Answer the question

Mathematics

1 answer:

ahrayia [7]3 years ago

6 0

Answer:

<h2>Perimeter: 16x + 16</h2><h2>Area: 40x + 15</h2>

Step-by-step explanation:

$l-\text{length}\\\\w-\text{width}\\\\\text{The formula of a perimeter of a rectangle:}\ P=2(l+w)\\\text{The formula of an area of a rectangle:}\ A=lw\\\\\text{We have}\ l=8x+3\ \text{and}\ w=5.\\\text{Substitute:}\\\\P=2(8x+3+5)\\P=2(8x+8)\qquad\text{use the distributive property:}\ a(b+c)=ab+ac\\P=(2)(8x)+(2)(8)\\\boxed{P=16x+16}\\\\A=(8x+3)(5)\qquad\text{use the distributive property}\\A=(8x)(5)+(3)(5)\\\boxed{A=40x+15}$

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4 years ago

Find the center of mass of the wire that lies along the curve r and has density =4(1 sin4tcos4t)

dolphi86 [110]

The mass of the wire is found to be 40π√2 units.

To calculate the mass of the wire which runs along the curve r ( t ) with the density function δ=5.

The general formula is,

Mass = $\int_a^b \delta\left|r^{\prime}(t)\right| d t$

To find, we must differentiate this same given curve r ( t ) with respect to t to estimate |r'(t)|.

The given integration limits in this case are a = 0, b = 2π.

Now, as per the question;

The equation of the curve is given as;

r(t) = (4cost)i + (4sint)j + 4tk

Now, differentiate this same given curve r ( t ) with respect to t.

$\begin{aligned}\left|r^{\prime}(t)\right| &=\sqrt{(-4 \sin t)^2+(4 \cos t)^2+4^2} \\&=\sqrt{16 \sin ^2 t+16 \cos ^2 t+16} \\&=\sqrt{16\left(\sin t^2+\cos ^2 t\right)+16}\end{aligned}$

Further simplifying;

$\begin{aligned}&=\sqrt{16(1)+16} \\&=\sqrt{16+16} \\&=\sqrt{32} \\\left|r^{\prime}(t)\right| &=4 \sqrt{2}\end{aligned}$

Now, use integration to find the mass of the wire;

$\begin{aligned}&=\int_a^b \delta\left|r^{\prime}(t)\right| d t \\&=\int_0^{2 \pi} 54 \sqrt{2} d t \\&=20 \sqrt{2} \int_0^{2 \pi} d t \\&=20 \sqrt{2}[t]_0^{2 \pi} \\&=20 \sqrt{2}[2 \pi-0] \\&=40 \pi \sqrt{2}\end{aligned}$