The scale factor is a ratio. The surface area of the real ramp is 2.92 yards².
What is the scale factor?
The ratio between comparable measurements of an item and a representation of that thing is known as a scale factor in arithmetic.
A.) Let the dilation factor be 'x'.
As it is given that the surface area of the pre-image is 54 cm², while the area of the image is 1350 cm². And as we know that the dilation factor is applicable to the dimensions, therefore,
[tex]\text{Area of pre-image}\times x^2 = \text{Area of Image}\\\\x^2 = \dfrac{1350}{54}\\\\x^2 = 25\\\\x=5[/tex]
Now, as the height of the pre-image is 4.7cm, therefore, the height of the image can be written as,
[tex]\text{Height of pre-image}\times x = \text{Height of Image}\\\\ \text{Height of Image} = 4.7 \times 5 = 23.5\rm\ cm[/tex]
B.) As we know that the height of the real ramp is 0.5 yards, while the height of the drawing is 6 cm, therefore, the scale factor can be written as,
[tex]\rm Scale\ factor = \dfrac{\text{Heightof Image}}{\text{Height of pre-image}}=\dfrac{0.5\ yards}{6\ cm} = \dfrac{1}{12}[/tex]
Now, the area of the pre-image can be written as,
[tex]\rm \text{Area of pre-image} = 2triangles+ 2 rectangle + 1 rectangle\ at\ base[/tex]
[tex]= 2(0.5 \times 6 \times 16) + 2(10 \times 9) + (16 \times 9)\\\\= 2(48)+2(90)+144\\\\= 96 + 180 + 144\\\\ = 420\ cm^2[/tex]
Further, the area of the actual ramp can be written as,
[tex]{\text{Area of pre-image}} \times \rm Scale\ factor = \text{Area of Image}\\\\ \text{Area of Image}= 420 \times (\dfrac{1}{12})^2 \\\\ \text{Area of Image}= 420 \times (\dfrac{1}{144})=2.92\ yards^2[/tex]
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