Lesson 2: Pythagorean Theorem Problem Solving - Multiple Calculations in Same Problem

   Constructing Knowledge

There are situations where multiple calculations are needed to solve a problem. Breaking the problem into smaller parts, and solving each part separately, will enable these types of problems to be solved.

For instance, being able to visualize the different right triangles in a rectangular prism is important for solving different dimensions of the prism.



   Multimedia

A video describing a multiple step Pythagorean Theorem problem is provided.



EXAMPLE

Hayley ordered a ringette stick and some other equipment online. Her purchase is being shipped in a rectangular box. If the stick is 180 cm long, and the shipping box has a length of 175 cm and a width of 20 cm, how tall must the box be in order for the stick to fit diagonally in the box?



Solution


This problem can be solved in steps.

This diagram contains a triangle representing the length of the stick and the height of the box which is the unknown. However, the diagonal along the bottom of the box is also unknown. As such, a second triangle must be solved first.

This diagram contains a triangle that can be used to solve for the diagonal along the bottom of the box because the length and width of the box were given in the problem. This will be the first calculation.

Solve for the bottom diagonal

Step 1: Identify the given and the required values.

  • The length of the box is 175 cm
  • The width is 20 cm
  • The diagonal along the bottome of the box is unknown.

Step 2: Draw and label the diagram.

Extract the portion of the picture that will enable the problem to be solved.



Step 3 and 4: Write the formula then substitute values and solve.

\(\begin{align} l^2+w^2&=d^2 \\ \\ 175^2+20^2&=d^2 \\ \\ 30\,625+400&=d^2 \\ \\ 31\,025&=d^2 \\ \\ \sqrt{31\,025}&=\sqrt{d^2} \\ \\ 176.14&=d \\ \end{align}\)

The diagonal along the bottom of the box is approximately 176.14 cm.

Step 5: Review the answer

The diagonal is the hypotenuse and the longest side so the answer is reasonable.

Solve for the height of the box

Step 1: Identify the given and the required values.

  • The length of the diagonal along the bottom of the box is 176.14 cm. This will be one of the legs of the new triangle.
  • The ringette stick is 180 cm long and it is the hypotenuse.
  • The height of the box is the unknown.

Step 2: Draw and label the diagram.

Extract the portion of the picture that will enable the problem to be solved.



Step 3 and 4: Write the formula, then substitute values and solve.

\(\begin{align} d^2+h^2&=s^2 \\ \\ 176.14^2+h^2&=180^2 \\ \\ 31\,025.3+h^2&=32\,400 \\ \\ \cancel{31\,025.3-{\color{red}{31\,025.3}}}+h^2&=32\,400-{\color{red}{31\,025.3}} \\ \\ h^2&=1374.7 \\ \\ \sqrt{h^2}&=\sqrt{1374.7} \\ \\ h&=37.1 \\ \end{align}\)

The box will need to be at least 37.1 cm high in order for the stick to fit.

Step 5: Review the answer.

The height is less than the diagonal (hypotenuse), so the answer is reasonable.


Now, it is your turn! Complete the questions in your Chapter 3, Lesson 2 Practice Makes Perfect that refer to Multiple Calculations in Same Problem.



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