Answer:
The base of the triangle is decreasing at a rate of 1.4167 inch/hour.
Step-by-step explanation:
Area of a triangle:
The area of a triangle of base b and height h is given by:
![A = bh](https://tex.z-dn.net/?f=A%20%3D%20bh)
In this question:
We have to derivate the equation of the area implicitly in function of time. So
![\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%20b%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%5Cfrac%7Bdb%7D%7Bdt%7D)
The altitude of a triangle is increasing at a rate 2 inch/hour while the area of the triangle is decreasing at a rate of 0.5 square inch per hour.
This means that:
![\frac{dh}{dt} = 2, \frac{dA}{dt} = -0.5](https://tex.z-dn.net/?f=%5Cfrac%7Bdh%7D%7Bdt%7D%20%3D%202%2C%20%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%20-0.5)
At what rate is the base of the triangle is changing when the altitude is 6 inch and the area is 24 square inch?
This is
when ![h = 6](https://tex.z-dn.net/?f=h%20%3D%206)
Area is 24, so the base is:
![A = bh](https://tex.z-dn.net/?f=A%20%3D%20bh)
![24 = 6b](https://tex.z-dn.net/?f=24%20%3D%206b)
![b = \frac{24}{6} = 4](https://tex.z-dn.net/?f=b%20%3D%20%5Cfrac%7B24%7D%7B6%7D%20%3D%204)
Then
![\frac{dA}{dt} = b\frac{dh}{dt} + h\frac{db}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%20b%5Cfrac%7Bdh%7D%7Bdt%7D%20%2B%20h%5Cfrac%7Bdb%7D%7Bdt%7D)
![-0.5 = 4(2) + 6\frac{db}{dt}](https://tex.z-dn.net/?f=-0.5%20%3D%204%282%29%20%2B%206%5Cfrac%7Bdb%7D%7Bdt%7D)
![6\frac{db}{dt} = -8.5](https://tex.z-dn.net/?f=6%5Cfrac%7Bdb%7D%7Bdt%7D%20%3D%20-8.5)
![\frac{db}{dt} = -\frac{8.5}{6}](https://tex.z-dn.net/?f=%5Cfrac%7Bdb%7D%7Bdt%7D%20%3D%20-%5Cfrac%7B8.5%7D%7B6%7D)
![\frac{db}{dt} = -1.4167](https://tex.z-dn.net/?f=%5Cfrac%7Bdb%7D%7Bdt%7D%20%3D%20-1.4167)
The base of the triangle is decreasing at a rate of 1.4167 inch/hour.
The fall of Constantinople of was 597 years ago
g - 3/7 = 2/7 is an equation.
![\displaystyle\bf\\g-\frac{3}{7}=\frac{2}{7}\\ \\g=\frac{2}{7}+\frac{3}{7}=\frac{2+3}{7}=\frac{5}{7}\\\\\boxed{\bf g=\frac{5}{7}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cbf%5C%5Cg-%5Cfrac%7B3%7D%7B7%7D%3D%5Cfrac%7B2%7D%7B7%7D%5C%5C%20%5C%5Cg%3D%5Cfrac%7B2%7D%7B7%7D%2B%5Cfrac%7B3%7D%7B7%7D%3D%5Cfrac%7B2%2B3%7D%7B7%7D%3D%5Cfrac%7B5%7D%7B7%7D%5C%5C%5C%5C%5Cboxed%7B%5Cbf%20g%3D%5Cfrac%7B5%7D%7B7%7D%7D)
Answer & Explanation:
A triangular prism has three rectangular sides and two triangular faces. To find the area of the rectangular sides, use the formula A = lw, where A = area, l = length, and h = height. To find the area of the triangular faces, use the formula A = 1/2bh, where A = area, b = base, and h = height.
Answer:
1 ft
Step-by-step explanation:
From the diagram;
![\triangle ADE\sim \triangle ABC](https://tex.z-dn.net/?f=%5Ctriangle%20ADE%5Csim%20%5Ctriangle%20ABC)
Therefore
![\frac{AB}{AD}=\frac{AC}{AE}](https://tex.z-dn.net/?f=%5Cfrac%7BAB%7D%7BAD%7D%3D%5Cfrac%7BAC%7D%7BAE%7D)
Also, AB=8+2=10ft, AD=8ft, AE=4ft. We substitute to get:
![\frac{10}{8}=\frac{AC}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B10%7D%7B8%7D%3D%5Cfrac%7BAC%7D%7B4%7D)
Simplify to get:
![AC = 5](https://tex.z-dn.net/?f=AC%20%3D%205)
But AC=AE+EC
This implies that,
5=4+EC
EC=5-1
EC= 1 ft