Math Problem Statement

an archer shoots an arrow to a height (meters) given by the equation y=-5t^2+18t-0.25, where t is the time in seconds. a target sits on a hill represented by the equation y=0.75x-1. at what height will the arrow strike the target and how long will it take?

Solution

We are given two equations:

  1. The height of the arrow as a function of time: y=5t2+18t0.25y = -5t^2 + 18t - 0.25
  2. The height of the target on a hill as a function of the horizontal distance: y=0.75x1y = 0.75x - 1

To find out when and at what height the arrow strikes the target, we need to set the two equations equal to each other since they represent the same height yy when the arrow hits the target. However, we first need to express both equations in terms of the same variable.

Since the first equation is in terms of tt (time), and the second one is in terms of xx (horizontal distance), we need to relate xx to tt. Assuming the horizontal motion of the arrow has constant velocity (neglecting air resistance), the horizontal distance xx can be written as: x=vxtx = v_x \cdot t where vxv_x is the horizontal velocity of the arrow.

Unfortunately, the problem does not provide a horizontal velocity vxv_x. Do you have any information on that, or would you like to assume a value for vxv_x, which we can use to proceed?

Alternatively, I can explain how to proceed generally with a variable horizontal velocity and show how the height can be calculated once the velocity is known. Let me know how you'd like to proceed!

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Linear Equations
Projectile Motion

Formulas

Quadratic equation: y = -5t^2 + 18t - 0.25
Linear equation: y = 0.75x - 1
Horizontal distance: x = v_x * t

Theorems

Quadratic Formula
Concept of Constant Velocity

Suitable Grade Level

Grades 9-12