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Pavlova-9 [17]
3 years ago
13

In the figure, side AB is given by the expression 5x+5x+3, and side BC is 3x+92x−4. The simplified expression for the area of re

ctangle ABCD is ________ and the restriction on x is_____
Mathematics
2 answers:
miss Akunina [59]3 years ago
5 0
Area = side AB * side BC  =  (5x + 5x + 3)(3x + 92x - 4)

= (10x + 3)(95x - 4) =  950x^2 - 40x + 285x - 12

=  950x^2 + 245x - 12  Answer

This cannot be negative  so restriction on x  is  950x^2 + 245x > 12
That is x > 0.0421
almond37 [142]3 years ago
5 0

Answer:

A=950x^2+245-12\\\\\{x\in R: (\frac{4}{95}

Step-by-step explanation:

The area of a rectangle is given by:

A=w*h

Where:\\\\w=width\\h=height

Let:

h=\overline{\rm AB}=5x+5x+3=10x+3\\\\and\\\\w=\overline{\rm BC}=3x+92x-4=95x-4

So, the area is given by:

A=(10x+3)*(95x-4)=950x^2-40x+285x-12=950x^2+245x-12

It wouldn't make sense if the result leads us to an area equal to 0 or to a negative area, therefore:

A=950x^2+245x-12>0

Solving for x using the quadratic formula:

x=\frac{-b\pm \sqrt{b^2-4ac} }{2a} =\frac{-245\pm \sqrt{245^2-4(950)(-12)} }{2(950)} =\frac{-245\pm \sqrt{60025+45600} }{1900}\\\\x=\frac{-245\pm \sqrt{105625} }{1900}=\frac{-245\pm 325}{1900}\\\\Hence\\\\x>\frac{4}{95} \\\\and\\\\x

Therefore, the area is given by:

A=950x^2+245-12\\\\\{x\in R: (\frac{4}{95}

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or  (18.124,0) and ( -6.124,0) are the required points.

Step-by-step explanation:

Let (x,0) be the point on x -axis that are 14 units from the point (6,-7) .

Then by distance formula , we have

\sqrt{(x-6)^2+(0-(-7))^2}=14\ \ \ [\ \because distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}]

Taking square on both the sides , we get

(x-6)^2+7^2=14^2\\\\\Rightarrow\ x^2+6^2-2(6)x+49=196\\\\\Rightarrow\ x^2+36-12x=147\\\\\Rightarrow\ x^2-12x=111\\\\\Rightarrow\ x^2-12x-111=0

Using quadratic formula : x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

x=\dfrac{12\pm\sqrt{(-12)^2-4(1)(-111)}}{2}\\\\\Rightarrow\ x=\dfrac{12\pm\sqrt{144+444}}{2}\\\\\Rightarrow\ x=\dfrac{12\pm\sqrt{588}}{2}\\\\\Rightarrow\ x=\dfrac{12\pm\sqrt{2^2\times7^2\times3}}{2}\\\\\Rightarrow\ x=\dfrac{12\pm14\sqrt{3}}{2}\\\\\Rightarrow\ x=6\pm7\sqrt{3}

so, (6+7\sqrt{3},0)\text{ and }(6-7\sqrt{3},0) are the required points.

since \sqrt{3}=1.732

so, (6+7(1.732),0)\text{ and }(6-7(1.732),0) are the required points.

i.e. (18.124,0) and ( -6.124,0) are the required points.

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