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3 years ago

Work out cube root 1331 : reciprocal of 0.2 give your answer in the form n:1

Mathematics

1 answer:

faust18 [17]3 years ago

6 0

Answer:

2.2 : 1

Step-by-step explanation:

cube root 1331 : reciprocal of 0.2

give your answer in the form n:1

The cube root of 1331

(1331)^1/3 = 11

Reciprocal of 0.2 = 1 /0.2 = 5

Hence, the ratio Can be expressed as :

11 : 5

In the form n : 1

divide both sides by 5

11/5 : 5 /5

2.2 : 1

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Suppose that \nabla f(x,y,z) = 2xyze^{x^2}\mathbf{i} + ze^{x^2}\mathbf{j} + ye^{x^2}\mathbf{k}. if f(0,0,0) = 2, find f(1,1,1).

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The simplest path from (0, 0, 0) to (1, 1, 1) is a straight line, denoted $C$ , which we can parameterize by the vector-valued function,

$\mathbf r(t)=(1-t)(\mathbf i+\mathbf j+\mathbf k)$

for $0\le t\le1$ , which has differential

$\mathrm d\mathbf r=-(\mathbf i+\mathbf j+\mathbf k)\,\mathrm dt$

Then with $x(t)=y(t)=z(t)=1-t$ , we have

$\displaystyle\int_{\mathcal C}\nabla f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}\nabla f(x(t),y(t),z(t))\cdot\mathrm d\mathbf r$

$=\displaystyle\int_{t=0}^{t=1}\left(2(1-t)^3e^{(1-t)^2}\,\mathbf i+(1-t)e^{(1-t)^2}\,\mathbf j+(1-t)e^{(1-t)^2}\,\mathbf k\right)\cdot-(\mathbf i+\mathbf j+\mathbf k)\,\mathrm dt$

$\displaystyle=-2\int_{t=0}^{t=1}e^{(1-t)^2}(1-t)(t^2-2t+2)\,\mathrm dt$

Complete the square in the quadratic term of the integrand: $t^2-2t+2=(t-1)^2+1=(1-t)^2+1$ , then in the integral we substitute $u=1-t$ :

$\displaystyle=-2\int_{t=0}^{t=1}e^{(1-t)^2}(1-t)((1-t)^2+1)\,\mathrm dt$

$\displaystyle=-2\int_{u=0}^{u=1}e^{u^2}u(u^2+1)\,\mathrm du$

Make another substitution of $v=u^2$ :

$\displaystyle=-\int_{v=0}^{v=1}e^v(v+1)\,\mathrm dv$

Integrate by parts, taking

$r=v+1\implies\mathrm dr=\mathrm dv$

$\mathrm ds=e^v\,\mathrm dv\implies s=e^v$

$\displaystyle=-e^v(v+1)\bigg|_{v=0}^{v=1}+\int_{v=0}^{v=1}e^v\,\mathrm dv$

$\displaystyle=-(2e-1)+(e-1)=-e$

So, we have by the fundamental theorem of calculus that

$\displaystyle\int_C\nabla f(x,y,z)\cdot\mathrm d\mathbf r=f(1,1,1)-f(0,0,0)$

$\implies-e=f(1,1,1)-2$

$\implies f(1,1,1)=2-e$

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3 years ago

If m angle GEF is thirteen less than five times m angle DEG and m angle DEF = 149º, find m angle GEF.

Nina [5.8K]

Answer:

122*

122 degrees

Step-by-step explanation:

m∠GEF is 13 less than 5 times m∠DEG and m∠DEF = 149*

Solution:

As per given data,

m∠GEF = 5m∠DEG - 13* … (i)

m∠DEF = 149* -> m∠GEF + m∠DEG = 149* .. (ii)

Substituting value of m∠GEF in (ii)

We get,

(5m ∠DEG - 13*) + m∠DEG = 149*

6m ∠DEG - 13* = 149*

6m ∠DEG = 149* + 13* = 162*

m∠DEG = * = 27*

Substituting value of m∠DEG in (i)

We get,

m∠GEF = 5(27*) - 13*

m∠GEF = 135* - 13* = 122*

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3 years ago