Math Problem Statement

A staircase is 30.4 -feet high and 77.5 -feet at the base. Determine the angle from its base to the top. Round to the nearest tenth of a degree. 

Solution

To determine the angle from the base of the staircase to the top, we can use trigonometry, specifically the tangent function. The tangent of the angle θ\theta can be found using the formula:

tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Here, the "opposite" side is the height of the staircase (30.4 feet), and the "adjacent" side is the base of the staircase (77.5 feet). So, the equation becomes:

tan(θ)=30.477.5\tan(\theta) = \frac{30.4}{77.5}

Now, we solve for θ\theta by taking the arctangent (inverse tangent) of both sides:

θ=arctan(30.477.5)\theta = \arctan\left(\frac{30.4}{77.5}\right)

Let's calculate this value.The angle from the base of the staircase to the top is approximately 21.421.4^\circ when rounded to the nearest tenth of a degree.

Would you like more details on this calculation or have any other questions?

Here are some related questions you might find interesting:

  1. How would the angle change if the height of the staircase were doubled?
  2. What is the length of the staircase (hypotenuse) given the current dimensions?
  3. How does the angle affect the steepness and safety of the staircase?
  4. What is the importance of this angle in architecture and construction?
  5. How would the base length need to change to achieve a 30-degree angle?

Tip: Knowing how to use trigonometric functions can help solve many real-world problems involving right triangles!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle
Angles

Formulas

tan(θ) = opposite / adjacent
θ = arctan(opposite / adjacent)

Theorems

Trigonometric Functions

Suitable Grade Level

Grades 9-11