+2446 \(\boxed{1}\hspace{3mm}y=x^2-3x+4\\ \boxed{2}\hspace{3mm}x+y=4\)
In the system of equations, the first equation is already solver for, so I can substitute its value into the second question.
| \(y=\textcolor{red}{x^2-3x+4};\\ x+\textcolor{red}{y}=4\) | Use substitution to get rid of one variable. | ||||
| \(x+\textcolor{red}{x^2-3x+4}=4\) | Combine like terms and subtract 4 from both sides. | ||||
| \(x^2-2x=0\) | Factor the GCF of the left-hand side of the equation: x. | ||||
| \(x(x-2)=0\) | Set both factors equal to 0 and solve. | ||||
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Let's substitute both possibilities for y:
| \(\boxed{2}\hspace{3mm}x+y=4\) | Let's substitute in the first possibility for x: x=0. |
| \(0+y=4\) | |
| \(y_1=4\) | |
| \(\boxed{2}\hspace{3mm}x+y=4\) | Substitute for the second option: x=2. |
| \(2+y=4\) | |
| \(y_2=2\) | |
Here are the answers in coordinate form: \((0,4)\) and \((2,2)\)
.+2446 1) It turns out that someone has already asked this question already, and I just happen to remember this post from February 27 of this year. I see no reason to make a duplicate answer.
Here is the link: https://web2.0calc.com/questions/help_80894
2a) The following equation below shows the general formula for a function that has exponential growth or decay.
\(P(t)=a*b^t\)
\(P(t)\) represents the function that yields the population of the town t years after 2010.
a = initial population of town
b = rate of exponential growth or decay
Since a is the initial population of the town, a=8500.
b, in this case, represents the portion of the population that remains as a fraction or decimal. The population begins at 100%. If the population decreases by 4.5% every year, then 100%-4.5% or 95.5% represents the percentage of the population that remains. As a decimal, this would be written as 0.955.
Now that both of this function are known, we can create an equation to find the population t years after 2010.
\(P(t)=8500(0.955)^t\)
2b) Since I have generated the equation for you, do you think that you could solve the next one? Just substitute into the formula and solve.
+2446 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.
