Question

practice 4 (from unit 5, lesson 8)
technology required.
a bagel shop makes bagels that serve 1 person each. for a party, the baker is asked to make a large bagel in the same shape that will serve 25 people.
a. by what scale factor will the bagel need to be dilated? round your answer to the nearest tenth.
type your answer in the box.
about

b. the bagels are topped with a thin layer of cream cheese. assume the thickness of the cream cheese is the same on both bagels. how many times more cream cheese will be required for the dilated bagel as for the original? round your answer to the nearest tenth.
type your answer in the box.
about times more cream cheese
how did i do?

Explanation:

Part (a)

Step1: Relate serving size to area scale factor

The serving size is related to the area of the bagel (since it's a 2D shape for the base, and serving 1 person vs 25 people means the area scales by a factor related to the number of people). The area of a similar figure scales with the square of the scale factor \( k \). So if the original area serves 1 person and the new serves 25, the area scale factor is 25. So we have \( k^2 = 25 \) (since area scales as \( k^2 \) for dilation).

Step2: Solve for the scale factor \( k \)

To find \( k \), take the square root of 25. \( k=\sqrt{25} = 5\)? Wait, no, wait. Wait, actually, the volume (or area, but if it's a bagel, maybe area of the top? Wait, the problem says "serve 25 people". If each bagel serves 1 person, the area (or volume) of the bagel is proportional to the number of people it serves. So if the original has area (or volume) \( A \) for 1 person, the new has area \( 25A \) for 25 people. For a dilation, the area scales as \( k^2 \), so \( k^2=\frac{25A}{A}=25 \), so \( k = \sqrt{25}=5 \)? Wait, but that's exact. But maybe I misread. Wait, no, maybe it's a 3D object? Wait, a bagel is a torus, but maybe we can consider it as a 2D shape for the top (the area to put cream cheese) or 3D volume. Wait, the first part (a) is about dilating the bagel to serve 25 people. If each bagel is a 2D shape (like a circle, since bagel is a torus, but maybe approximated as a circle for area), then the area scales with \( k^2 \). So if original area \( A \) serves 1 person, new area \( A' \) serves 25 people, so \( \frac{A'}{A}=25 \), so \( k^2 = 25 \), so \( k = \sqrt{25}=5 \). But that's exact. Wait, maybe the problem is about volume? Wait, no, the first part (a) is about dilating the bagel. Let's re - read: "By what scale factor will the bagel need to be dilated?". So dilation factor for linear dimensions. If the number of people served is proportional to the volume (since a bagel is a 3D object, volume of the bagel itself, or the area of the top for cream cheese? Wait, no, the first part (a) is about the bagel's size (the bagel itself, to serve 25 people). If each bagel (original) has volume \( V \) for 1 person, the new bagel has volume \( 25V \) for 25 people. For a dilation, volume scales as \( k^3 \), and linear dimensions scale as \( k \). Wait, maybe the problem considers the bagel as a 2D shape (area) for the amount it can serve? Wait, this is confusing. Wait, the problem says "serve 1 person each" and "serve 25 people". So the area (or volume) is proportional to the number of people. If it's a 2D area (like the surface area of the bagel's top), then area scales with \( k^2 \), so \( k^2 = 25 \), \( k = 5 \). But that's exact. But maybe the problem is about a 3D object, and the number of people served is proportional to the volume. Then volume scales as \( k^3 \), so \( k^3=25 \), so \( k=\sqrt[3]{25}\approx2.9 \). Oh! Wait, that must be it. I made a mistake earlier. I assumed area, but maybe it's volume. Because a bagel is a 3D object, and the amount it can serve (the amount of food) is proportional to its volume. So if the original volume \( V \) serves 1 person, the new volume \( V' \) serves 25 people, so \( \frac{V'}{V}=25 \). For a dilation, the volume scales as \( k^3 \) (where \( k \) is the linear scale factor), so \( k^3 = 25 \). Then \( k=\sqrt[3]{25}\approx2.9 \) (rounded to the nearest tenth). Let's calculate \( \sqrt[3]{25} \). \( 2.9^3=2.9\times2.9\times2.9 = 2.9\times8.41 = 24.389\approx25 \), so \( \sqrt[3]{25}\approx2.9 \). Ah, that makes sense. So my initial mistake…

Step1: Relate cream cheese to area (or volume?)

The cream cheese is a thin layer on the bagel. If the thickness is the same, the amount of cream cheese is proportional to the area of the surface it's applied to. For a dilation, the area of a 2D surface (like the top of the bagel) scales with \( k^2 \), where \( k \) is the linear scale factor. We found \( k\approx2.9 \) from part (a).

