Answer:
a) 1/50
b) 21 m²
Step-by-step explanation:
Given a scale drawing of a garden is a right triangle with legs 24 cm and 7 cm, and the length of the hypotenuse of the actual garden is 12.5 m, you want to know the scale factor of the drawing and the area of the actual garden.
(a) Scale factor
The scale factor of a drawing is generally the ratio of the drawing dimension to the actual dimension. Here, we need to know the length of the hypotenuse on the drawing in order to determine the scale factor.
The hypotenuse is given by the Pythagorean theorem:
c² = a² +b²
c = √(a² +b²) = √(7² +24²) = √(49 +576) = √625 = 25 . . . . . cm
The ratio of hypotenuse lengths is ...
[tex]\dfrac{\text{$25$ cm}}{\text{$12.5$ m}}=\dfrac{\text{$0.25$ m}}{\text{$12.5$ m}}=\dfrac{25}{1250}=\boxed{\dfrac{1}{50}}[/tex]
(b) Area
Each actual dimension is 50 times the drawing dimension. That means the actual area will be 50² times the drawing area:
A = (1/2)(0.24 m)(0.07 m)×2500 = 0.0084×2500 m² = 21 m²
The area of the actual flower garden is 21 square meters.
__
Additional comment
When the scale factor is expressed as a fraction without units, the dimensions used to compute it must have the same units. That is why we converted the drawing dimensions from centimeters to meters.
We did the same for the area calculation because the scale factor is unitless, and we wanted the area in square meters, not square centimeters.
