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3 years ago

When a rectangle is dilated, how do the perimeter and area of the rectangle change?

1 answer:

4 0

A transformation of a figure in which all of the dimensions of the figure are multiplied by the same scale factor is called a dilation.

Effect of Dilation on Perimeter

Whenever a figure is dilated by a scale factor, the perimeter of the figure changes according to the same scale factor.

Effect of Dilation on Area

When a figure is dilated by a scale factor of $\frac{a}{b}$ , the area of the figure is dilated by a scale factor of $\frac{a^{2}}{b^{2}}$

Example-

Lets imagine a rectangle with length 10 cm and width 8 cm

So area becomes = 10*8 = 80 square cm

Lets suppose the rectangle is reduced by a scale factor of $\frac{1}{2}$ to produce a new rectangle.

So we will find the square of scale factor = $(\frac{1}{2})^{2}=\frac{1}{4}$

Now to find the area of the new rectangle(dilated one) we will multiply the area of the original rectangle by 1/4

= $80*\frac{1}{4}=20$

Hence, area becomes 20 square cm.

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Write down the explicit solution for each of the following: a) x’=t–sin(t); x(0)=1

Kay [80]

Answer:

a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)

Step-by-step explanation:

Let's solve by separating variables:

$x'=\frac{dx}{dt}$

a) x’=t–sin(t), x(0)=1

$dx=(t-sint)dt$

Apply integral both sides:

$\int {} \, dx=\int {(t-sint)} \, dt\\\\x=\frac{t^2}{2}+cost +k$

where k is a constant due to integration. With x(0)=1, substitute:

$1=0+cos0+k\\\\1=1+k\\k=0$

Finally:

$x=\frac{t^2}{2} +cos(t)$

b) x’+2x=4; x(0)=5

$dx=(4-2x)dt\\\\\frac{dx}{4-2x}=dt \\\\\int {\frac{dx}{4-2x}}= \int {dt}\\$

Completing the integral:

$-\frac{1}{2} \int{\frac{(-2)dx}{4-2x}}= \int {dt}$

Solving the operator:

$-\frac{1}{2}ln(4-2x)=t+k$

Using algebra, it becomes explicit:

$x=2+ke^{-2t}$

With x(0)=5, substitute:

$5=2+ke^{-2(0)}=2+k(1)\\\\k=3$

Finally:

$x=2+3e^{-2t}$

c) x’’+4x=0; x(0)=0; x’(0)=1

Let $x=e^{mt}$ be the solution for the equation, then:

$x'=me^{mt}\\x''=m^{2}e^{mt}$

Substituting these equations in c)

$m^{2}e^{mt}+4(e^{mt})=0\\\\m^{2}+4=0\\\\m^{2}=-4\\\\m=2i$

This becomes the solution m=α±βi where α=0 and β=2

$x=e^{\alpha t}[Asin\beta t+Bcos\beta t]\\\\x=e^{0}[Asin((2)t)+Bcos((2)t)]\\\\x=Asin((2)t)+Bcos((2)t)$

Where A and B are constants. With x(0)=0; x’(0)=1:

$x=Asin(2t)+Bcos(2t)\\\\x'=2Acos(2t)-2Bsin(2t)\\\\0=Asin(2(0))+Bcos(2(0))\\\\0=0+B(1)\\\\B=0\\\\1=2Acos(2(0))\\\\1=2A\\\\A=\frac{1}{2}$

Finally:

$x=\frac{1}{2} sin(2t)$

7 0

3 years ago

Write a equation that is parallel to 5x-y=4

ozzi

Let's have this equation equal y first. All we have to do is add y to both sides and subtract 4 from both sides.
5x-4=y
Now, to get a parallel line, the y intercept has to change. The y intercept (the -4 in the equation) can be an infinite amount of number but -4.
Let's choose 6. We would have 5x+6=y or you could put it in the way the other equation was, 5x-y=-6.

6 0

3 years ago

Please help me , i don't know this it's 7th grade math

scZoUnD [109]

Answer:

C. 1920

Step-by-step explanation:

25*8*10

1920

8 0

2 years ago

Read 2 more answers

When given an algebraic expression involving subtraction why is it best to rewrite the expression using addition before identify

torisob [31]

When given an algebraic expression involving subtraction why is it best to rewrite the expression using addition before identifying the terms is because you can consider the basic rule of operations such as the PEMDAS or simply the MDAS in computing and calculating equations. Hence Addition comes before Subtraction. For example given -3 + 5, you can arrange this set into a simpler one such as 5 – 3 = 2. You aim to have a positive sum rather than a negative value sum.

7 0

3 years ago

Please help with 16 & 17

son4ous [18]

Answer:

16. Parallelogram

17. x=3, y=9

Step-by-step explanation:

16. You should graph it to get a visual. But using the given points, you can tell that the slopes for the lines AC and BD are the same, thus meaning those lines are parallel. The same for lines AD and BC.

17. 5x+2=17, so x= 3

3y-6=21, so y=9

5 0

3 years ago