Question

the radius, r, of a sphere is 7.2 mm. calculate the sphere’s volume, v. use the value 3.14 for π, and round your answer to the nearest tenth. (do not round any intermediate computations.)

Explanation:

Step1: Recuerda la fórmula del volumen de una esfera

La fórmula para el volumen \( V \) de una esfera es \( V = \frac{4}{3}\pi R^3 \), donde \( R \) es el radio de la esfera.

Step2: Sustituye los valores conocidos

Dado que \( R = 7.2 \) mm y \( \pi = 3.14 \), sustituimos estos valores en la fórmula:
\[ V = \frac{4}{3} \times 3.14 \times (7.2)^3 \]
Primero, calculamos \( (7.2)^3 \):
\[ 7.2^3 = 7.2 \times 7.2 \times 7.2 = 51.84 \times 7.2 = 373.248 \]
Luego, multiplicamos por \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \times 3.14 = \frac{12.56}{3} \approx 4.1867 \]
Ahora, multiplicamos este resultado por \( 373.248 \):
\[ V \approx 4.1867 \times 373.248 \]
Calculamos esta multiplicación:
\[ 4.1867 \times 373.248 \approx 1562.6 \]
(Verificando: \( \frac{4}{3} \times 3.14 \times 373.248 = \frac{4 \times 3.14 \times 373.248}{3} = \frac{4690.56}{3} = 1563.52 \)? Espera, quizás un cálculo más preciso:

\( 7.2^3 = 7.2 \times 7.2 = 51.84; 51.84 \times 7.2 = 373.248 \)

Luego, \( \frac{4}{3} \times 3.14 = 4.186666... \)

Multiplicar por 373.248: \( 4.186666... \times 373.248 \)

\( 4 \times 373.248 = 1492.992 \)

\( 0.186666... \times 373.248 \approx 0.186666 \times 373.248 \approx 69.6 \)

Sumando: \( 1492.992 + 69.6 \approx 1562.592 \), que redondeado a la décima es 1562.6? Espera, quizás cometí un error. Vamos a calcular de nuevo:

\( V = \frac{4}{3} \times 3.14 \times 7.2^3 \)

\( 7.2^3 = 7.2 \times 7.2 \times 7.2 = 51.84 \times 7.2 = 373.248 \)

\( \frac{4}{3} \times 3.14 = 4.186666... \)

\( 4.186666... \times 373.248 = 4.186666... \times 373.248 \)

Calculando 373.248 × 4 = 1492.992

373.248 × 0.186666... = 373.248 × (11/60) ≈ 373.248 × 0.183333... ≈ 68.38

Entonces total ≈ 1492.992 + 68.38 ≈ 1561.372? No, quizás la fórmula es correcta. O tal vez el error está en la redondeo.

Wait, la fórmula es \( V = \frac{4}{3}\pi r^3 \). Entonces:

\( r = 7.2 \)

\( r^3 = 7.2 \times 7.2 \times 7.2 = 373.248 \)

\( \frac{4}{3} \times 3.14 = 4.186666... \)

Multiplicar: 4.186666... × 373.248 = ?

Calculando 373.248 × 4 = 1492.992

373.248 × 0.186666... = 373.248 × (11/60) ≈ 373.248 × 0.183333... = 373.248 / 5.4545... ≈ 68.4

Entonces 1492.992 + 68.4 = 1561.392, que redondeado a la décima es 1561.4? No, esto es confuso. Vamos a usar una calculadora más precisa:

\( \frac{4}{3} \times 3.14 \times 7.2^3 \)

\( 7.2^3 = 373.248 \)

\( \frac{4}{3} \times 3.14 = 4.186666... \)

\( 4.186666... \times 373.248 = 4.186666 \times 373.248 \)

Calculando: 373.248 × 4 = 1492.992; 373.248 × 0.186666 = 373.248 × (11/60) = (373.248 × 11)/60 = 4105.728 / 60 = 68.4288

Entonces total es 1492.992 + 68.4288 = 1561.4208, que redondeado a la décima es 1561.4? Pero quizás me equivoqué en el radio. Wait, el radio es 7.2 mm. Let's check with another approach:

\( V = \frac{4}{3} \pi r^3 \)

\( r = 7.2 \), \( \pi = 3.14 \)

\( r^3 = 7.2 * 7.2 * 7.2 = 373.248 \)

\( \frac{4}{3} * 3.14 = 4.186666... \)

\( 4.186666... * 373.248 = 4.186666 * 373.248 \)

Let's compute 373.248 * 4 = 1492.992

373.248 * 0.186666 = 373.248 * (1/5.357) ≈ 69.6

Wait, maybe the correct calculation is:

373.248 * 4.186666 = 373.248 * (4 + 0.186666) = 373.248*4 + 373.248*0.186666 = 1492.992 + 69.6 = 1562.592, which rounds to 1562.6.

Ah, sí, porque 0.186666... es 11/60 ≈ 0.183333? No, 4/3 is 1.333..., so 4/3 * 3.14 = 4.186666..., which is correct. So 373.248 * 4.186666 ≈ 373.248 * 4.1867 ≈ let's do 373.248 * 4 = 1492.992, 373.248 * 0.1867 ≈ 373.248 * 0.1 = 37.3248, 373.248 * 0.08 = 29.85984, 373.248 * 0.0067 ≈ 2.5007, so total 37.3248 + 29.85984 = 67.18464 + 2.5007 ≈ 69.685, so 1492.992 + 69.685 ≈ 1562.677, which rounds to 1562.7? Wait, no, 1562.677 rounded to the nearest tenth is 1562.7? Wait, 0.677 is closer to 0.7? No, 0.677 is 0.6 + 0.077, so the tenths place is 6, the hundredths is 7, which is more than 5, so we round up the tenths place: 6 becomes 7. So 1562.7?

Wait, let's do the multiplication precisely:

4.186666... * 373.248

= (4 + 0.186666...) * 373.248

= 4*373.248 + 0.186666...*373.248

= 1492.992 + (373.248 * 11/60)

= 1492.992 + (4105.728 / 60)

= 1492.992 + 68.4288

= 1561.4208

Wait, now I'm confused. There's a mistake here. Let's use a calculator:

7.2 cubed is 7.2*7.2=51.84, 51.84*7.2=373.248. Correct.

Then, 4/3 * 3.14 * 373.248 = (4 * 3.14 * 373.248) / 3

Calculate numerator: 4*3.14=12.56; 12.56*373.248=?

12.56*300=3768

12.56*73.248=12.56*70=879.2; 12.56*3.248≈40.80

So 879.2 + 40.80=920

Total numerator: 3768 + 920=4688? Wait, no, 12.56*373.248:

373.248 * 10 = 3732.48

373.248 * 2 = 746.496

373.248 * 0.56 = 373.248 * 0.5 = 186.624; 373.248 * 0.06=22.39488; total 186.624+22.39488=209.01888

So total: 3732.48 + 746.496 = 4478.976 + 209.01888 = 4687.99488

Then divide by 3: 4687.99488 / 3 = 1562.66496, which rounds to 1562.7 when rounded to the nearest tenth.

Ah, there we go. So the correct calculation is:

12.56 * 373.248 = 4687.99488

Divide by 3: 4687.99488 / 3 = 1562.66496, which rounds to 1562.7 (since the hundredth digit is 6, which is more than 5, so we round up the tenth digit: 6 becomes 7).

So the volume is approximately 1562.7 cubic millimeters.

Answer:

1562.7