Example Question - calculate height
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Calculation of Cylinder Dimensions Using Given Volume
Let the height of the cylinder be \( h \) and the radius be \( r \). The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \). The circumference \( C \) of the base of the cylinder is given by \( C = 2\pi r \). We are given: \( V = 12,936 \) cm\(^3\) \( \pi = \frac{22}{7} \) Firstly, we express \( r \) in terms of \( C \): <p>\( C = 2\pi r \Rightarrow r = \frac{C}{2\pi} \)</p> Substitute \( r \) into the volume formula and solve for \( h \): <p>\( V = \pi \left(\frac{C}{2\pi}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{2 \cdot \frac{22}{7}}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{7C}{44}\right)^2 h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{44/7}\right)^{\!2} h \)</p> <p>\( 12,936 = \frac{22}{7} \left(\frac{C}{4}\right)^{\!2} h \)</p> <p>\( 12,936 = \frac{22}{7} \cdot \frac{C^2}{16} \cdot h \)</p> <p>\( 12,936 \cdot \frac{16}{22} = C^2 h \)</p> <p>\( 12,936 \cdot \frac{16}{22} = C^2 h \)</p> <p>\( 12,936 \cdot \frac{7 \cdot 16}{22} = 7C^2 h \)</p> <p>\( 12,936 \cdot \frac{112}{22} = 7C^2 h \)</p> Now, let's solve for \( h \) in terms of \( C \): <p>\( h = \frac{12,936 \cdot \frac{112}{22}}{7C^2} \)</p> <p>\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)</p> Finally, we substitute the given value of the volume to find \( h \): <p>\( h = \frac{12,936 \cdot 112}{22 \cdot 7C^2} \)</p>
Calculating the Height of Statue of Liberty
In the image, you're provided with a right triangle where the length of the hypotenuse is 340 feet and the length of the base is 150 feet. You need to find the height of the Statue of Liberty, which corresponds to the vertical side of the triangle. To solve for the height, you can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The theorem is expressed as: a² + b² = c² Here, we are looking for the length of a (the height), while we know the lengths of b (the base, 150 feet) and c (the hypotenuse, 340 feet). We can rearrange the theorem to solve for a: a² = c² - b² Plug in the given lengths: a² = (340)² - (150)² a² = 115,600 - 22,500 a² = 93,100 Now, take the square root of both sides to find the value of a: a = √93,100 a ≈ 305 feet So, the height of the Statue of Liberty is approximately 305 feet to the nearest foot.
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