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Sample Question 11:
Step 1: Understand that in a cone, the radius and height vary inversely with increasing height. This means the volume of the cone changes as the inverse of the cube of the percentage increase in height. This relationship can be represented as: \(V_I * \text{Height}^3 = V_N\).
Step 2: Given the condition that the top \(\frac{1}{8}\) of the volume of the mountain is above water, you can equate this to the cubed value of Height to get: \(\frac{1}{8} = \text{Height}^3\).
Step 3: Solving for Height gives you \(\text{Height} = \frac{1}{2}\).
Step 4: To find the actual height in feet that is above water, multiply the total height of the mountain (8000 feet) by the value you solved for in step 3, giving you: \(8000 * \frac{1}{2} = 4000\) feet.
Step 5: So, the depth of the ocean at the base of the mountain (or how much of the mountain is submerged) is 4000 feet, which corresponds to answer choice \(\textbf{(A)}\ 4000\).
A cone-shaped mountain has its base on the ocean floor and has a height of 8000 feet. The top \(\frac{1}{8}\) of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
\(\textbf{(A)}\ 4000 \qquad \textbf{(B)}\ 2000(4-\sqrt{2}) \qquad \textbf{(C)}\ 6000 \qquad \textbf{(D)}\ 6400 \qquad \textbf{(E)}\ 7000\)
Answer Keys
Question 11: ASolutions
Question 11Step 1: Understand that in a cone, the radius and height vary inversely with increasing height. This means the volume of the cone changes as the inverse of the cube of the percentage increase in height. This relationship can be represented as: \(V_I * \text{Height}^3 = V_N\).
Step 2: Given the condition that the top \(\frac{1}{8}\) of the volume of the mountain is above water, you can equate this to the cubed value of Height to get: \(\frac{1}{8} = \text{Height}^3\).
Step 3: Solving for Height gives you \(\text{Height} = \frac{1}{2}\).
Step 4: To find the actual height in feet that is above water, multiply the total height of the mountain (8000 feet) by the value you solved for in step 3, giving you: \(8000 * \frac{1}{2} = 4000\) feet.
Step 5: So, the depth of the ocean at the base of the mountain (or how much of the mountain is submerged) is 4000 feet, which corresponds to answer choice \(\textbf{(A)}\ 4000\).