Let's focus on calculating the diagonal of Models 1 and 2 using the provided heights and radii. The vertical cross-sections of these cylindrical models form right-angled triangles, where the height and radius are the legs, and the diagonal is the hypotenuse.
### For Model 1:
- Height ([tex]\(h_1\)[/tex]): 48 cm
- Radius ([tex]\(r_1\)[/tex]): 14 cm
- Diagonal given ([tex]\(d_1\)[/tex]): 50 cm
To verify if the given diagonal matches the height and radius, we use the Pythagorean theorem:
[tex]\[ d_1 = \sqrt{h_1^2 + r_1^2} \][/tex]
Plugging in the values:
[tex]\[ d_1 = \sqrt{48^2 + 14^2} \][/tex]
[tex]\[ d_1 = \sqrt{2304 + 196} \][/tex]
[tex]\[ d_1 = \sqrt{2500} \][/tex]
[tex]\[ d_1 = 50 \text{ cm} \][/tex]
Thus, for Model 1, the calculated diagonal is 50 cm, which matches the given diagonal.
### For Model 2:
- Height ([tex]\(h_2\)[/tex]): 35 cm
- Radius ([tex]\(r_2\)[/tex]): 6 cm
- Diagonal given ([tex]\(d_2\)[/tex]): 37 cm
Similarly, we use the Pythagorean theorem to find the diagonal:
[tex]\[ d_2 = \sqrt{h_2^2 + r_2^2} \][/tex]
Plugging in the values:
[tex]\[ d_2 = \sqrt{35^2 + 6^2} \][/tex]
[tex]\[ d_2 = \sqrt{1225 + 36} \][/tex]
[tex]\[ d_2 = \sqrt{1261} \][/tex]
[tex]\[ d_2 \approx 35.51 \text{ cm} \][/tex]
Thus, for Model 2, the calculated diagonal is approximately 35.51 cm, which is slightly less than the given diagonal of 37 cm.
### Summary:
- Model 1: The calculated diagonal is 50 cm, which matches the given diagonal.
- Model 2: The calculated diagonal is approximately 35.51 cm, which does not exactly match the given diagonal of 37 cm.
This step-by-step analysis confirms the match for Model 1 and highlights the slight discrepancy for Model 2.
