Question

a pyramid art installation has a surface area of 24 m². an artist creates replicas with scale factors of 1/8, 1/10, and 1/12. what is the surface area of each replica? match each scale factor with its correct surface area. scale factor of 1/8 scale factor of 1/10 scale factor of 1/12

Explanation:

Step1: Recall the relationship between scale - factor and surface - area

The ratio of the surface - areas of two similar solids is equal to the square of the ratio of their corresponding linear dimensions (scale - factor). If the scale - factor is $k$, the ratio of the surface - area of the replica ($A_{r}$) to the surface - area of the original ($A_{o}$) is $k^{2}$, i.e., $A_{r}=k^{2}A_{o}$.

Step2: Calculate for scale - factor $k = \frac{1}{8}$

Given $A_{o}=24\ m^{2}$ and $k=\frac{1}{8}$, then $A_{r}=(\frac{1}{8})^{2}\times24=\frac{1}{64}\times24=\frac{24}{64}= 0.375\approx0.38\ m^{2}$.

Step3: Calculate for scale - factor $k=\frac{1}{10}$

Given $A_{o}=24\ m^{2}$ and $k = \frac{1}{10}$, then $A_{r}=(\frac{1}{10})^{2}\times24=\frac{1}{100}\times24 = 0.24\ m^{2}$.

Step4: Calculate for scale - factor $k=\frac{1}{12}$

Given $A_{o}=24\ m^{2}$ and $k=\frac{1}{12}$, then $A_{r}=(\frac{1}{12})^{2}\times24=\frac{1}{144}\times24=\frac{24}{144}\approx0.17\ m^{2}$.

Answer:

scale factor of $\frac{1}{8}$: $0.38\ m^{2}$
scale factor of $\frac{1}{10}$: $0.24\ m^{2}$
scale factor of $\frac{1}{12}$: $0.17\ m^{2}$