then they are congruent)
Step 8. △HKJ ~ △LNP( AA similarity theorem)
Hence proved.
Thus, the missing step in 2 is (Triangle angle sum theorem)
Answer:
D) triangle angle sum theorem
Step-by-step explanation:
Just finished the test!!
Use the Distributive Property to find (z−5)(z+3).
The Distributive Property allows you to multiply the terms within (z−5)(z+3) to obtain z² + 3z - 5z - 15. After combining like terms, the final result is z² - 2z - 15.
Explanation:To use the Distributive Property to find the product of (z−5)(z+3), you multiply each term in the first parenthesis by each term in the second parenthesis. Here are the steps:
Multiply z by z, which is z².
Multiply z by +3, which gives you +3z.
Multiply -5 by z, which results in -5z.
Finally, multiply -5 by +3, which is -15.
After multiplication, you combine like terms:
z² + 3z - 5z - 15
Combine +3z and -5z to get -2z
So, the final result is z² - 2z - 15.
Complete the statements below that show y = 8x2 + 32x + 17 being converted to vertex form. Factor out the leading coefficient. y = 8(x2 + 4x) + 17 Write in vertex form.
Answer:
4, -32
2, -15
Step-by-step explanation:
Trust me bro.
Which of the polygons listed below have at least three angles?
I Triangles
II Quadrilaterals
III Pentagons
IV Hexagons
A. III and IV
B. II, III, and IV
C. I, II, III, and IV
D. IV