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How would you go about solving this? Please provide formulae, rules, explanation / working out . Thank you in advance!

100 POINTS ENSURE YOUR ANSWER IS CORRECT BEFORE POSTING How would you go about solving this Please provide formulae rules explanation working out Thank you in class=

Respuesta :

roohafzah

Answer :

  • k = 13.8%

Explanation :

let the volume of statue A be V and that of statue B be v by

thus,

  • V = x units^3
  • v = x units^3 - 0.20 units^3
  • v = 0.80x units^3

now,

consider the surface area

let the surface area of A be SA units^2 and that of B be sa units^2

then,

  • (0.80x/x)^(1/3) = (sa/SA)^(1/2)
  • (4/5)^(1/3) = (sa/SA)^(1/2)
  • (sa/SA) = ((4/5)^(1/3))^2
  • (sa/SA) = 0.862
  • sa = 0.862SA
  • sa = 86.2% of SA

thus,

  • k = 100% - 86.2%
  • k = 13.8%

thus, the surface area of statue B is 13.8% less than the surface area of statue A.

msm555

Answer:

k% = 13.8%

Step-by-step Explanation:

To work out the value of [tex]\sf k[/tex], representing the percentage decrease in surface area of statue B ([tex]\sf SA_B[/tex]) compared to statue A ([tex]\sf SA_A[/tex]), given that the volume of statue B is 20% less than the volume of statue A, we can follow these steps:

Volume Relationship:

Given that the volume of statue B ([tex]\sf V_B[/tex]) is 20% less than the volume of statue A ([tex]\sf V_A[/tex]), we have:

[tex]\sf V_B = V_A - 20\% \textsf{ of } V_A [/tex]

[tex]\sf V_B = V_A - \dfrac{20}{100} \times V_A [/tex]

[tex]\sf V_B = V_A - 0.2 V_A [/tex]

[tex]\sf V_B = (1-0.2) V_A [/tex]

[tex]\sf V_B = 0.8V_A [/tex]

Surface Area Relationship for Similar Figures:

For similar figures, the ratio of volumes ([tex]\sf \dfrac{V_B }{ V_A}[/tex]) is related to the cube of the linear scale factor ([tex]\sf k[/tex]), and the ratio of surface areas ([tex]\sf \dfrac{SA_B }{ SA_A}[/tex]) is related to the square of the linear scale factor ([tex]\sf k[/tex]).

Therefore,

[tex]\sf \dfrac{V_B}{V_A} = \left( \dfrac{k}{1} \right)^3 = k^3 [/tex]

[tex]\sf \dfrac{SA_B}{SA_A} = \left( \dfrac{k}{1} \right)^2 = k^2 [/tex]

Calculate [tex]\sf k[/tex]:

Substitute [tex]\sf V_B = 0.8 \times V_A[/tex] into the volume ratio equation:

[tex]\sf \dfrac{0.8 \cancel{V_A}}{\cancel{V_A}} = k^3 [/tex]

[tex]\sf 0.8 = k^3 [/tex]

Taking the cube root of both sides to solve for [tex]\sf k[/tex]:

[tex]\sf k = \sqrt[3]{0.8} \approx 0.9283177667 [/tex]

Calculate [tex]\sf k\%[/tex] (Percentage Decrease in Surface Area):

Now, calculate [tex]\sf k\%[/tex] which represents the percentage decrease in surface area:

[tex]\sf k\% = (1 - k^2) \times 100\% [/tex]

[tex]\sf k\% = (1 - (0.9283177667)^2) \times 100\% [/tex]

[tex]\sf k\% = (1 - 0.861773876) \times 100\% [/tex]

[tex]\sf k\% = 0.138226124 \times 100\% [/tex]

[tex]\sf k\% = 13.8226124\% [/tex]

[tex]\sf k\% \approx \boxed{13.8\%}\textsf{(in 3 significant figures)}[/tex]

Therefore, [tex]\sf k \approx \boxed{13.85\%}[/tex], indicating that the surface area of statue B is approximately 13.8% less than the surface area of statue A, consistent with the given volume relationship between the two statues.

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