Q 2. - Calculus

Question

Examples: Construct examples of the thing described in the following. Try to find examples that are different than any in the reading. 

(a) Two triangles that are similar to the right triangle with legs of length $1$ and hypotenuse $\sqrt{2}$.

(b) Two triangles with a hypotenuse of length $5$.

(c) Two cylinders with a volume of $100$ cubic units. 

Step-by-Step Solution

Answer

1. This results in a square with sides that are 1 and diagonals that are 1.
2. A rectangle with four sides, three widths, and five diagonals.
3. A cylinder having a $200$cubic unit capacity, a radius of 1, and a height of $63.6$.

Part (a) Step 1: Given information

Two triangles with hypotenuses of $\sqrt{2}$ and legs of length one, comparable to the right triangle.

Part (a) Step 2: Explanation

A square with a side length of one and a diagonal length of one is the result.

Part (b) Step 1: Given information

Two triangles with a $5$-length hypotenuse.

Part (b) Step 2: Explanation

Each triangle will have sides that are $3$ and $4$ and a hypotenuse that is $5$ in length.

This results in a rectangle with dimensions of length $4$, width $3$, and diagonal $5$.

Part (C) Step 1: Given information

Two $100$cubic units-per-cubic-inch cylinders.

Part (c) Step 2: Explanation

A cylinder with a capacity of $100$cubic units can produce a cylinder with a radius of 1 and a height of $31.8$ according to the math below.

$$\begin{gathered} V=\pi r^{2}h \\ 100=\left(\frac{22}{7}\right)r^{2}h \\ {r}^{2}h=\frac{700}{22} \\ {r}^{2}h=31.8 \\ 1.(31.8)=31.8 \end{gathered}$$

Thus, a large cylinder of radius one and height $63.6$ with a volume of 200 cubic units can be created from two cylinders of radius 1 and height 31.8.

Hence,

$$\begin{gathered} V=\pi r^{2}h \\ V=\frac{22}{7}·1^2·(63.6) \\ V=\frac{1399.2}{7} \\ V=199.89 \\ V=200 \end{gathered}$$