Answer:
The area of the end is 1755.44 ft²
Therefore, the maximum error in calculating area of the end is 8.3592
Step-by-step explanation:
Consider the provided information.
The length of the base is measured to be 35 feet, with a maximum error in measurement of 1 inch,
[tex]1\ inch = \frac{1}{12}\ feet[/tex]
The area of shape = Area of square + Area of equilateral triangle
[tex]A=x^2+\frac{\sqrt{3}}{4}x^2[/tex]
Substitute x=35 in above.
[tex]A=(35)^2+\frac{\sqrt{3}}{2}\times (35)^2=1755.44[/tex]
Hence, the area of the end is 1755.44 ft²
Differentiate [tex]A=x^2(1+\frac{\sqrt{3}}{4})[/tex] with respect to x as shown:
[tex]dA=2x\ dx(1+\frac{\sqrt{3}}{4})[/tex]
Substitute dx =1/12 and x=35
[tex]dA=2(35)\ \frac{1}{12}(1+\frac{\sqrt{3}}{4})=8.3592[/tex]
Therefore, the maximum error in calculating area of the end is 8.3592
The end of a house has the shape of a square surmounted by an equilateral triangle. If the length of the base is measured to be 35 feet, with a maximum error in measurement of 1 inch, The area of the end is calculated as 1226.433 ft², and by using the differentials to estimate the maximum error in the calculation of the area, the maximum error is 8.326 ft².
To find the area of the end and use the differential to estimate the maximum error in the calculation of the area, we have:
The area of the end is to be calculated by using the formula:
[tex]\mathbf{A = \dfrac{\sqrt{3}}{4} x^2 + x^2}[/tex]
[tex]\mathbf{A = \Big(\dfrac{\sqrt{3}}{4} +1^2 \Big)+35^2}[/tex]
[tex]\mathbf{A = \Big(\dfrac{\sqrt{3}}{4} +1^2 \Big)+35^2}[/tex]
[tex]\mathbf{A = \Big(1.433 \Big)+35^2}[/tex]
A = 1226.433 ft²
in order to estimate the maximum error, we need to determine the derivative of the area of the end which we earlier mentioned to be;
[tex]\mathbf{A = \dfrac{\sqrt{3}}{4} x^2 + x^2}[/tex]
Taking the derivative with respect to A.
[tex]\mathbf{dA = \Big( \dfrac{\sqrt{3}}{4} \times 2x + 2x \Big) \ dx}[/tex]
[tex]\mathbf{dA = \Big( \dfrac{\sqrt{3}}{4}+1 \Big) \ 2x dx}[/tex]
∴
[tex]\mathbf{dA = \Big( \dfrac{\sqrt{3}}{4}+1 \Big) \ 2(35) \Big(\pm \dfrac{1}{12} \Big)}[/tex]
[tex]\mathbf{dA = \Big( 1.433 \Big) \ (70) \Big(\pm 0.083 \Big)}[/tex]
[tex]\mathbf{dA = 8.326 \ ft ^2}[/tex]
Therefore, we can conclude that the area of the end is calculated as 1226.433 ft², and by using the differentials to estimate the maximum error in the calculation of the area, the maximum error is 8.326 ft².
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