jfan17

Sorry the answer should be :

\((6\sqrt2 * 6\sqrt2)/2 = 36\)

since I mixed up the leg length for the hypotenuse length. The length of each leg is \(6\sqrt2\), not 6

Because \(\angle Q = \angle R\), they must be acute angles(because if they were the right angle, then two right angles would already be 180 degrees, making the triangle degenerate). Name the angle measurement of one of these 2 angles x. We can write the expression:

90 + x + x = 180(the internal angle sum of a triangle).

90 + 2x = 180

2x = 180

x = 45

That means that we have a 45-45-90 triangle at hand, which happens to be a special right triangle. The ratio of a 45-45-90 triangle is:

\(1,1,\sqrt2\) where the legs are 1, and the hypotenuse is \(\sqrt2\)(Remember this! It will make math with certain right triangles a lot easier!)

We know that angle P must be the right angle, because angle Q and R are both 45 degrees.

We then have that the hypotenuse PR is equal to \(6\sqrt2\)

Because the ratios of a 45-45-90 triangle are 1,1,\(\sqrt2\), we know that right triangle PQR is scaled by a factor of 6 compared to a 45-45-90 of leg length 1.

That means the leg lengths of our triangle are:

6 * 1 = 6

Now that we have the length of the legs as 6, we just need to find the area.

The area is equivalent to the (height * base) /2. Since we have both legs, we can just set one of them as the height, and the other the base(since they share a 90 degree angle). We then have our area as:

(6*6)/2 = 18

Name your desired number as x. If 20% of x is equal to 81, we can write:

\(0.20x = 81\)(do you see how I got here?)

The way you reach this expression is because 20% is equivalent to 20 * 1/100, which is just 20/100, which is equivalent to 0.20

Then, we realize that if we multiply this by 5, we just get:

\(x= 405\)

That gives us our desired answer.

The x intercept is just the point on the line at which the line intersects the x axis. In other words, the y value is equal to 0(can you see why?). When the Y value is equal to 0, that means that the line has to intersect the x axis because the x axis is at a y value of 0. Similarly for the y intercept, it's defined as the point that intersects the y axis on a given line. That means that the x value is 0, because the y axis is located at x = 0. That means we just have to look for the points that intersect the x and y axes respectively. We see that there is one point on this graph that intersects both, which is directly at the origin. This gives us an x intercept of 0, and a y intercept of 0, because the coordinates of the origin are (0,0)

no problem

Notice that the legs of the right triangle are in a ratio of 3 : 4 . If you remember anything about right triangles, you'll know the "3 - 4 -5" right triangle, which is a right triangle of side lengths 3,4, and 5(is there another way to phrase this ? lol). We can see that this right triangle of legs 20 and 15 inches is a "scaled up" version of the 3 4 5 right triangle, and is in fact, similar to the 3-4-5 triangle(by a ratio of 5). As such, we know that the hypotenuse of this triangle must be 5 * 5 = 25 inches long. Alternatively, we can just use pythagorean theorem to find the length of the hypotenuse of our triangle. Using pythagorean theorem, we have where h is the hypotenuse:

\(h^2 = 15^2 + 20^2 = 225 + 400 = 625\)

\(h = \sqrt{625} = 25\)

That means that the perimeter of said right triangle is:

\(15+20+25 = 60\)

Because the square and right triangle both have equal perimeters, we can name the side length of the square x

We can then write the equation:

\(4x = 60\), because the square has 4 sides of length x, which gives us its perimeter of x + x + x + x = 4x

Dividing by 4 on both sides, we get: 

\(x = 15\)

The area of a square is equal to 

x² given a side length x, so we then have:

\(15^2 = 225\) in² as the area of our square

Sorry I don't have an accompanying diagram(I'm not too tech savvy with programs that others are more familiar with like desmos, etc.), but I'll try my best to explain my solution.

We draw a large circle, and within it, another smaller circle that shares the same center(that's the definition of concentric more or less). Then, we draw a chord(a line segment with both endpoints on the circle) across the large circle that touches the smaller circle at exactly one spot. After we've done this, we can then draw a line from the center of the smaller circle to the point of tangency, creating a 90 degree angle. By definition, if a line is tangent to a circle, it forms a 90 degree angle with the radius drawn to that point.

Here's a proof of the following done by Euclid:

https://www.quora.com/Why-does-the-tangent-of-a-circle-make-an-angle-of-90%C2%B0-with-the-radius-drawn-from-the-point-of-contact

Next, we notice that we can draw in two radii of the larger circle, both of which intersect an endpoint of the chord. We then form two right triangles that have the same hypotenuse and same leg(which is the radius of the smaller circle). By pythagorean theorem, it stands to reason that if the hypotenuse and leg of a right triangle are congruent to another hypotenuse leg pair, then both right triangles are congruent. For this reason, we know the legs of the two right triangles are equal to each other, meaning that the point of tangency is indeed the midpoint of the chord drawn.

