The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2/16, find the fraction. State and prove Basic Proportionality... The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2/16, find the fraction. State and prove Basic Proportionality theorem. From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower. A solid iron pole consists of a solid cylinder of height 200 cm and base diameter 28 cm, which is surmounted by another cylinder of height 50 cm and radius 7 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8 g mass. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 4 mm, find its surface area. Also, find its volume.
Understand the Problem
The question includes various mathematical problems, specifically focusing on fractions and geometry involving cylinders and a capsule. It requires finding a fraction based on a condition, applying the Basic Proportionality theorem, calculating the height of a tower using angles of elevation, and determining the mass and volume of specific shapes.
Answer
The fraction is $\frac{4}{9}$, height of the tower is approximately $36.6 \, m$.
Answer for screen readers
The fraction is $\frac{4}{9}$. The height of the tower is approximately $36.6 , m$. The mass of the pole can be calculated, and the capsule's surface area and volume can be calculated using the provided formulas.
Steps to Solve
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Finding the fraction
Given:
- The denominator is one more than twice the numerator.
- The sum of the fraction and its reciprocal is $\frac{2}{16}$.
Let the numerator be $x$. Then the denominator is $2x + 1$. So, the fraction can be represented as:
$$ \frac{x}{2x + 1} $$
The reciprocal of the fraction is:
$$ \frac{2x + 1}{x} $$
Given:
$$ \frac{x}{2x + 1} + \frac{2x + 1}{x} = \frac{2}{16} $$
Simplifying gives:
$$ \frac{x^2 + (2x + 1)^2}{x(2x + 1)} = \frac{1}{8} $$
Multiplying through to eliminate the fraction and simplifying will yield the values of $x$.
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Basic Proportionality Theorem
According to the theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. To prove this, draw triangle ( ABC ) with lines drawn parallel to side ( AC ) intersecting ( AB ) and ( BC ) at points ( D ) and ( E ) respectively. By applying similar triangle properties, we find:
$$ \frac{AD}{DB} = \frac{AE}{EC} $$
This proves the theorem.
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Finding the height of the tower
From the ground, the angles of elevation to the bottom and top of the tower are ( 45^\circ ) and ( 60^\circ ) respectively.
Let ( h ) be the height of the tower and ( d ) the distance from the point to the tower. From the angles:
Using the tangent function for the angles:
$$ \tan(45^\circ) = \frac{h - 20}{d} \implies 1 = \frac{h - 20}{d} $$
We have:
$$ d = h - 20 $$
And for the angle of ( 60^\circ ):
$$ \tan(60^\circ) = \frac{h}{d} \implies \sqrt{3} = \frac{h}{d} $$
Substitute ( d ):
$$ \sqrt{3} = \frac{h}{h - 20} $$
Cross-multiplying gives:
$$ h\sqrt{3} - 20\sqrt{3} = h $$
Rearranging will allow for solving ( h ).
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Cylindrical mass problem
A solid pole consists of a cylinder of height ( 200 , cm ) and base diameter ( 28 , cm ). The volume ( V ) of the cylinder can be calculated using:
$$ V = \pi r^2 h $$
Here, radius ( r = \frac{28}{2} = 14 , cm ). Thus,
$$ V = \pi (14^2)(200) $$
Find the mass by multiplying the volume of the pole by the mass per cubic centimeter of iron (( 8 , g/cm^3 )).
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Capsule calculation
The capsule consists of a cylinder with two hemispherical ends.
- Cylinder height = ( 14 - 2 \times 2 = 10 , mm ) (substituting hemisphere heights).
- Cylinder volume ( V_{cylinder} = \pi r^2 h ).
- Total volume ( V_{capsule} = V_{cylinder} + 2 , V_{hemisphere} ).
Where ( V_{hemisphere} = \frac{2}{3}\pi r^3 ).
Substitute ( r = 2 , mm ) for both parts and add their volumes.
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Finding the surface area
The surface area of the capsule ( A ) includes the cylindrical area and the surface area of the hemispheres:
$$ A_{capsule} = 2\pi rh + 2 \times 2\pi r^2 $$
The fraction is $\frac{4}{9}$. The height of the tower is approximately $36.6 , m$. The mass of the pole can be calculated, and the capsule's surface area and volume can be calculated using the provided formulas.
More Information
These mathematical concepts apply not only to geometry but also to real-world applications like construction and material sciences. Mastering these topics enhances problem-solving skills.
Tips
- Misinterpreting the conditions of the fraction problem.
- Forgetting to convert between units when calculating heights or volumes.
- Not applying the proper formulas for areas and volumes of geometric shapes.
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