L1 Mathematical Reasoning
Question Bank
Candace is practicing her cricket throws. The height, $h$ (in metres), of a ball she throws vertically into the air is given by the equation:
$h = -5t^2 + 5t + 10$
where $t$ is the time in seconds after the ball is thrown.
What is the maximum height the ball reaches?
Rawiri is investigating a number puzzle. He wants to find two numbers that have a sum of 20 but also have the largest possible product.
To start his investigation, he lets the first number be $x$. Since the two numbers must add up to 20, the second number must be $20-x$. He then sets up a table to see how the product changes for different values of $x$.
Here is the start of his table:
| First Number, $x$ | Second Number, $20-x$ | Product, $P$ |
|---|---|---|
| 1 | 19 | 19 |
| 2 | 18 | 36 |
| 3 | 17 | 51 |
Carry on Rawiri's investigation by continuing the table. Use your table and a graph to find the two numbers that give the maximum possible product.
Sheryl wants to build a spacious new enclosure for her chickens to let them roam safely in her backyard. She has 20m of fence and plans to build the rectangular enclosure up against an existing shed, so she only needs to fence three sides.
Use tables and graphs to investigate how changing the dimensions of the enclosure affects its area. Use your table and graph to give recommendations on the dimensions she should use to give her chickens the largest possible space. Give at least 3 comments in your recommendation.
Sophie is investigating a number puzzle. She wants to find two numbers that have a difference of 12 but have the smallest possible sum of their squares.
To start her investigation, she lets the first number be $x$. Since the two numbers must have a difference of 12, the second number must be $x-12$. She then sets up a table to see how the sum of their squares ($S$) changes for different values of $x$.
Here is the start of her table:
| First Number ($x$) | Second Number ($x-12$) | Sum of Squares ($S$) |
|---|---|---|
| 1 | -11 | 122 |
| 2 | -10 | 104 |
| 3 | -9 | 90 |
Carry on Sophie's investigation by continuing the table. Use your table and a graph to find the two numbers and the smallest possible sum of their squares.
A roofing company is making gutters from a flat rectangular sheet of metal that is 40 cm wide. They will fold up the edges on each side by a height of $x$ cm to form an open-top gutter with a rectangular cross-section, as shown in the diagram below.
What value of $x$ will maximize the cross-sectional area, allowing the gutter to carry the most water? You should justify your answer with a graph.
A concert venue manager is trying to find the best ticket price to maximize revenue. Currently, they sell 500 tickets when the price is $65. After some research, they estimate that for each $5 increase in price, they will sell 20 fewer tickets.
To figure out the ideal price, the manager starts an investigation. They use $x$ to represent the number of $5 price increases and $y$ to represent the total revenue.
Here is the start of their table:
| Num of price increases, $x$ | Ticket Price ($) | Number of Tickets Sold | Total Revenue, $y$ |
|---|---|---|---|
| 1 | $70 | 480 | $33,600 |
| 2 | $75 | 460 | $34,500 |
Complete the concert manager's investigation by continuing the table. Based on your findings, state the ideal ticket price the manager should set. Support your recommendation with a graph showing the relationship between the number of price increases, $x$, and the total revenue, $y$, and include the equation that models this relationship.
A farmer is designing a new layout for his animals and wants to be as efficient as possible with his fencing. He decides to set up two identical, side-by-side rectangular pens against an existing hedge, as shown in the diagram. The pens will share a middle boundary to save on materials.
The farmer has 18m of fence available to build the enclosure. To figure out the best design, he calls the length of the fence extending from the hedge, $x$.
(i) Find an expression for the total area, $A$, of the two combined pens in terms of $x$.
(ii) Use tables and graphs to investigate the effect that changing $x$ has on the total area. Use your investigation to recommend the ideal dimensions for the pens.
Aroha wants to build a new super enclosure for her rabbits. The photo is her current enclosure. The new design will extend beyond the house as shown in the plan below. She has 16m of material to build the three new sides of the enclosure (shown in brown on the plan). The side against the house does not need any material.
(i) Find an expression for the area, $A$, of the enclosure in terms of $x$.
(ii) Use your expression from part (i) to investigate the ideal dimensions for the rabbit enclosure that will maximize its area. Support your answer with a table of values and a graph.
An agricultural scientist is trying to maximize crop yield. She finds that if she plants 62 soybean plants in a small plot, the average yield per plant is 150 pods. She knows from experience that for each additional plant she adds to the plot, the yield of every plant is reduced by 2 pods due to competition for nutrients.
To find the ideal number of plants, the scientist starts an investigation. She uses $x$ to represent the number of additional plants and $y$ to represent the total number of pods from the plot.
Here is the start of her table:
| Additional Plants ($x$) | Total Plants | Pods per Plant (pods) | Total Pods ($y$) |
|---|---|---|---|
| 1 | 63 | 148 | 9,324 |
| 2 | 64 | 146 | 9,344 |
Complete the scientist's investigation by continuing the table to find the maximum possible yield. Based on your findings, state the ideal number of plants she should grow. Support your recommendation with a graph showing the relationship between the number of additional plants ($x$) and the total pods ($y$), and include the equation that models this relationship.