Hey guest! In this problem, we're going to get involved with some systems of equations. First, we have that Allysa bought two different types of chickens: Americana and Delaware, Let's name the quantity that she purchased of each of these chickens \(a \) and \(d\) respectively. We have that she purchased 12 chickens total, so we can write the equation:

\(a + d = 12\)

The problem then tells us that Americana chickens cost $3.75 each, and Delaware Chickens cost $2.50 each. It also gives us that the total amount of money that Allysa spent was 35$. We can then write:

\(3.75a + 2.50d = 35\)

The reasoning for this equation is because each Americana is 3.75, so 3.75 * the number of Americanas purchased represents the total amount of money she spent on Americanas, and likewise for Delaware chickens. 

Now, let's multiply our first equation by 2.50, and then subtract it from our second equation to get the value of a.

We get:

\(2.50a + 2.50d = 12 * 2.5 = 30\)

Subtract this from our second equation, and we get:

\(1.25a = 5\)

Dividing by 1.25 on both sides, we get:

\(a = \frac5{1.25} = 4\)

Now that we have the number of Americans purchased, a, we can just plug this back into our first equation \(a+d = 12\)

\(4 + d = 12\)

Subtracting 4 on both sides, we get:

\(d = 8 \)

Hey guest! I'm assuming you meant \(37 \frac14 * 32\) for this problem.

To get the answer for this one, I'd imagine there are a variety of solutions, but rather than converting \(37 \frac14\) into a fraction and then multiplying that by 32(that would be quite cumbersome in my opinion), what I would do is split the mixed number into two parts. A whole number, and a fractional part(The reasoning behind this is not random!).

We can write the following equivalence:

\(37 \frac14 * 32 = (37 + \frac14) * 32\)

We then distribute this into:

\((37 + \frac14) * 32 = (37*32) + (\frac14 * 32)\)

Do you see now why this method makes this particular problem easy? If you haven't noticed, the right side, which is \((\frac14 * 32)\), reduces very nicely, as 4 is a factor of 32(32 is 4 * 8). This leaves us with:

\((37 * 32) + (8) \)

Plugging this in a calculator, this converts to:

\((1184) + (8) = 1192\) as our final answer

Hey guest! Let's start with some basic background knowledge of types of interest:

there are two main types, simple and compound. With simple interest, given a quantity:

\(x\), and in interest rate of \(r\%\) per year, we have that every year, the interest amount is:

^{\(x * r\%\)}. In other words, the amount paid every year(if we're talking about an interest on payment) is the exact same constant value.

Compound interest however, given the same variables, is a bit different.

Compound interest is "compounded"(hence the name) and added on to the previous year(or whatever metric of time used). The formula for compound interest is then typically defined:

\(A = P(1+\frac{r}{n})^{nt}\)

Where A = Amount paid

P = The principal, or the initial amount first paid 

R = Interest rate as a decimal

N = The number of times the interest is compounded per unit of time "t"

T = Time(in this example, years).

Here's a great website for reference on compound interest:

https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php

Getting to the problem itself:

A)

The interest that Sue gets with an investment of 2300 pounds and an interest rate of 2.4% a year over 3 years is:

\(A = 2300(1+\frac{0.024}1)^{3*1} = 2300(1.024)^3 = 2300(1.073741824) = 2469.6061952\)

For Bill, he invests 1800 pounds at an interest of 3.4% per year.

Our equation is then:

\(A = 1800(1+\frac{0.034}1)^{3*1} = 1800(1.034)^3 = 1800(1.105507304) = 1989.9131472\)

Of course, both of these answers are using calculators.

B)

To solve this, we figure out the compound interest for Bill after 2 years, and then multiply that amount by (1 + 0.04), which represents the 4 percent compound interest rate added on to that.

Sue:

We have already that Sue has an interest payment of 2469.6061952 after 3 years

Now, moving on to Bill:

By our previous formula:

\(\)

\(A = 1800(1.034)^2 = 1800(1.069156) = 1924.4808\)

We then multiply this value by 1.04 to represent a 4 percent interest rate(4% = 0.04, so 1+ 0.04 represents a 4 percent compounded interest).

\(1924.4808 * 1.04 = 2001.460032\)

Clearly, this is less than Sue's amount after 3 years, so the answer is then Sue