# Tutor profile: Jonathan H.

## Questions

### Subject: Geometry

Take a square of any side length. On each side, mark the midpoint, halfway between both ends. Now connect those four midpoints, forming a smaller square within the first. Find the ratio of the area of the bigger square to the smaller square.

It would be very possible to calculate the ratio of side lengths and use area formulas to deduce the ratio of the areas of the two squares. A more intuitive way to look at it is to split up the diagram in to smaller triangles. Notice that the space between the smaller and bigger squares consists of four triangles of the same size, one in each corner. If you look closer at the smaller square in the middle drawing diagonals from each corner to the opposite corners, you can see that it too can be split up in to four triangles of equal size. Therefore, the large square is made up of eight of these triangles, and the smaller one is composed of four. 8/4 is 2, so the ratio of the areas is 2:1.

### Subject: Chemistry

Which of the following will have the smallest atomic radius: F-, Ne, or Na+?

Na+ will have the smallest atomic radius. Each of the three species described possesses the exact same electron configuration, 1s2 2s2 2p6. The sodium ion, however, has the nucleus with the largest positive charge because of its higher atomic number. The largest positive charge will thus have the greatest pull on electrons, bringing them closer to the nucleus, giving Na+ the smallest radius.

### Subject: Calculus

A car's velocity in meters per second at a certain time t can be expressed as a function of t: 2t^2+5t. If the car starts from rest at time zero, how far does the car travel in the first four seconds. (t=0-4).

If we graph the velocity function with respect to time t, the area under the curve above the t axis between t=0 and t=4 will represent the total displacement of the car during that time interval. This can be calculated by taking the integral from 0 to 4 of the function 2t^2+5t. Taking this integral gives 82.67 meters. Thus the car travels a net distance of 82.67 meters in the first four seconds.

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