+2446 I'm pretty sure that GST stands for Goods and Services Tax. It's a Canadian tax, so the rest of the world may be unfamiliar with such:
Anyway, let's calculate the amount paid for a DVD. To find it, find how much the discount affected the price of the DVD:
| \($12.50*0.45=5.625\) | This is how much the discount affected the price. Subtract 5.625 from the original price of the DVD. |
| \(12.50-5.625=$6.875\) | In the context of money, money is always rounded to the nearest hundredth place, so round this number to that decimal. |
| \($6.88\) | |
Okay, now we know what the cost of the DVD is after discount is applied to the price. Now, let's add the discount. Normally, you would multiply 6.88 by 5% and then add it to 6.88, but there is actually a faster method that requires one fewer step: Multiply 6.88*1.05. See how that saves a step? Let's try it together:
| \($6.88*1.05=$7.224\) | Yet again, round to the nearest hundredth. |
| \($7.22\) | You are done! This is your final cost for the DVD. |
+2446 Use the law of cosines to find the missing angle measures. I'll use a picture to illustrate the law of cosines. I think is makes it easier to understand:
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/lcos.gif
Let's say that the lowercase letters have lengths. I'll arbitrarily assign them for you:
a=10
b=15
c=19
Using this information, simply substitute into the law of cosines formula. You will need a calculator that can calculate trigonometric functions:
| \(c^2=a^2+b^2-2ab\cos C\) | Simply plug in the known values and solve for the missing one. |
| \(19^2=10^2+15^2-2(10)(15)\cos C\) | Because you can use a calculator, I would not simplify anything yet. Subtract 10^2+15^2 on both sides. |
| \(19^2-10^2-15^2=-2(10)(15)\cos C\) | Divide by -2(10)(15) on both sides |
| \(\frac{19^2-10^2-15^2}{-2(10)(15)}=\cos C\) | Use the inverse cosine to isolate C. |
| \(C=\cos^{-1}(\frac{19^2-10^2-15^2}{-2(10)(15)})\) | Use your calculator to evaluate this monstrosity |
| \(C\approx96.89^\circ\) | Repeat this process for the other missing angle measures. |
One last warning before you go! Be sure that your calculator is on degree mode when doing the final calculation. Otherwise, your answer will be represented differently. In radian mode, \(\cos^{-1}(\frac{19^2-10^2-15^2}{-2(10)(15)})\approx1.69\). It is not wrong but a triangle's angle measure is usually represented in degrees--not radians. If your calculator can only preform a calculation like this in radian mode, multiply your answer by \(\frac{180}{\pi}\) to convert.
+2446 The parabola focus is the point wherein the distance to a point on a parabola is equidistant to the distance to the directrix!
To find the focus, convert the quadratic to vertex form, \(y=a(x-h)^2+k\) where \((h,k+\frac{1}{4a})\) is the focus. Let's try and do this:
| \(y=\frac{1}{8}x^2+4x+20\) | This is the original quadratic equation. In order ro convert the quadratic to the desired form above, we need to use a method called "completing the square." First, subtract 20 on both sides. |
| \(y-20=\frac{1}{8}x^2+4x\) | Multiply by 8 on both sides to get rid of the pesky fraction |
| \(8y-160=x^2+32x\) | This is where completing the square comes in handy. Do the linear x-term and half it. Take that quantity and square it. Add it to both sides. |
| \(8y-160+(\frac{32}{2})^2=x^2+32x+(\frac{32}{2})^2\) | Simplify both sides of the equation |
| \(8y+96=x^2+32x+256\) | What's the point of doing all this work? Well, the right hand side is a perfect square trinomial. |
| \(8y+96=(x+16)^2\) | Subtract 96 on both sides of the equation |
| \(8y=(x+16)^2-96\) | Divide by 8 on both sides |
| \(y=\frac{1}{8}(x+16)^2-12\) | |
Our quadratic equation is finally in vertex form. Now, we can find the focus by using the formula I mentioned above, \((h,k+\frac{1}{4a})\). Let's plug those values into this quadratic equation. First, identify what h, k, and a are.
h=-16
k=-12
a=1/8
Let's plug these values in:
| \((-16,-12+\frac{1}{4(\frac{1}{8})})\) | Do 4*1/8 first. |
| \((-16,-12+\frac{1}{\frac{1}{2}})\) | I'll use a fraction rule that states that \(\frac{a}{\frac{b}{c}}=\frac{ac}{b}\) |
| \((-16,-12+2)\) | Continue simplifying. |
| \((-16,-10)\) | |
Now, you are finally done. The point of the focus is \((-16,-10)\).
+2446 I'm posting this solution as an alternate method to finding the vertex of a quadratic equation. Either method, presented by hecticlar or me, are acceptable methods.
Finding the vertex of a parabola is actually simple, or, at least, I think so. First, find the line of symmetry by using this formula:
\(\frac{-b}{2a}\)
However, we must identify what a and b stand for. Let's look at our quadratic function and analyze it. Here it is:
\(y=-\frac{1}{4}x^2+4x-19\)
As a review, a is the coefficient of the quadratic term, and b is the coefficient of the linear term. Let's plug it into the formula above, \(\frac{-b}{2a}\).
| \(\frac{-4}{2(-\frac{1}{4})}\) | Let's solve this expression by simplifying the denominator first. |
| \(\frac{-4}{-\frac{1}{2}}\) | I'll use a fraction rule that states that \(\frac{a}{\frac{b}{c}}=\frac{ac}{b}\). Let's apply it! |
| \(\frac{-4*2}{-1}\) | Simplify this |
| \(8\) | |
This is not our answer. The vertex is the point where either the minimum or maximum is on a parabola. The point we have found is the line that divides the parabola in half. To find the corresponding y-coordinate, substitute \(8\) into the function. Let's do that:
| \(y=-\frac{1}{4}x^2+4x-19\) | Anywhere you see an x substitute in an 8 in its place. |
| \(y=-\frac{1}{4}(8)^2+4(8)-19\) | According to order of operations do the exponent operations first |
| \(y=-\frac{1}{4}*64+4*8-19\) | Continue to simplify until you get the y-coordinate. |
| \(y=-16+4*8-19\) | Do 4*8 next, of course. |
| \(y=-16+32-19\) | |
| \(y=-3\) | |
After all of this, we have determined that the vertex is \((8,-3)\). This is your answer.
+2446 I am only showing this alternate solution because I think it is a nicer way of solving it than what the guest provided. His/her method, though, is sound and is perfectly fine.
| \([(1.08)(1+x)]^{\frac{1}{2}}-1=10\) | Add 1 on both sides |
| \([(1.08)(1+x)]^{\frac{1}{2}}=11\) | Simplify \((1.08)(1+x)\) by distributing the 1.08 to the 1 and the x. |
| \((1.08+1.08x)^{\frac{1}{2}}=11\) | Taking something to the power of 1/2 is the same as square root, so let's write it like that |
| \(\sqrt{1.08+1.08x}=11\) | Square both sides to get rid of the radical |
| \(1.08+1.08x=121\) | Subtract 1.08 on both sides |
| \(1.08x=119.92\) | Divide by 1.08 on both sides to isolate x. |
| \(x=\frac{119.92}{1.08}\) | Multiply the numerator and denominator by 100 to make them whole numbers |
| \(x=\frac{11992}{108}\div\frac{4}{4}\) | Divide the numerator and denominator by its GCF, 4, to put the improper fraction in simplest terms. |
| \(x=\frac{2998}{27}=111.\overline{037037}\) | You are done! This is your final answer! |
