Which choice describes a kite? A. a quadrilateral that is a parallelogram and has two pairs of congruent angles B. a quadrilater
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Answer:
<em>1. A sperm whale is </em><em>3</em><em> </em><em>orders of magnitude</em><em> heavier than a La Plata river dolphin; 2. The radius of the larger ball is </em><em>one (1) order of magnitude</em><em> bigger than the radius of the smaller ball; 3. The volume of the larger ball is </em><em>3 orders of magnitude</em><em> bigger than the volume of the smaller ball</em>.
Step-by-step explanation:
If we expressed a number as:
(1)
Where
(2)
or
(3)
Then, <em>b</em> represents the <em>order of magnitude </em>of such a number (<em>Order of magnitude (2020), </em>in Wikipedia).
The order of magnitude can be defined as "...the smallest power of ten needed to represent a quantity" (Weisstein, Eric W. "Order of Magnitude". From MathWorld--A Wolfram Web Resource).
Having gathered all this information, we can proceed as follows:
<h3>First case</h3>The<em> La Plata river dolphin</em> weighs between 30 and 50kg and the sperm whale weighs between 35,000 and 40,000kg.
Then, considering (1) and (3) to express the dolphin and whale's weight (since in this way the order of magnitude is the same as the exponent part in the <em>scientific notation</em>):
Since the range for the weights are in the same order of magnitude for both dolphin and whale (considering the definition above):
Then
Thus
<em>A sperm whale is </em><em>3</em><em> </em><em>orders of magnitude</em><em> heavier than a La Plata river dolphin.</em>
<h3>Second case</h3>Following the same reasoning, we can conclude that <em>the radius of the larger ball is </em><em>one (1) order of magnitude</em><em> bigger than the radius of the smaller ball:</em>
For this case, we need to calculate <em>the volume of a sphere</em> for both radii (1cm and 10cm).
The volume of a sphere is
Then, the volume of the <em>ball of radius 1cm</em> is:
And, the volume of the <em>ball of radius 10cm</em> is:
Thus
As a result, <em>the volume of the larger ball is </em><em>3 orders of magnitude</em><em> bigger than the volume of the smaller ball</em>.
So, we know the center is at -3,-1, ok
hmmm what's the radius anyway? well, we know that there's a point at 1,2 that is on the circle's path...hmmmm what's the distance from the center to that point? well, is the radius, let's check then.
![\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ -3}}\quad ,&{{ -1}})\quad % (c,d) &({{ 1}}\quad ,&{{ 2}}) \end{array}\qquad % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2} \\\\\\ r=\sqrt{[1-(-3)]^2+[2-(-1)]^2}\implies r=\sqrt{(1+3)^2+(2+1)^2} \\\\\\ r=\sqrt{16+9}\implies r=\sqrt{25}\implies r=5](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bdistance%20between%202%20points%7D%5C%5C%20%5Cquad%20%5C%5C%0A%5Cbegin%7Barray%7D%7Blllll%7D%0A%26x_1%26y_1%26x_2%26y_2%5C%5C%0A%25%20%20%28a%2Cb%29%0A%26%28%7B%7B%20-3%7D%7D%5Cquad%20%2C%26%7B%7B%20-1%7D%7D%29%5Cquad%20%0A%25%20%20%28c%2Cd%29%0A%26%28%7B%7B%201%7D%7D%5Cquad%20%2C%26%7B%7B%202%7D%7D%29%0A%5Cend%7Barray%7D%5Cqquad%20%0A%25%20%20distance%20value%0Ad%20%3D%20%5Csqrt%7B%28%7B%7B%20x_2%7D%7D-%7B%7B%20x_1%7D%7D%29%5E2%20%2B%20%28%7B%7B%20y_2%7D%7D-%7B%7B%20y_1%7D%7D%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0Ar%3D%5Csqrt%7B%5B1-%28-3%29%5D%5E2%2B%5B2-%28-1%29%5D%5E2%7D%5Cimplies%20r%3D%5Csqrt%7B%281%2B3%29%5E2%2B%282%2B1%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0Ar%3D%5Csqrt%7B16%2B9%7D%5Cimplies%20r%3D%5Csqrt%7B25%7D%5Cimplies%20r%3D5)
so, what's the equation of a circle with center at -3, -1 and a radius of 5?
![\bf \textit{equation of a circle}\\\\ (x-{{ h}})^2+(y-{{ k}})^2={{ r}}^2 \qquad \begin{array}{lllll} center\ (&{{ h}},&{{\quad k}})\qquad radius=&{{ r}}\\ &-3&-1&5 \end{array} \\\\\\\ [x-(-3)]^2+[y-(-1)]^2=5^2\implies (x+3)^2+(y+1)^2=25](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bequation%20of%20a%20circle%7D%5C%5C%5C%5C%20%0A%28x-%7B%7B%20h%7D%7D%29%5E2%2B%28y-%7B%7B%20k%7D%7D%29%5E2%3D%7B%7B%20r%7D%7D%5E2%0A%5Cqquad%20%0A%5Cbegin%7Barray%7D%7Blllll%7D%0Acenter%5C%20%28%26%7B%7B%20h%7D%7D%2C%26%7B%7B%5Cquad%20%20k%7D%7D%29%5Cqquad%20%0Aradius%3D%26%7B%7B%20r%7D%7D%5C%5C%0A%26-3%26-1%265%0A%5Cend%7Barray%7D%20%0A%5C%5C%5C%5C%5C%5C%5C%0A%5Bx-%28-3%29%5D%5E2%2B%5By-%28-1%29%5D%5E2%3D5%5E2%5Cimplies%20%28x%2B3%29%5E2%2B%28y%2B1%29%5E2%3D25)
hmmm what's the radius anyway? well, we know that there's a point at 1,2 that is on the circle's path...hmmmm what's the distance from the center to that point? well, is the radius, let's check then.
so, what's the equation of a circle with center at -3, -1 and a radius of 5?