Answer:
The equation
will be a parabola when it is graphed
b must be equal to -6 for infinitely many solutions for system of equations
and ![-3 x+\frac{1}{2} y=-3](https://tex.z-dn.net/?f=-3%20x%2B%5Cfrac%7B1%7D%7B2%7D%20y%3D-3)
<u>Solution:
</u>
Need to calculate value of b so that given system of equations have an infinite number of solutions
![\begin{array}{l}{y=6 x+b} \\\\ {-3 x+\frac{1}{2} y=-3}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7By%3D6%20x%2Bb%7D%20%5C%5C%5C%5C%20%7B-3%20x%2B%5Cfrac%7B1%7D%7B2%7D%20y%3D-3%7D%5Cend%7Barray%7D)
Let us bring the equations in same form for sake of simplicity in comparison
![\begin{array}{l}{y=6 x+b} \\\\ {\Rightarrow-6 x+y-b=0 \Rightarrow (1)} \\\\ {\Rightarrow-3 x+\frac{1}{2} y=-3} \\\\ {\Rightarrow -6 x+y=-6} \\\\ {\Rightarrow -6 x+y+6=0 \Rightarrow(2)}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7By%3D6%20x%2Bb%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow-6%20x%2By-b%3D0%20%5CRightarrow%20%281%29%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow-3%20x%2B%5Cfrac%7B1%7D%7B2%7D%20y%3D-3%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow%20-6%20x%2By%3D-6%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow%20-6%20x%2By%2B6%3D0%20%5CRightarrow%282%29%7D%5Cend%7Barray%7D)
Now we have two equations
![\begin{array}{l}{-6 x+y-b=0\Rightarrow(1)} \\\\ {-6 x+y+6=0\Rightarrow(2)}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B-6%20x%2By-b%3D0%5CRightarrow%281%29%7D%20%5C%5C%5C%5C%20%7B-6%20x%2By%2B6%3D0%5CRightarrow%282%29%7D%5Cend%7Barray%7D)
Let us first see what is requirement for system of equations have an infinite number of solutions
If
and
are two equation
then the given system of equation has no infinitely many solutions.
In our case,
![\begin{array}{l}{a_{1}=-6, \mathrm{b}_{1}=1 \text { and } c_{1}=-\mathrm{b}} \\\\ {a_{2}=-6, \mathrm{b}_{2}=1 \text { and } c_{2}=6} \\\\ {\frac{a_{1}}{a_{2}}=\frac{-6}{-6}=1} \\\\ {\frac{b_{1}}{b_{2}}=\frac{1}{1}=1} \\\\ {\frac{c_{1}}{c_{2}}=\frac{-b}{6}}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7Ba_%7B1%7D%3D-6%2C%20%5Cmathrm%7Bb%7D_%7B1%7D%3D1%20%5Ctext%20%7B%20and%20%7D%20c_%7B1%7D%3D-%5Cmathrm%7Bb%7D%7D%20%5C%5C%5C%5C%20%7Ba_%7B2%7D%3D-6%2C%20%5Cmathrm%7Bb%7D_%7B2%7D%3D1%20%5Ctext%20%7B%20and%20%7D%20c_%7B2%7D%3D6%7D%20%5C%5C%5C%5C%20%7B%5Cfrac%7Ba_%7B1%7D%7D%7Ba_%7B2%7D%7D%3D%5Cfrac%7B-6%7D%7B-6%7D%3D1%7D%20%5C%5C%5C%5C%20%7B%5Cfrac%7Bb_%7B1%7D%7D%7Bb_%7B2%7D%7D%3D%5Cfrac%7B1%7D%7B1%7D%3D1%7D%20%5C%5C%5C%5C%20%7B%5Cfrac%7Bc_%7B1%7D%7D%7Bc_%7B2%7D%7D%3D%5Cfrac%7B-b%7D%7B6%7D%7D%5Cend%7Barray%7D)
As for infinitely many solutions ![\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}](https://tex.z-dn.net/?f=%5Cfrac%7Ba_%7B1%7D%7D%7Ba_%7B2%7D%7D%3D%5Cfrac%7Bb_%7B1%7D%7D%7Bb_%7B2%7D%7D%3D%5Cfrac%7Bc_%7B1%7D%7D%7Bc_%7B2%7D%7D)
![\begin{array}{l}{\Rightarrow 1=1=\frac{-b}{6}} \\\\ {\Rightarrow6=-b} \\\\ {\Rightarrow b=-6}\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bl%7D%7B%5CRightarrow%201%3D1%3D%5Cfrac%7B-b%7D%7B6%7D%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow6%3D-b%7D%20%5C%5C%5C%5C%20%7B%5CRightarrow%20b%3D-6%7D%5Cend%7Barray%7D)
Hence b must be equal to -6 for infinitely many solutions for system of equations
and
<span><span>8<span>c^2</span></span><span>(<span>3−<span>c^4</span></span>)</span></span><span>=<span><span>(<span>8<span>c^2</span></span>)</span><span>(<span>3+<span>−<span>c^4</span></span></span>)</span></span></span><span>=<span><span><span>(<span>8<span>c^2</span></span>)</span><span>(3)</span></span>+<span><span>(<span>8<span>c^2</span></span>)</span><span>(<span>−<span>c^4</span></span>)</span></span></span></span><span>=<span><span>24<span>c^2</span></span>−<span>8<span>c^6</span></span></span></span><span>=<span><span>−<span>8<span>c^6</span></span></span>+<span>24<span>c^<span>2</span></span></span></span></span>
Answer:
13 inches
Step-by-step explanation:
Use the formula P = 2l + 2w to find width.
80 = 2(27) + 2w
80 = 54 + 2w
80 - 54= 2w
26 = 2w
w = 26/2
w = 13
The width of the rectangle is 13 inches.
Answer:
1) 32
2) 8 yards
Step-by-step explanation:
1. We must first subtract the base area of the pyramid from the total surface area to get the lateral surface area:
![LA=2225-25^2=1600](https://tex.z-dn.net/?f=LA%3D2225-25%5E2%3D1600)
The lateral surface area is 4 times the area of one the congruent triangles.
![LA=4\cdot \frac{1}{2}\cdot 25\cdot x](https://tex.z-dn.net/?f=LA%3D4%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%2025%5Ccdot%20x)
![\implies 1600=50x](https://tex.z-dn.net/?f=%5Cimplies%201600%3D50x)
![\implies \frac{1600}{50}=\frac{50x}{50}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cfrac%7B1600%7D%7B50%7D%3D%5Cfrac%7B50x%7D%7B50%7D)
![32=x](https://tex.z-dn.net/?f=32%3Dx)
Therefore the height of the slant surface is 32 yards
2) The surface area of a cone is
, where l is the slant height.
We substitute the surface area S.A=151.58 and
to obtain:
![151.58=3.14\cdot 4^2+3.14\cdot 4 l](https://tex.z-dn.net/?f=151.58%3D3.14%5Ccdot%204%5E2%2B3.14%5Ccdot%204%20l)
![151.58=50.24+12.56l](https://tex.z-dn.net/?f=151.58%3D50.24%2B12.56l)
![101.34=12.56l](https://tex.z-dn.net/?f=101.34%3D12.56l)
![\frac{101.34}{12.56}=l](https://tex.z-dn.net/?f=%5Cfrac%7B101.34%7D%7B12.56%7D%3Dl)
l=8.06
To the nearest whole number, the slant height is 8 yards