ETERNITY

Problem 003

Looking back at our previous solution, we find that the time it takes for the ball to hit the ground is 3.01 seconds. Since we know Range is the initial x velocity times time (I'm assuming air resistance can be neglected), we can plug our values in to the equation, getting $v_{xi}=\dfrac{R}{t}=\dfrac{33m}{3.01s}=10.96m/s$.

Hello notrandel! Haven't seen a physics problem for a long time. I might be a little rusty, so be sure to check my work before submitting!

Problem 002 (Where is 001? Or did you already finish that?)

First, let's draw a Free Body Diagram. Always do this at the beginning of a problem. (I'm not doing it here for the sake of time, but you can draw one on your paper.) The ball is thrown horizontally, so there is no initial vertical motion. So, we know that $v_{yi}=0$. We also have our vertical displacement, -44.3, and also the acceleration of gravity, -9.8 (not putting units for the sake of neatness). We can plug this into our 2nd Kinematic Equation, or $\Delta x=v_0 t+\dfrac{1}{2}at^2$, which gives, using the quadratic formula to solve for $t$, 3.01s. We can eliminate the negative solution since it doesn't make sence in the context of the problem.

Hello Guest.

Yes, I believe there will be, as there has been every single year on AoPS (Art of Problem Solving.)

Hope this helped! 

Hello GamemasterX40!  This is my attempt of solving this problem:

"Ms. Thayer bought 6 items for her physics class at a mean cost of 8.40 dollars per item. When she added a dial caliper for 46.50 dollars to her purchases, the new mean for all 7 items became about... Explain your thinking."

SOLUTION (and explanation):

We know that the mean is the average of a set of numbers. With this information in mind, we know that the total cost of the 6 items Ms Thayer bought for her class was: \[6 \times 8.40 = 50.40 \text{ dollars.}\] Now, Ms. Thayer bought a 7th item, the dial caliper, for 46.50 dollars. Adding this to our total, we get $50.40+46.50 = 96.90 \text{ dollars.}$ Again, we knot that the mean is the average of a set of numbers. That means the mean, or in the context of the problem, the average price, of the 7 objects is: \[\frac{96.90}{7}=13.8428571429... \approx 13.84 \text{ dollars.}\]

I hope this helped! 

No problem! Always happy to help :D

I usually prefer using the vertex method, where all you have to do is rearrange the equation.

But this is nice! This person will now know two methods to do these problems! 

All right... Here is my attempt:

We need to rewrite this equation in vertex form. First, we isolate the $x$ to give $x = y^2 - 8y + 13$. Completing the square in $y$, we get \[x = (y - 4)^2 - 3.\]

The graph of this equation is a rightward-opening parabola, so the vertex is the leftmost point, not the top or bottom point as usual.


Since $(y-4)^2$ is nonnegative, the value of $x$ is smallest when $y=4$, which makes $x=-3$. Therefore, the vertex of the parabola is $\boxed{(-3,4)}$.

Hope this helped! 

Hello Guest.

For a), It is could be congruent, since AAA is not a congruence, but a similarity.

For b), it is not congruent, since the triangle is not an iscoceles triangle and we have te values of two not-corresponding sides.

I hope this helped! 

Hello Guest.

"Andrew is building a scale model of an elephant. He determines that a real elephant is 10 feet tall and 14 feet long. If Andrew wants his model to be 8 inches tall, how many inches long should his model be?"

SOLUTION:

We can use ratios to help us solve this problem here.

We already know the ratio for height, which is 10:8. Now, we just have to set the width of his drawing to $x$ and write out an equivalent ratio for that: \[\frac{10}{8}=\frac{14}{x}.\] Simplifying and cross multiplying, we get $\boxed{x=\frac{56}{5}}$

I hope this helped!