Question

С земли бросили мяч высоко вверх. Его высоту в метрах над поверхностью земли через t секунд описывает функция h(t)= 30t−5t2. 1. На какую наибольшую высоту от поверхности земли взлетит мяч? Мяч взлетит на высоту метр(-ов, -а). 2. Через сколько секунд мяч упадёт на землю? Мяч упадет на землю через секунд(-ы).

Answer 1

Ответ:

Объяснение:

1) h(t)= 30t−5t²=-5(t²-6t)=-5(t²-6t+9)+45=-5(t-3t)²+45

так как -5(t-3t)²≤0 ⇒hmax=45 м

2) h=0

h(t)= 30t−5t²=-5t(t-6)=0

t₁=0 (момент бросания)

t₂=6 секунд

Answer 2

Problem Analysis

We are given a function that describes the height of a ball thrown into the air as a function of time. We need to find the maximum height reached by the ball and the time it takes for the ball to hit the ground.

Solution

To find the maximum height reached by the ball, we need to determine the vertex of the parabolic function h(t) = 30t - 5t^2. The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic equation.

In this case, a = -5, b = 30, and c = 0. Plugging these values into the formula, we can find the time at which the ball reaches its maximum height.

To find the time it takes for the ball to hit the ground, we need to find the value of t when h(t) = 0.

Let's solve these two parts of the problem step by step.

Finding the Maximum Height

The function h(t) = 30t - 5t^2 represents the height of the ball above the ground at time t. To find the maximum height, we need to find the vertex of this parabolic function.

The vertex of a parabola in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)). In this case, a = -5 and b = 30.

Using the formula, we can find the time at which the ball reaches its maximum height:

t = -b / (2a)

Substituting the values of a and b, we get:

t = -30 / (2 * -5) t = -30 / -10 t = 3

Therefore, the ball reaches its maximum height at t = 3 seconds.

To find the maximum height, we substitute this value of t into the function h(t):

h(3) = 30 * 3 - 5 * 3^2 h(3) = 90 - 5 * 9 h(3) = 90 - 45 h(3) = 45

Therefore, the ball reaches a maximum height of 45 meters above the surface of the Earth.

Answer 1: The ball reaches a maximum height of 45 meters above the surface of the Earth.

Finding the Time of Impact

To find the time it takes for the ball to hit the ground, we need to find the value of t when h(t) = 0.

Setting h(t) = 0, we can solve for t:

30t - 5t^2 = 0

Factoring out t, we get:

t(30 - 5t) = 0

This equation is satisfied when either t = 0 or 30 - 5t = 0.

Solving the second equation, we get:

30 - 5t = 0 5t = 30 t = 6

Therefore, the ball hits the ground at t = 6 seconds.

Answer 2: The ball hits the ground after 6 seconds.

Summary

To summarize, the ball reaches a maximum height of 45 meters above the surface of the Earth and hits the ground after 6 seconds.

Please let me know if I can help you with anything else.