Planning a room and a garden: what is worth it, and estimating

The plane and space, grade 10 (3 units) Lesson 14 of 15Learning

Planning a room and a garden: what is worth it, and estimating

The steps of planning

Planning a room or a garden uses all that this course has taught:

a plan – a drawing with the measurements;
perimeter – for a fence, skirting, a frame;
area – for grass, tiling, paint;
waste and rounding up – how much to really buy;
comparing offers – what is cheaper in the end;
estimating – a quick calculation with round numbers, to check the exact one.

Example: two offers for a fence

Two companies offer to build a fence:
Company A: 95 ₪ a metre, the fitting included.
Company B: 70 ₪ a metre, and another 1,200 ₪ for fitting, whatever the length.
The example below finds at what length the two offers are equal, and when each is worth it.

A fence of 40 m: A – $\displaystyle {95\cdot40}={3{,}800}$ ₪; B – $\displaystyle {70\cdot40+1{,}200}={4{,}000}$ ₪. A is cheaper.
A fence of 60 m: A – $\displaystyle {95\cdot60}={5{,}700}$ ₪; B – $\displaystyle {70\cdot60+1{,}200}={5{,}400}$ ₪. B is cheaper.
The length $\displaystyle x$ at which the prices are equal: $\displaystyle {95x}={70x+1{,}200}$ , so $\displaystyle {25x}={1{,}200}$ , $\displaystyle {x}={48}$ m.
For a fence shorter than 48 m company A is worth it, and for a longer one company B: the longer the fence, the more metres the fixed fitting price is spread over.

Exercise 1

Two gardeners offer to lay grass:
Gardener A: 30 ₪ per m².
Gardener B: 22 ₪ per m², and another 480 ₪ to take away the rubbish.
Find the area, in m², at which the two offers cost the same.

a

b

Which gardener is worth it for a garden of 100 m²? Explain.

The plan of a garden

Below is the plan of a garden behind a house. The measurements are in metres.
The garden is a rectangle of 14 m by 10 m. The thick line at the top is the wall of the house.
In the corner by the house there is a paved patio, a square of 4 m by 4 m.
The garden has a round ornamental pond with a diameter of 2 m.
The rest of the garden will be grass, and around it a fence on the three sides that are not the house’s wall (the dashed line).

Exercise 2

Use the plan of the garden.

a

The length of the fence, in metres

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The area of the grass – without the patio and the pond – in m²

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Tiles of 40 cm by 40 cm for the patio, with 10% waste: how many tiles to buy?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

d

One of the companies from the example builds the fence. Which company is worth it, and by how many shekels?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A bedroom is 3.9 m by 5.1 m.
A square metre of parquet, fitting included, costs 148 ₪.

a

Estimate, with round numbers, the area of the room and the price of the parquet.

b

The exact price, in shekels

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 10 (3 units)

Perimeters

The perimeter of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2(a+b)$
The perimeter of a square – $\displaystyle a$ the side
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4a$
Pi
$\displaystyle \pi\approx3.14$
The circumference – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$

Routes: Pythagoras, distance, speed and time

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2$
The hypotenuse
$\displaystyle c$ $\displaystyle =$ $\displaystyle \sqrt{a^2+b^2}$
A leg
$\displaystyle a$ $\displaystyle =$ $\displaystyle \sqrt{c^2-b^2}$
Speed – $\displaystyle s$ the distance, $\displaystyle t$ the time
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{s}{t}$
Distance
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$
Time
$\displaystyle t$ $\displaystyle =$ $\displaystyle \frac{s}{v}$

Areas

The area of a rectangle
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle A$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle A$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$
A square metre in square centimetres (a dunam is 1000 m²)
$\displaystyle 1\text{ m}^2=100\cdot100\text{ cm}^2=10000\text{ cm}^2$

Tilings

The sum of the angles of a polygon with $\displaystyle n$ sides
$\displaystyle 180^{\circ}\cdot(n-2)$
The angle of a regular polygon with $\displaystyle n$ sides
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \frac{180^{\circ}\cdot(n-2)}{n}$
The angles around a vertex of a tiling
$\displaystyle \alpha_1+\alpha_2+\ldots=360^{\circ}$