Question 5 A steep mountain is inclined 74 degree to the horizontal and rises to a height of $3400 \mathrm{ft}$ above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point $830 \mathrm{ft}$ out in the plain from the base of the mountain. Find the shortest length of cable needed. Your answer is $\mathrm{ft}$; Submit Question

Step 1 ：We are given that the height of the mountain is \(3400 \, \text{ft}\) and the distance in the plain from the base of the mountain is \(830 \, \text{ft}\).

Step 2 ：We need to find the shortest length of the cable needed to reach the top of the mountain from the point in the plain.

Step 3 ：We can use the Pythagorean theorem to calculate the length of the cable. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as \(c = \sqrt{a^2 + b^2}\), where \(c\) is the length of the cable, \(a\) is the height of the mountain, and \(b\) is the distance in the plain from the base of the mountain.

Step 4 ：Substituting the given values into the formula, we get \(c = \sqrt{3400^2 + 830^2}\).

Step 5 ：Calculating the above expression, we find that the shortest length of cable needed is approximately \(\boxed{3499.84 \, \text{ft}}\).