If a Viking ship sails due North from a latitude of 63°푁to 71°푁, how far has the ship traveled? Assume the radius of the Earth is 6,378,000meters. Note: the ship has sailed through an angle of 8°, as measured from the center of the Earth. Round to the nearest whole number.
This questions may seem difficult, but we can actually use proportions to solve for the distance that the ship has traveled. In this problem, using the unit circle, we know that we are solving for an arc length. With knowledge of the unit circle, we can use the proportion: (arc length/circumference)=(degree/360). We know the circumference of a circle is 2(pi)(r), and we have the radius. it is 6,378,000 meters. If we plug this into our circumference equation we get 2(pi)(6,378,000). We also have our degree, it the difference between the two latitudes in the question. 71 degrees-63 degrees=8 degrees. Now, we can plug all of this information into our first equation and solve for the arc length. (arc length/2(pi)(6,378,000))=(8/360). You can rearrange this as arc length= ((8/360)2pi)x6,378,000. This gets the thing you're solving on its own. Also note that the stuff inside the first parentheses gives you a measure of the angle in radians, and this is similar to the equation (radians)(radius)=arc length. You should get arc length= 890537 meters (remember your units!)
If sinA=5/6 and cosA=3/6, then what does tanA=?
For this question, you need to remember a fun acronym called SOHCAHTOA. This is used for triangles, and it stands for Sine=opposite/hypotenuse, Cosine=adjacent/hypotenuse, and Tangent=opposite/adjacent. if we know sine and cosine, we can figure out tangent by solving for the o, a, and h. If sin=o/a, then our o is 5 and our h is 6 based off of the information given to us in the question. The question also gives us a, because we know that cosine=a/h, so based off of the question, our a=3. Now, we have all of the variables, so we can solve for tanA. Tangent=o/a, so we can plug in our solved values we found in the question using our knowledge of triangles and the handy-dandy acronym. tanA=5/3.
If x and y both satisfy the equations 9x+2y=8 and 3x+4y=2, what does y=?
You can solve for y using a system of equations with elimination. We want to solve for y, so in order to do so, we need to get y by itself. And to do that, you must cancel out the x's. Elimination allows you to do so. Line the two equations up like so: 9x+2y=8 3x+4y=2 We need to multiply the second equation by a number, so that when we subtract it from the first equation the x's cancel. In this case, multiplying the second equation by 3 would help us. 9x+2y=8 3(3x+4y)=2(3) Remember to multiply both sides of the equation so you aren't changing the balance. After distributing the 3, you can set it up like a subtraction problem. 9x+2y=8 -(9x+12y)=6 Here, you can see that the 9x cancels with the other 9x on the bottom. Then, you can simplify further by subtracting the y's and the integer on the right. 2y-12y=-10y, and 8-6=2. Now you have one equation and one variable, -10y=2. You still need to get y by itself, so you can divide both sides by -10, and get y=-2/10, or simplified further, y=-1/5.