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Vera_Pavlovna [14]
3 years ago
13

Find r(t) if r'(t) = 5t4i + 6t5j + t k and r(1) = i + j.

Mathematics
1 answer:
Vladimir [108]3 years ago
8 0
\mathbf r(t)=\displaystyle\int(5t^4\,\mathbf i+6t^5\,\mathbf j+t\,\mathbf k)\,\mathrm dt
\mathbf r(t)=(t^5+C_1)\,\mathbf i+(t^6+C_2)\,\mathbf j+\left(\dfrac12t^2+C_3\right)\,\mathbf k

Given \mathbf r(1)=\mathbf i+\mathbf j, we have

\begin{cases}1^5+C_1=1\\\\1^6+C_2=1\\\\\dfrac121^2+C_3=0\end{cases}\implies C_1=0,C_2=0,C_3=-\dfrac12

so that the particular solution is

\mathbf r(t)=t^5\,\mathbf i+t^6\,\mathbf j+\dfrac12(t^2-1)\,\mathbf k
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Answer:

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Step-by-step explanation:

Given data

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Answer:

a) The volume in cubic inches is 36000

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Step-by-step explanation:

* Lets study the information of the problem to solve it

- The dimensions of the piece of cardboard are 90 inches by 70 inches

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∴ The new dimensions of the piece of cardboard are;

90 - (15 × 2) = 90 - 30 = 60 inches

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- The dimensions of the box will be:

# Length = 60 inches

# width = 40 inches

# height = 15 inches

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V = Length × width × height

∵ The length = 60 inches

∵ The width = 40 inches

∵ The height = 15 inches

∴ V = 60 × 40 × 15 = 36000 inches³

a) The volume in cubic inches is 36000

* Now lets revise how to change from inch to feet

- 1 foot = 12 inches

∵ 1 foot = 12 inches

∴ 1 foot³ = (12)³ inches³

∴ 1 foot³ = 1728 inches³

∵ The volume of the box is 36000 inches³

∴ The volume of the box in cubic feet = 36000 ÷ 1728 = 125/6  

b) The volume in cubic feet is 125/6

* Now lets revise how to change from feet to yard

- 1 yard = 3 feet

∵ 1 yard = 3 feet

∴ 1 yard³ = (3)³ feet³

∴ 1 yard³ = 27 feet³

∵ The volume of the box is 125/6 feet³

∴ The volume of the box in cubic yard = 125/6 ÷ 27 = 125/162  

c) The volume in cubic yard is 125/162

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Answer:

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