Question

4 The ratio of corresponding dimensions of two similar solids is 3/4. The surface
area of the first solid is 96 m². Its volume is 720 m³. Find the surface area and
volume of the second solid. Round each answer to the nearest tenth, if
necessary. Show your work

5 A globe in a brass stand has a diameter of 40 in. What is the volume of the
globe to the nearest cubic inch? Show your work.

Answer 1

4.

The ratio is 3/4 for corresponding dimensions, the ratio of their surface areas is equal to the square of this ratio:

$\sf (\dfrac{3}{4})^2=\dfrac{S}{96}$

Simplify the exponent:

$\sf \dfrac{9}{16}=\dfrac{96}{S}$

Cross multiply:

$\sf 9S=1536$

Divide 9 to both sides:

$\sf S\approx 170.7~m^2$

So the surface area of the second solid is 54 square meters.

The ratio is 3/4 for corresponding dimensions, the ratio of their volumes is equal to the cube of this ratio:

$\sf (\dfrac{3}{4})^3=\dfrac{720}{V}$

Simplify the exponent:

$\sf \dfrac{27}{64}=\dfrac{720}{V}$

Cross multiply:

$\sf 27V=46080$

Divide 64 to both sides:

$\sf V\approx 1706.7~m^3$

5.

A globe is a sphere, use the formula for the volume of a sphere:

$\sf V=\dfrac{4}{3}\pi r^3$

The radius is half of the diameter, so the radius here is 40/2 = 20. Plug it in the formula, use 3.14 to approximate for Pi:

$\sf V=\dfrac{4}{3}(3.14)(20)^3$

Simplify the exponent:

$\sf V=\dfrac{4}{3}(3.14)(8000)$

Multiply:

$\sf V\approx \boxed{\sf 33,493~in^3}$

Answer 2

Question 4:

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Ratio of Dimension to Area
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Dimension : Area
$\frac{3}{4} \ : \ (\frac{3}{4} )^2$
$\frac{3}{4} \ : \frac{9}{16}$

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Find Area
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Let x be the surface area of the second solid
$\frac{9}{16} = \frac{96}{x}$
$9x = 96 \times 16$
$9x = 1536$
$x = 1536 \div 9$
$x =170.7 m^2 (\ nearest \ tenth )$

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Answer: 170.7 m²
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---------------------------------------------------
Ratio of Dimension to Volume
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Dimension : Volume
$\frac{3}{4} \ : \ (\frac{3}{4} )^3$
$\frac{3}{4} \ : \ \frac{27}{64}$

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Find Volume
----------------------------------------------------
Let x be the surface are of the second solid
$\frac{27}{64} = \frac{720}{x}$
$27x = 720 \times 64$
$9x = 46080$
$x = 146080 \div 27$
$x =1706.7 m^2 (nearest \ hundredth )$

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Answer: 1706.7 m³
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Question 5
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Find Radius
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Radius = Diameter ÷ 2
Radius = 40 ÷ 2
Radius = 20

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Find volume of the globe, which is a sphere
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$Volume \ of \ sphere \ = \frac{4}{3} \pi x^{3}$

$Volume \ of \ sphere \ = \frac{4}{3} \pi (20)^{3}$

$Volume \ of \ sphere \ = 33510.3 in^3 \ ( \ nearest \ hundredth )$

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Answer: 33510.3 in³
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