chem tech makes paper cones used to filter liquids in lab experiments. to make paper cones, the company starts with the net of the shape, where the slant line is needed. the company wants the height of the cones to be 6 cm, if they make the diameter 13 cm, what must the slant height be? use the Pythagorean Theorem and round the answer to the nearest tenth.

Question

-11.5 cm
-8.8 cm
-2.5 cm
-14.3 cm

GPT-4o mini · Answer

To find the slant height of the cone, we can use the Pythagorean Theorem.

For a cone, the slant height (\( s \)), height (\( h \)), and radius (\( r \)) are related by the formula:

\[ s^2 = r^2 + h^2 \]

Given:

Height (\( h \)) = 6 cm
Diameter = 13 cm, so the radius (\( r \)) = Diameter / 2 = 13 cm / 2 = 6.5 cm

Now, we can plug in the values into the Pythagorean Theorem:

\[ s^2 = (6.5)^2 + (6)^2 \]

Calculating the squares:

\[ s^2 = 42.25 + 36 \] \[ s^2 = 78.25 \]

Now, we find \( s \) by taking the square root:

\[ s = \sqrt{78.25} \approx 8.8 \text{ cm} \]

Rounding to the nearest tenth gives us:

8.8 cm

Thus, the slant height must be 8.8 cm.

Answer

melissa has a gerbil that lives in a cage in the shape of a rectangular prism. she wants to put a ramp in the cage to give her gerbil exercise. the cage has a length of 70 cm, a width of 35 cm, and a height of 50 cm. how long does the ramp need to fit diagonally in the change? round to the nearest tenth.
-92.9 cm
-116.3 cm
-60.2 cm
-78.6 cm

GPT-4o mini · Answer

To find the length of the diagonal ramp that fits inside the rectangular cage, we can use the formula for the diagonal \( d \) of a rectangular prism, given by:

\[ d = \sqrt{l^2 + w^2 + h^2} \]

where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Given:

Length (\( l \)) = 70 cm
Width (\( w \)) = 35 cm
Height (\( h \)) = 50 cm

Now, we can substitute the values into the formula:

\[ d = \sqrt{(70)^2 + (35)^2 + (50)^2} \]

Calculating the squares:

\[ (70)^2 = 4900 \] \[ (35)^2 = 1225 \] \[ (50)^2 = 2500 \]

Now, add these values together:

\[ d = \sqrt{4900 + 1225 + 2500} \] \[ d = \sqrt{8625} \]

Now, we compute the square root:

\[ d \approx 92.9 \text{ cm} \]

So, the length of the ramp that needs to fit diagonally in the cage is approximately:

92.9 cm