Respuesta :
Answer:
Part A:
Length: [tex]\sf \dfrac{80}{3} [/tex] feet
Height: [tex]\sf \dfrac{45}{3} [/tex] feet
Part B:
[tex]\sf \textsf{Area}_{\textsf{new}} = 72 \textsf{ square feet} [/tex]
[tex]\sf \textsf{Area}_{\textsf{new}} = 200 \textsf{ square feet} [/tex]
Part C:
[tex]\sf \dfrac{\textsf{Area}_{\textsf{original}}}{\textsf{Area}_{\textsf{new}}} =\left(\dfrac{3}{5}\right)^2 [/tex]
Step-by-step explanation:
Part A: Dimensions of the New Deck
Given that the new deck will be a 3:5 scaled version of the original deck, we can use the scale factor to find the dimensions.
The scale factor is [tex]\sf \dfrac{5}{3} [/tex] because the new deck is 5 parts compared to the original deck's 3 parts.
So, for each dimension of the original deck:
- Length: [tex]\sf 16 [/tex] feet
- Height: [tex]\sf 9 [/tex] feet
We'll multiply each dimension by the scale factor to find the dimensions of the new deck.
For the length of the new deck:
[tex]\sf \textsf{Length}_{\textsf{new}} = 16 \times \dfrac{5}{3} \\\\ = \dfrac{16 \times 5}{3} \\\\= \dfrac{80}{3} \textsf{ feet} [/tex]
For the height of the new deck:
[tex]\sf \textsf{Height}_{\textsf{new}} = 9 \times \dfrac{5}{3} \\\\= \dfrac{9 \times 5}{3}\\\\ = \dfrac{45}{3} \textsf{ feet} [/tex]
So, the dimensions of the new deck are:
Length: [tex]\sf \dfrac{80}{3} [/tex] feet
Height: [tex]\sf \dfrac{45}{3} [/tex] feet
[tex]\hrulefill[/tex]
Part B: Area of the Original and New Decks
The area of a triangle is given by the formula:
[tex]\sf \textsf{Area} = \dfrac{1}{2} \times \textsf{base} \times \textsf{height} [/tex]
For the original deck:
[tex]\sf \textsf{Area}_{\textsf{original}} = \dfrac{1}{2} \times 16 \times 9 \\\\= 72 \textsf{ square feet} [/tex]
For the new deck:
[tex]\sf \textsf{Area}_{\textsf{new}} = \dfrac{1}{2} \times \dfrac{80}{3} \times \dfrac{45}{3} \\\\= \dfrac{1}{2} \times \dfrac{80 \times 45}{3 \times 3}\\\\ = \dfrac{1}{2} \times \dfrac{3600}{9} = \dfrac{3600}{18} \\\\= 200 \textsf{ square feet} [/tex]
So,
[tex]\sf \textsf{Area}_{\textsf{new}} = 72 \textsf{ square feet} [/tex]
[tex]\sf \textsf{Area}_{\textsf{new}} = 200 \textsf{ square feet} [/tex]
[tex]\hrulefill[/tex]
Part C: Comparing the Ratio of Areas to the Scale Factor
The ratio of the areas of the original deck to the new deck is:
[tex]\sf \dfrac{\textsf{Area}_{\textsf{original}}}{\textsf{Area}_{\textsf{new}}} = \dfrac{72}{200} [/tex]
To simplify the ratio, we find the greatest common divisor (GCD) of 72 and 200, which is 8.
[tex]\sf \dfrac{72}{200} = \dfrac{72 \div 8}{200 \div 8} = \dfrac{9}{25} [/tex]
So, the ratio of the areas of the original deck to the new deck is [tex]\sf \dfrac{9}{25}[/tex].
Comparing this to the scale factor (3:5), we see that the ratio of the areas matches the square of the scale factor. This is expected because the area is a two-dimensional measure, so when we scale the dimensions linearly, the area is scaled by the square of the scale factor.
