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Kabir A.
Prospective Graduate Student in Mechanical Engineering; 160+ on both Math and Verbal sections of the GRE
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SAT
TutorMe
Question:

The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium and one cup of juice contains 261 mg of calcium. Which of the following inequalities represents the possible number of cups of milk $$m$$ and cups of juice $$j$$ a 20-year-old could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?

Kabir A.
Answer:

If you have $$m$$ cups of milk, then the total amount of calcium resulting from said milk is equivalent to $$299m$$ mg of calcium (and $$261j$$ mg of calcium from one glass of juice). The total amount of calcium from $$m$$ cups of milk and $$j$$ cups of juice is $$299m+261j$$; the resulting inequality is $$299m+261j\geq1000$$.

GRE
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Question:

The total amount of Judy’s water bill for the last quarter of the year was $40.50. The bill consisted of a fixed charge of$13.50 plus a charge of $0.0075 per gallon for the water used in the quarter. For how many gallons of water was Judy charged for the quarter? Kabir A. Answer:$40.50 - $13.50 removes the fixed charge from the total bill and leaves you with$27. To find the total number of gallons used, divide 27 by .0075 to come up with 3600 gallons of water used.

Mechanical Engineering
TutorMe
Question:

A solid aluminum cylinder (with a radius of 10 mm and length of 10 cm) is rigidly fixed to the side of a desk. If I hang a 10 kg weight off of the end of this cylinder, will it yield? If not, how much deflection will there be? (For reference, take the yield strength of aluminum as 100 MPa and the Young's modulus as 70 GPa).

Kabir A.
Answer:

To answer the first part of this question, one must calculate the maximum stress at any point on the beam. We can use the equation $$\sigma_{max} = y*M/I$$ (where $$\sigma_{max}$$ is maximum stress, $$M$$ is bending moment, $$y$$ represents the distance from the beam's neutral axis (which, in this case, is just the radius of the beam), and $$I$$ represents the cross-sectional inertia of the beam) to figure this out. The largest moment on the beam will be $$10 kg * 9.81 m/s^2 * 10 cm = 9.81 N*m$$ . The inertia of the beam is equal to $$\pi r^4/2$$, which in this case will be $$1.57*10^{-8}$$ $$m^4$$. Knowing this, we can solve for $$\sigma_{max}$$ to be 6.248 MPa. There are many different possible yield strengths for aluminum depending on the alloy, but taking into account the numbers from above, it is clear that this specimen will not yield. The simplest way to estimate the overall deflection is to take into account that this cylinder is cantilevered. Knowing this, we can write the maximum deflection $$\delta_{max} = FL^3/3EI$$, where $$F$$ is applied force, $$L$$ is the cylinder's length, and $$E$$ is the Young's modulus. We know all of these quantities already, and can calculate the maximum deflection at the tip of the beam to be $$2.975 * 10^{-5} m$$, or a little under 30 microns.

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