AzizHusain

E = 1/2mv^2

First, multiply both sides by 2 to get rid of the 1/2 on the right-hand side of the equation: 2E = mv^2

Next, divide by v^2 to obtain: 2E/v^2 = m.

A square foot is a unit of area and simply expressed as ft^2 for simplification. 1 sq ft or ft^2 would be a square that is 1 foot by 1 foot since 1*1 = 1 ft^2. Each "side" or length & width would be 1 feet.

We begin by squaring both sides to get rid of the square roots: sqrt(6x - 15) = sqrt(4x + 7) --> 6x - 15 = 4x + 7

Next, we get all the x terms on one side and all the regular numbers to the other side by adding 15 to both sides and subtracting 4x from both sides: By doing this, we get 2x = 22.

Now, we are left with something easy and we can see that 22/11 = x --> x = 11.

Do a quick check to see if x = 11 satisfies the condition:

sqrt(6*(11) - 15) = sqrt(4*(11) + 7) 

sqrt(66 - 15) = sqrt(44 + 7)

sqrt(51) = sqrt(51) --> Yes.

We can use proportions to solve this problem. 

Proportions are useful since they can be used to relate units and any missing pieces can be found.

In this case, we need to have this format: _ KG / _ N = _ KG / _ N

So far, we have this much information: 7 KG / 35 N = 8 KG / _ N

If we cross multiply, we obtain: (7 KG* _ N) = (8 KG*35 N) --> (7KG* _ N) = (280 KGN)

Dividing by 7 KG, we cancel the KG's out and obtain: 40 N.

Note: If 7 kg = 35 N, 8 kg = 40 N, 9 kg = 45, etc. We can see a pattern (For every 1 KG increase, we see an increase of 5 N). Furthermore, you could just multiply the mass by 5 to obtain the N, the formula would be z(KG) = z*5(N) where z = the mass. So for 90 KG, we would have 90*5 = 450 N.

Interesting nicknames and fun post. It is nice to relax at the end of the day or whenever and enjoy something new. I have never been on such an active forum before. Also, you posted some cool pictures which were all incredibly funny and awesome! 

Ohh, I see the simplification part now! Thanks, Alan! I overlooked splitting the numerator into two.

sqrt(-49) = 7i

sqrt(-64) = 8i

We have imaginary numbers since square roots of negative numbers cannot be real numbers. We use the "i" notation instead.

7i*8i = 56(i^2)*

*The multiplication of i by i = i*i = i^2 = -1

We get 56(-1) = -56.

Let's make a system of equations:

x = cost of sausage sandwiches and y = cost of milkshakes

(1) 3x + y = 8.25

(2) x + 2y = 5.25

We can solve for x or y in either equation and substitute it to the other. 

Let's solve for y in equation(2).

x + 2y = 5.25, subtract x from both sides: 5.25 - x = 2y

Next, solve for y by dividing both sides by 2 = 2.625 - (1/2)x = y. 

Now, substitute this - 2.625 - (1/2)x = y into equation(1) for y.

3x + y = 8.25 --> 3x + 2.625 - (1/2)x = 8.25. Now solve for x.

Collect like terms - 3x - (1/2)x = 2.5x and subtract 2.625 from both sides to obtain: 2.5x = 5.625. Divide both sides by 2.5 to obtain x = 2.25 or $2.25 for the cost of one sausage sandwich.

Substitute the value of x = 2.25 into either equation(1) or (2) to solve for y, the price of one milkshake.

3x + y = 8.25 --> 3(2.25) + y = 8.25 --> 6.75 + y = 8.25. Subtract 6.75 from both sides to obtain y = 1.50 or $1.50 for the cost of one milkshake.

Now, do a check to see if the values for x and y make sense.

3x + y = 8.25 --> 3(2.25) + 1.50 ?= 8.25 --> Yes.

x + 2y = 5.25 --> 2.25 + 2(1.50) ?=5.25 --> Yes.

Therefore, the cost of one sausage sandwich is $2.25 and the cost of one milkshake is $1.50.

Since we are not given a value for the variable, "a", I assume we have to somehow simplify this expression.

tan^2a-1/1+tana = tan^(2a-1) / 1 + tan(a) OR tan^(2a) - 1 / 1 + tan(a) OR tan^(2)a - 1 / 1 + tan(a) OR tan^(2)(a-1) / 1 + tan(a). The first two possiblities are weird because the "2a" is a variable and is included in the power of the tan. The last one is unlikely leaving the third form, which is one that makes the most sense.

tan^(2)a - 1 / 1 + tan(a): Simplifying this is tricky since the "tan^(2)a" and "tan(a)" are both very different. We would need a bit more information such as a value for the variable, "a", or any other extra hints provided.

For g(x)=4^x, notice that we can say y = g(x) = 4^x or y is a function of x. The same goes for y = f(x) = log(5)x. 

For g(x)=4^x: To find the x-intercept, we need to set y = 0 --> y = g(x) = 0 = 4^x or simply, 0 = 4^x. Now, if we plug in any values not includng 0 for x such as x = -5, -4, -3, -1, 1, 2, etc. we get y-values that are NOT 0. Therefore, the x-intercept does not exist. If we graph g(x)=4^x, we notice that as the x-values reach negative infinity, the y-values are closer and closer to the 0 value but never 0. Therefore, there is no x-intercept.

To find the y-intercept, we need to set x = 0 --> y= g(x) = 4^0 = 1. Any number to a power of 0 is equal to 1. Now, notice that if we tried plugging in x = 0 first when we were finding the x-intercept, we would have already found the y-intercept without noticing! Therefore, the y-intercept = 1 at the point (0,1).

f(x)=log(5)x: To find the x-intercept, we need to set y = 0 --> y = f(x) = 0 = log(5)x. Since this is a log function, it is best to observe the behavior of the graph since plugging in points may be a bit too tedious. It is best to use the properties of logs to find the intercept(s). 0 = log(5)x. The log, in this case, is in the log based 10 form (log 10). So by properties of logs, we can do the following: 10^0 = 10^(log10(5)x) --> 1 = 5x [10 to the power of log based 10 cancels out leaving the 5x] --> Now, the algebra is simple and we divide both sides by 5 to obtain x = 1/5 = 0.2. However, by looking at the graph, we notice that the graph is symmteric. Therefore, the x-intercepts would be at 0.2 and -0.2 at the points (0.2, 0) & (-0.2, 0).

To find the y-intercept, we need to set x = 0 --> y = f(x) = log(5)0 = log(0). Notice that we have y = log 0 and there is no x anywhere which means there is no y-intercept since x = 0 becomes log 0 which is a continous function when graphed, the y-value goes to negative infinity because of the nature of the log 0 graph. There is no y-intercept.