+2442 Hello, Guest!
A formula that will be important for this problem is the volume formula for a cylinder. It is the following:
\(V_{\text{cylinder}}=\pi r^2h\)
Let's compute the volume of the original right circular cylinder:
| \(V_{\text{original}}=\pi r_{\text{old}}^2 h_{\text{old}}\) | The original volume is the one where radius and height remained unchanged. |
Now, let's consider a volume wherein the variables are tweaked somewhat.
| \(r_{\text{new}}=r_{\text{old}}-20\%*r_{\text{old}}\) | As a decimal, 20%=0.2 |
| \(r_{\text{new}}=r_{\text{old}}-0.2r_{\text{old}}=0.8r_{\text{old}}\) | Now, let's find how the height was affected. |
| \(h_{\text{new}}=h_{\text{old}}+25\%*h_{old}\) | As a decimal, 25%=0.25 |
| \(h_{\text{new}}=h_{\text{old}}+0.25h_{\text{old}}=1.25h_{\text{old}}\) | Now, we have tweaked both variables to fit the description in the original problem. |
Now, let's find the volume of the new right cylinder:
| \(V_{\text{new}}=\pi r_{\text{new}}^2h_{\text{new}}\) | Plug in the known values for the radius and height. |
| \(V_{\text{new}}=\pi (0.8r_{\text{old}})^2*1.25h_{\text{old}}\) | The only thing left to do is simplify. |
| \(V_{\text{new}}=\pi *0.64r_{\text{old}}^2*1.25h_{\text{old}}\) | |
| \(V_{\text{new}}=0.8\pi r_{\text{old}}^2h_{\text{old}}\) | |
Now the only thing left to do is to calculate the percent change. I want you to try to do that. See what you can do. Check in with me if you would like.
+2442 I am not sure why the answerers have treated an expression as an equation.
Anyway, as the other answerers have pointed out, the trinomial \(-x^2-14x-24\) is factorable. When factoring, I have one suggestion: Manipulate the expression so that the quadratic term (the x^2-term) is positive.
| \(-x^2-14x-24\) | Factor out a negative one so that the quadratic term is positive. |
| \(-(\textcolor{red}{x^2+14x+24})\) | Now, factor the quadratic highlighted in red. Using the "AC" method, ac=24 and b=14. 12 and 2 are the only pair of numbers that multiply to obtain 24 and add to obtain 14. |
| \(-(x+12)(x+2)\) | There you go! You have factored this polynomial. |
+2442 Hello, Lightning!
The quadratic formula is a formula that solves for the roots of any quadratic. Let's apply it to the quadratic \(4x^2+7x+k\).
| \(a=4, b=7, c=k;\\ x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Substitute in the appropriate values into the formula. |
| \(x_{1,2}=\frac{-7\pm\sqrt{7^2-4*4*k}}{2*4} \) | Simplify. |
| \(x_{1,2}=\frac{-7\pm\sqrt{49-16k}}{8}\) | |
Obviously, we do not know what k is, but we do know that the roots of the quadratic with the unknown k are \(x_{1,2}=\frac{-7\pm i\sqrt{15}}{8}\).
Do you think you can take it on from here?
