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Chapter 4: Problem 55

Multiple Choice What is the maximum area of a right triangle with hypotenuse 10? \(\begin{array}{llll}{\text { (A) } 24} & {\text { (B) } 25} & {\text { (C) } 25 \sqrt{2}} & {\text { (D) } 48} & {\text { (E) } 50}\end{array}\)

Short Answer

Expert verified
The maximum area of a right triangle with hypotenuse 10 is 25 square units. So, option (B) is the correct answer.

Step by step solution

01

Finding the length of the sides

Firstly, it can be noticed that for the area of a right triangle to be maximum, base and height should be of the same length because in this way the product will be maximized. Therefore, assuming a right triangle with the same base and height, we use Pythagorean theorem to find the lengths of these sides. Thus, if \( a \) is the length of the base and the height, \( a^2 + a^2 = 10^2 \) or \( 2a^2 = 100 \) or \( a^2 = 50 \). Hence, \( a = \sqrt{50} \).
02

Calculating the maximum area

With both base and height equal to \( \sqrt{50} \), the area, \( A \), of the right triangle can be found by using the formula \( A = 1/2 \times \text{base} \times \text{height} = 1/2 \times \sqrt{50} \times \sqrt{50} = 1/2 \times 50 = 25 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Theorem
In a right triangle, the Pythagorean Theorem is fundamental for understanding the relationship among the three sides. The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:
  • If the sides of the right triangle are \(a\) and \(b\), and the hypotenuse is \(c\), then
\[c^2 = a^2 + b^2\]This theorem helps in calculating unknown sides of a right triangle, especially when applied to real-world problems and various geometry exercises. Without the Pythagorean Theorem, determining the sides, as seen in the original exercise, would be exceedingly challenging.
Triangle Area
The area of a triangle, specifically a right triangle, is determined using a straightforward formula: one-half the product of its base and height. For a right triangle, the two legs serve as the base and height. The formula is:
  • Area \(A = \frac{1}{2} \times \text{base} \times \text{height}\)
To maximize the area of a right triangle, one strategy is to have its base and height of equal length when the hypotenuse is fixed. This is derived from the Pythagorean Identity, which maximizes the enclosed area when the sides are equal for a given perimeter or hypotenuse length. For example, in the problem, if the base and height are close in value, the calculated area can reach its maximum, making it an interesting application of geometry.
Geometry Concepts
Geometry involves studying various properties and measures of shapes and figures, including triangles, which are one of the simplest yet most fundamental shapes. Right triangles, especially, allow exploration of critical concepts like angles, symmetry, and the behavior of different geometric properties.
  • **Symmetry:** In right triangles with equal legs, symmetry ensures that calculations remain consistent and predictable.
  • **Angle Calculations:** Knowing one angle measurement helps deduce the rest, especially when one is a known right angle.
  • **Optimizations:** Geometry often involves maximizing or minimizing certain aspects, such as area or perimeter. Calculations based on the sides or angles lead to direct applications of geometric theorems and formulas.
By understanding these concepts, students can navigate exercises involving right triangles with greater confidence and apply these lessons to more complex geometrical interests, whether theoretical or practical.

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Most popular questions from this chapter

Quadratic Approximations (a) Let \(Q(x)=b_{0}+b_{1}(x-a)+b_{2}(x-a)^{2}\) be a quadratic approximation to \(f(x)\) at \(x=a\) with the properties: \(\begin{aligned} \text { i. } Q(a) &=f(a) \\ \text { ii. } Q^{\prime}(a) &=f^{\prime}(a) \\ \text { ii. } & Q^{\prime \prime}(a)=f^{\prime \prime}(a) \end{aligned}\) Determine the coefficients \(b_{0}, b_{1},\) and \(b_{2}\) (b) Find the quadratic approximation to \(f(x)=1 /(1-x)\) at \(x=0 .\) (c) Graph \(f(x)=1 /(1-x)\) and its quadratic approximation at \(x=0 .\) Then zoom in on the two graphs at the point \((0,1) .\) Comment on what you see. (d) Find the quadratic approximation to \(g(x)=1 / x\) at \(x=1\) Graph \(g\) and its quadratic approximation together. Comment on what you see. (e) Find the quadratic approximation to \(h(x)=\sqrt{1+x}\) at \(x=0 .\) Graph \(h\) and its quadratic approximation together. Comment on what you see. (f) What are the linearizations of \(f, g,\) and \(h\) at the respective points in parts \((b),(d),\) and \((e) ?\)

Free Fall On the moon, the acceleration due to gravity is 1.6 \(\mathrm{m} / \mathrm{sec}^{2} .\) (a) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later? (b) How far below the point of release is the bottom of the crevasse? (c) If instead of being released from rest, the rock is thrown into the crevasse from the same point with a downward velocity of \(4 \mathrm{m} / \mathrm{sec},\) when will it hit the bottom and how fast will it be going when it does?

Arithmetic Mean The arithmetic mean of two numbers \(a\) and \(b\) is \((a+b) / 2 .\) Show that for \(f(x)=x^{2}\) on any interval \([a, b],\) the value of \(c\) in the conclusion of the Mean Value Theorem is \(c=(a+b) / 2 .\)

The Effect of Flight Maneuvers on the Heart The amount of work done by the heart's main pumping chamber, the left ventricle, is given by the equation $$W=P V+\frac{V \delta v^{2}}{2 g}$$ where \(W\) is the work per unit time, \(P\) is the average blood pressure, \(V\) is the volume of blood pumped out during the unit of time, \(\delta("\) delta") is the density of the blood, \(v\) is the average velocity of the exiting blood, and \(g\) is the acceleration of gravity. When \(P, V, \delta,\) and \(v\) remain constant, \(W\) becomes a function of \(g,\) and the equation takes the simplified form $$W=a+\frac{b}{g}(a, b\( constant \))$$ As a member of NASA's medical team, you want to know how sensitive \(W\) is to apparent changes in \(g\) caused by flight maneuvers, and this depends on the initial value of \(g\) . As part of your investigation, you decide to compare the effect on \(W\) of a given change \(d g\) on the moon, where \(g=5.2 \mathrm{ft} / \mathrm{sec}^{2},\) with the effect the same change \(d g\) would have on Earth, where \(g=32\) \(\mathrm{ft} / \mathrm{sec}^{2} .\) Use the simplified equation above to find the ratio of \(d W_{\mathrm{moon}}\) to \(d W_{\mathrm{Earth}}\)

Tin Pest When metallic tin is kept below \(13.2^{\circ} \mathrm{C}\) it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious. And indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases it is reasonable to assume that the rate \(v=d x / d t\) of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, \(v\) may be considered to be a function of \(x\) alone, and $$v=k x(a-x)=k a x-k x^{2}$$ where \(\begin{aligned} x &=\text { the amount of product, } \\ a &=\text { the amount of substance at the beginning, } \\ k &=\text { a positive constant. } \end{aligned}\) At what value of \(x\) does the rate \(v\) have a maximum? What is the maximum value of \(v ?\)
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