The surface area S (in square centimeters) of a large dog
can be approximated by the model S = 11.2m^2/3 where m is the mass (in grams) of the dog. A Labrador retriever’s mass is about three times the mass of a Scottish terrier. Is its surface area also three times that of a Scottish terrier?
To determine whether the surface area of a Labrador retriever is three times that of a Scottish terrier, we need to compare their respective surface areas using the given model.
Let's assume the mass of a Scottish terrier is represented by m.
According to the given information, the mass of a Labrador retriever is three times the mass of a Scottish terrier. Therefore, the mass of a Labrador retriever is 3m.
Using the model S = 11.2m^(2/3), let's calculate the surface area of a Scottish terrier:
S(Scottish terrier) = 11.2m^(2/3)
Now, let's calculate the surface area of a Labrador retriever:
S(Labrador retriever) = 11.2(3m)^(2/3)
= 11.2(9m^2)^(1/3)
= 11.2 * 3^(2/3) * m^(2/3)
= 11.2 * 3^(2/3) * [m^(2/3)] * 1
Since S(Labrador retriever) = 11.2 * 3^(2/3) * [m^(2/3)], we can see that the surface area of a Labrador retriever contains an additional factor of 3^(2/3) compared to the surface area of a Scottish terrier.
To determine if the surface area is three times that of a Scottish terrier, we need to compare this factor, 3^(2/3), to 3.
Calculating 3^(2/3):
3^(2/3) ≈ 1.817
Since 3^(2/3) ≈ 1.817 is not equal to 3, we can conclude that the surface area of a Labrador retriever is not three times that of a Scottish terrier.
To determine if the surface area of a Labrador retriever is three times that of a Scottish terrier, we can compare the surface area equations for both dogs.
Given that the surface area (S) of a large dog can be approximated by the model S = 11.2m^(2/3), where m is the mass of the dog in grams.
Let's assume the mass of the Scottish terrier is represented by m_scottish.
Therefore, the surface area of the Scottish terrier (S_scottish) can be calculated as S_scottish = 11.2 * m_scottish^(2/3).
Since the mass of the Labrador retriever is three times that of the Scottish terrier, we can represent it as m_labrador = 3 * m_scottish.
Now, let's calculate the surface area of the Labrador retriever (S_labrador) using the given equation. Substitute the mass of the Labrador retriever into the equation:
S_labrador = 11.2 * (3 * m_scottish)^(2/3)
To simplify the equation, we can apply the power rule: (a * b)^n = a^n * b^n.
S_labrador = 11.2 * 3^(2/3) * (m_scottish)^(2/3)
Now, let's compare the surface area equations for the Scottish terrier and Labrador retriever:
S_labrador = 11.2 * 3^(2/3) * (m_scottish)^(2/3)
S_scottish = 11.2 * (m_scottish)^(2/3)
We can see that the only difference is the constant factor of 3^(2/3) in the Labrador retriever's equation compared to the Scottish terrier's equation.
S_labrador = (3^(2/3)) * S_scottish
Therefore, the surface area of a Labrador retriever is proportional to the surface area of a Scottish terrier by a factor of (3^(2/3)), which is approximately 1.817.
So, the surface area of a Labrador retriever is not exactly three times that of a Scottish terrier, but rather approximately 1.817 times (or 81.7% more) than the Scottish terrier's surface area.
Ratio = 11.2(3m)^(2/3) / 11.2m*(2/3) = 11.2*2.08m^(2/3) / 11.2*m^(2/3) = 2.08
Labrador S. = 2.08 times Sottish terrier S.