Respuesta :
Given =
Two similar pyramid have base area of 12.2 cm² and 16 cm².
surface area of the larger pyramid = 56 cm²
find out the surface area of the smaller pyramid
To proof =
Let us assume that the surface area of the smaller pyramid be x.
as surface area of the larger pyramid is 56 cm²
Two similar pyramid have base area of 12.2 cm² and 16 cm².
by using ratio and proportion
we have
ratio of the base area of the pyramids : ratio of the surface area of the pyramids
[tex]\frac{12.2}{16} : \frac{x}{56}[/tex]
x = 12.2 ×56×[tex]\frac{1}{16}[/tex]
by solvingthe above terms
we get
x =42.7cm²
Hence the surface area of the smaller pyramid be 42.7cm²
Hence proved
Similar figures have scaled dimensions. The surface area of the smaller pyramid is [tex]42.7 cm^2[/tex]
What are similar figures?
Similar figure are zoomed in or zoomed out (or just no zoom) version of each other. They are scaled version of each other, and by scale, we mean that each of their dimension(like height, width etc linear quantities) are constant multiple of their similar figure.
So, if a side of a figure is of length L units, and that of its similar figure is of M units, then:
- [tex]L = k \times M[/tex]
where 'k' will be called as scale factor.
The linear things grow linearly like length, height etc.
The quantities which are squares or multiple of linear things twice grow by square of scale factor. Thus, we need to multiply or divide by [tex]k^2[/tex] to get each other corresponding quantity from their similar figures' quantities.
- So area of first figure = [tex]k^2 \times[/tex] area of second figure
Similarly, increasing product derived quantities will need increased power of 'k' to get the corresponding quantity. Thus, for volume, it is k cubed. or
- Volume of first figure = [tex]k^3 \times[/tex] volume of second figure.
It is because we will need to multiply 3 linear quantities to get volume, which results in k getting multiplied 3 times, thus, cubed.
For this case, we have:
- The base area of first pyramid = 12.2 cm sq.
- The base area of the second pyramid = 16 cm sq.
Thus, since they're similar, let the scale factor be 'k' (remember that scale factors are always unitless).
Then, we get:
Base area of first pyramid= [tex]k^2 \times[/tex] Base area of second pyramid
or
[tex]12.2 = k^2 \times 16\\\\k^2 = \dfrac{12.2}{16} = \dfrac{6.1}{8}[/tex]
Now, we have surface area of larger pyramid (second figure as its base is bigger so all measurements must be bigger as they're similar) is of 56 cm sq.
Then, as we have:
Surface area of first pyramid= [tex]k^2 \times[/tex] Surface area of second pyramid
or
Surface area of first pyramid = [tex]\dfrac{6.1}{8} \times 56 = 42.7 \: \rm cm^2[/tex]
Thus, the surface area of the smaller pyramid is [tex]42.7 cm^2[/tex]
Learn more about similar figures here:
https://brainly.com/question/2786083