Step2: Calculate the area scale factor

So the area scale factor is \( k^2=(2.9)^2 = 8.41\approx8.4 \) (rounded to the nearest tenth). Alternatively, if we use the exact \( k = \sqrt[3]{25} \), then \( k^2=(\sqrt[3]{25})^2=\sqrt[3]{25^2}=\sqrt[3]{625}\approx8.5 \) (wait, let's calculate \( \sqrt[3]{625} \). \( 8.5^3 = 8.5\times8.5\times8.5=72.25\times8.5 = 614.125\), \( 8.6^3=8.6\times8.6\times8.6 = 73.96\times8.6=636.056 \). So \( \sqrt[3]{625}\approx8.5 \). Wait, there's a discrepancy because we used \( k = 2.9 \) (rounded) in part (a). Let's do it more accurately. From part (a), \( k=\sqrt[3]{25}\approx2.924017738 \). Then \( k^2=(2.924017738)^2\approx8.5498\approx8.5 \) when rounded to the nearest tenth. Wait, but if we use \( k = 2.9 \), \( k^2 = 8.41\approx8.4 \). Which is correct? Let's see: the volume scale factor is 25 (since 25 people). The area scale factor for the cream cheese: if the cream cheese is a thin layer, the volume of the cream cheese is approximately the area of the surface times the thickness (since thickness is small, it's like a 2D area times a small height, so volume of cream cheese \( V_{cc}=A\times t \), where \( A \) is the area and \( t \) is the thickness). So the ratio of the cream cheese volumes (new to original) is \( \frac{V_{cc}'}{V_{cc}}=\frac{A'\times t}{A\times t}=\frac{A'}{A}=k^2 \). But also, the volume of the bagel is proportional to \( k^3 \) (since it's a 3D object). But the number of people served is proportional to the bagel's volume, so \( k^3 = 25 \). Therefore, \( k^2=k^{3-\ 1}=k^3\times k^{-1}=25\times\frac{1}{k} \). Since \( k = \sqrt[3]{25} \), then \( k^2=\frac{25}{\sqrt[3]{25}}=\sqrt[3]{25^2}=\sqrt[3]{625}\approx8.5 \). Let's calculate \( \sqrt[3]{625} \):

We know that \( 8^3 = 512 \), \( 9^3=729 \). So \( \sqrt[3]{625} \) is between 8 and 9. Let's use linear approximation. Let \( f(x)=x^3 \), \( f'(x)=3x^2 \). We know that \( f(8.5)=8.5^3 = 614.125 \), \( f(8.6)=8.6^3 = 636.056 \). We need to find \( x \) such that \( x^3 = 625 \). The difference between 625 and 614.125 is \( 625 - 614.125 = 10.875 \). The difference between \( f(8.6) \) and \( f(8.5) \) is \( 636.056 - 614.125 = 21.931 \). So the fraction is \( \frac{10.875}{21.931}\approx0.496 \). So \( x\approx8.5 + 0.496\times(8.6 - 8.5)\approx8.5+0.0496\approx8.55 \). Wait, but maybe the problem expects us to use the scale factor from part (a) (rounded to the nearest tenth) and square it. If in part (a) we have \( k\approx2.9 \), then \( k^2 = 2.9\times2.9 = 8.41\approx8.4 \). But if we use the more accurate \( k=\sqrt[3]{25}\approx2.924 \), then \( k^2\approx8.5 \). Let's check the problem statement again. It says "Round your answer to the nearest tenth." Let's calculate \( (\sqrt[3]{25})^2 \):

\( \sqrt[3]{25}\approx2.924017738 \)

\( (2.924017738)^2=2.924017738\times2.924017738\approx8.5498\approx8.5 \)

So the more accurate answer is approximately 8.5.

(step - by - step for part b):

Step1: Determine the relationship between cream cheese and scale factor

The amount of cream cheese is proportional to the area of the surface it covers (since thickness is constant). For a dilated figure, the area scales with the square of the linear scale factor \( k \) (i.e., \( \text{Area ratio}=k^2 \)).

Step2: Use the scale factor from part (a)

From part (a), the linear scale factor \( k = \sqrt[3]{25}\approx2.924 \).

Step3: Calculate \( k^2 \)

Square the scale factor: \( k^2=(\sqrt[3]{25})^2=\sqrt[3]{25^2}=\sqrt[3]{625}\approx8.5 \) (rounded to the nearest tenth).

Answer:

(a):
\( 2.9 \)

Part (b)