Answer:
The length of each side of the 9 in
Step-by-step explanation:
The volume of the gift box is 972 in. The height of the gift box is 12 inches
and the area of the base is 81 in².
We want to determine the length of each side of the square base if the base shape is a square.
Recall that, the area of a square is
![{l}^{2}](https://tex.z-dn.net/?f=%20%7Bl%7D%5E%7B2%7D%20)
It was given that, the area of this base is 81 in²
![{l}^{2} = 81](https://tex.z-dn.net/?f=%20%7Bl%7D%5E%7B2%7D%20%20%3D%2081)
Take positive square root:
![l = \sqrt{81} = 9in](https://tex.z-dn.net/?f=l%20%3D%20%20%5Csqrt%7B81%7D%20%20%3D%209in)
As we know that the standard equation of circle is
, where <em>r</em> is the radius of circle and centre at <em>(h,k) </em>
Now , as the circle passes through <em>(2,9)</em> so it must satisfy the above equation after putting the values of <em>h</em> and <em>k</em> respectively
![{:\implies \quad \sf (3)^{2}+(4)^{2}=r^{2}}](https://tex.z-dn.net/?f=%7B%3A%5Cimplies%20%5Cquad%20%5Csf%20%283%29%5E%7B2%7D%2B%284%29%5E%7B2%7D%3Dr%5E%7B2%7D%7D)
After raising ½ power to both sides , we will get <em>r = +5 , -5</em> , but as radius can never be -<em>ve</em> . So <em>r = +</em><em>5</em><em> </em>
Now , putting values in our standard equation ;
<em>This is the required equation of </em><em>Circle</em>
Refer to the attachment as well !
Step-by-step explanation:
Since, roots are - 3 & 2
Therefore, (x + 3) & (x - 2) would be factors.
Leading coefficient is 5
Hence, required quadratic equation is:
![5(x + 3)(x - 2) = 0 \\ \therefore \: 5( {x}^{2} - 2x + 3x - 6) = 0\\ \therefore \: 5( {x}^{2} + x - 6) = 0\\ \therefore \: 5 {x}^{2} + 5x - 30 = 0\: \\ is \: the \: required \: quadratic \: equation.](https://tex.z-dn.net/?f=5%28x%20%2B%203%29%28x%20-%202%29%20%3D%200%20%5C%5C%20%20%20%5Ctherefore%20%5C%3A%20%205%28%20%7Bx%7D%5E%7B2%7D%20%20-%202x%20%2B%203x%20-%206%29%20%20%3D%200%5C%5C%20%20%5Ctherefore%20%5C%3A%20%205%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%20x%20-%206%29%20%20%3D%200%5C%5C%20%5Ctherefore%20%5C%3A%205%20%7Bx%7D%5E%7B2%7D%20%20%2B%205x%20-%2030%20%20%3D%200%5C%3A%20%20%5C%5C%20is%20%5C%3A%20the%20%5C%3A%20required%20%5C%3A%20%20%20quadratic%20%5C%3A%20equation.)
$5125.78. For simple questions like this you can search them on google and find answers very quickly so you dont waste points btw.