Question

volume

1. a rectangular prism is made of 64 linking cubes. what might the area of the base be? explain.

the area would be

1. the volume of a rectangular prism and its height are given. what is the area of the prisms base?

a)
v = 51 cm³ h = 3 cm
b)
v = 250 cm³ h = 10 cm
c)
v = 375 cm³ h = 4 cm

1. connor says the volume of a rectangular prism is 6 × area of the base since there are six faces.

is connor right all the time, some of the time, or never? explain.

1. what is the volume of each cylinder?

a) radius = 10 cm
12 cm

Explanation:

Step1: Recall volume formula for prism

The volume of a rectangular prism is $V = B \times h$, where $B$ is the base area, $h$ is height. Rearranged, $B = \frac{V}{h}$.
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Problem 1

Step1: Relate cubes to volume

Each cube = 1 unit³, so $V=64$.

Step2: Find possible base areas

Base area $B = \frac{64}{h}$, where $h$ is a positive integer divisor of 64. Possible $h$ values: 1,2,4,8,16,32,64.
Corresponding $B$: $\frac{64}{1}=64$, $\frac{64}{2}=32$, $\frac{64}{4}=16$, $\frac{64}{8}=8$, $\frac{64}{16}=4$, $\frac{64}{32}=2$, $\frac{64}{64}=1$.
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Problem 2a

Step1: Apply base area formula

$B = \frac{V}{h}$

Step2: Substitute values

$B = \frac{51}{3}$

Step3: Calculate result

$B=17$
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Problem 2b

Step1: Apply base area formula

$B = \frac{V}{h}$

Step2: Substitute values

$B = \frac{250}{10}$

Step3: Calculate result

$B=25$
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Problem 2c

Step1: Apply base area formula

$B = \frac{V}{h}$

Step2: Substitute values

$B = \frac{375}{4}$

Step3: Calculate result

$B=93.75$
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Problem 3

Step1: Recall prism volume formula

Volume $V = B \times h$, not related to face count.

Step2: Analyze Connor's claim

Connor confuses volume with surface area. Volume depends on base area and height, not number of faces.
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Problem 4a

Step1: Recall cylinder volume formula

$V = \pi r^2 h$, use $\pi=3.14$

Step2: Substitute values

$V = 3.14 \times 10^2 \times 12$

Step3: Calculate step-by-step

$10^2=100$; $3.14 \times 100=314$; $314 \times 12=3768$

Answer:

1. Possible base areas are $1\ \text{unit}^2$, $2\ \text{unit}^2$, $4\ \text{unit}^2$, $8\ \text{unit}^2$, $16\ \text{unit}^2$, $32\ \text{unit}^2$, $64\ \text{unit}^2$. These come from dividing the total volume (64 cubic units) by any positive integer height that divides 64 evenly.
2. a) $17\ \text{cm}^2$

b) $25\ \text{cm}^2$
c) $93.75\ \text{cm}^2$

1. Never. The volume of a rectangular prism is equal to the area of the base multiplied by the height, not 6 times the base area. The number of faces (6) is related to surface area, not volume.
2. a) $3768\ \text{cm}^3$