Question Video: Solving a Real-World Problem Using Integration Mathematics • Higher Education

A glassed, arched entry to a hotel follows the curve 𝑦 = (βˆ’1/2) (π‘₯ βˆ’ 9) (π‘₯ βˆ’ 3), where 𝑦 is the vertical height of the arch at a horizontal distance of π‘₯ meters from a point on the floor. If glass costs 1,190 US dollars per square meter, calculate the cost of the entrance.

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Video Transcript

A glassed, arched entry to a hotel follows the curve 𝑦 equals negative a half multiplied by π‘₯ minus nine multiplied by π‘₯ minus three, where 𝑦 is the vertical height of the arch at a horizontal distance of π‘₯ meters from a point on the floor. If glass costs 1,190 US dollars per square meter, calculate the cost of the entrance.

In this question, we are given a scenario which reflects the real world. There is a hotel which has an arched entry made of glass. The boundary of this arched entry traces the curve 𝑦 equals negative a half multiplied by π‘₯ minus nine multiplied by π‘₯ minus three, where 𝑦 is the vertical height of the arch at a horizontal distance of π‘₯ meters from a point on the floor which represents the origin. We know that the curve represented by the quadratic equation 𝑦 equals negative a half multiplied by π‘₯ minus nine multiplied by π‘₯ minus three is a parabola with a downward opening since the coefficient of the π‘₯ squared term in the equation, which is negative a half, is less than zero. We are asked to calculate the cost of the entrance, given that glass costs 1,190 US dollars per square meter.

The cost of the entrance will be given by the total area of the entrance in square meters multiplied by the cost of glass per square meter which is 1,119 US dollars. So, in order to answer the question, we need to work out the total area of the entrance in square meters. To do this, we will find the area between the curve 𝑦 equals negative a half π‘₯ minus nine π‘₯ minus three, which represents the boundary of the arch, and the line 𝑦 equals zero, which represents the points at which the vertical height of the arch is zero, i.e., the floor. Recall that we can do this by integrating the curve 𝑦 equals negative a half π‘₯ minus nine π‘₯ minus three with respect to π‘₯ between the values of π‘₯ at which the curve crosses the line 𝑦 equals zero. Let’s find the values π‘₯ one and π‘₯ two where the curve and the line meet.

To do this, we will solve the equations for the curve and the line simultaneously. Doing so, we obtain the equation zero equals negative a half π‘₯ minus nine π‘₯ minus three. Multiplying both sides of this equation by negative two, we obtain zero equals π‘₯ minus nine multiplied by π‘₯ minus three. Using the fact that if the product of two numbers is zero, then at least one of them is zero. We obtain that the equation zero equals π‘₯ minus nine multiplied by π‘₯ minus three implies that π‘₯ minus nine equals zero or π‘₯ minus three equal zero, which in turn implies that π‘₯ equals nine or π‘₯ equal three. So, we find that the two π‘₯-values where the curve and the line meet are π‘₯ equals nine and π‘₯ equals three.

We will take π‘₯ one equal to three and π‘₯ two equal to nine as three is smaller than nine. Now, let’s integrate the curve 𝑦 equals negative a half π‘₯ minus nine π‘₯ minus three with respect to π‘₯ between π‘₯ equals three and π‘₯ equals nine. Taking the factor of negative a half outside the integral as it is a constant and distributing the parentheses, we obtain that the integral is equal to negative a half multiplied by the integral from three to nine of π‘₯ squared minus 12π‘₯ plus 27 with respect to π‘₯.

We will evaluate this integral by applying the power rule for integration to each term in the function π‘₯ squared minus 12π‘₯ plus 27. The power rule for integration tells us that in order to integrate the term π‘Žπ‘₯ to the power of 𝑛 with respect to π‘₯, where π‘Ž and 𝑛 are constants, we keep the coefficient π‘Ž as it is, increase the exponent 𝑛 by one, and divide by the new exponent. We then add a constant of integration, 𝑐, if the integral is indefinite. However, as we are evaluating a definite integral, we won’t be needing this.

Using this formula, we have that the integral in question is equal to negative a half multiplied by π‘₯ cubed over three minus 12π‘₯ squared over two plus 27π‘₯ evaluated from three to nine. Which simplifies to negative a half multiplied by π‘₯ cubed over three minus six π‘₯ squared plus 27π‘₯ evaluated from three to nine. Evaluating between the limits and simplifying, we obtain negative a half multiplied by negative 36, which equals 18. So, the area of the arched entry is 18 square meters. Now, all we need to do to find the cost of the entrance is multiply 18 by 1,190. Doing so, we obtain 21,420. Therefore, the cost of the glassed, arched entrance of the hotel is 21,420 US dollars.

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